Preliminaries on Stochastic Calculus in Infinite Dimension

Abstract

We recall some basic notions of measure theory and give a short introduction to random variables and the theory of the Bochner integral.

1.1 Basic Probability

We recall some basic notions of measure theory and give a short introduction to random variables and the theory of the Bochner integral.

1.1.1 Probability Spaces, $\sigma$-Fields

Definition 1.1

($\pi$-system, $\sigma$-field) Consider a set $\Omega$ and denote by ${\mathcal P}({\Omega })$ the power set of $\Omega$.

• (i) A non-empty class of subsets of $\Omega$, $\mathscr {F}\subset {\mathcal P}({\Omega })$, is called a $\pi$-system if it is closed under finite intersections.

• (ii) A class of subsets of $\Omega$, $\mathscr {F}\subset \mathcal {P}(\Omega )$, is called a $\sigma$-field in $\Omega$ if $\Omega \in \mathscr {F}$ and $\mathscr {F}$ is closed under complements and countable unions.

• (iii) A class of subsets of $\Omega$, $\mathscr {F}\subset \mathcal {P}(\Omega )$, is called a $\lambda$-system if:

  • $\Omega \in \mathscr {F}$;

  • if $A, B\in \mathscr {F}, A\subset B$, then $B\setminus A\in \mathscr {F}$;

  • if $A_i\in \mathscr {F}, i=1,2,..., A_i\uparrow A$, then $A\in \mathscr {F}$.

If $\mathscr {G}$ and $\mathscr {F}$ are two $\sigma$-fields in $\Omega$ and $\mathscr {G} \subset \mathscr {F}$, we say that $\mathscr {G}$ is a sub-$\sigma$-field of $\mathscr {F}$. Given a class $\mathscr {C}\subset \mathcal {P}(\Omega )$, the smallest $\sigma$-field containing $\mathscr {C}$ is called the $\sigma$-field generated by $\mathscr {C}$. It is denoted by $\sigma (\mathscr {C})$ . A $\sigma$-field $\mathscr {F}$ in $\Omega$ is said to be countably generated if there exists a countable class of subsets $\mathscr {C}\subset \mathcal {P}(\Omega )$ such that $\sigma (\mathscr {C}) = \mathscr {F}$.

If $\mathscr {C}\subset \mathcal {P}(\Omega )$ and $A\subset \Omega$ we define $\mathscr {C}\cap A:=\{B\cap A:B\in \mathscr {C}\}$. We denote by $\sigma _A(\mathscr {C}\cap A)$ the $\sigma$-field of subsets of A generated by $\mathscr {C}\cap A$. It is easy to see that $\sigma _A(\mathscr {C}\cap A)=\sigma (\mathscr {C})\cap A$ (see, for instance, [18], p. 5).

For $A\subset \Omega$ we denote its complement by $A^c:=\Omega \setminus A$, and for $A, B\subset \Omega$ we denote their symmetric difference by $A\Delta B:=(A\setminus B)\cup (B\setminus A)$. We will write $\mathbb {R}^+=[0,+\infty ), \overline{\mathbb {R}}^{\;+}=[0,+\infty )\cup \{+\infty \}, \overline{\mathbb {R}}=\mathbb {R}\cup \{\pm \infty \}$.

Theorem 1.2

Let $\mathscr {G}$ be a $\pi$-system and $\mathscr {F}$ be a $\lambda$-system in some set $\Omega$, such that $\mathscr {G}\subset \mathscr {F}$. Then $\sigma (\mathscr {G})\subset \mathscr {F}$.

Proof

See [370], Theorem 1.1, p. 2. $\square$

Corollary 1.3

Let $\mathscr {G}$ be a $\pi$-system and $\mathscr {F}$ be the smallest family of subsets of $\Omega$ such that:

• $\mathscr {G}\subset \mathscr {F}$;

• if $A\in \mathscr {F}$ then $A^c\in \mathscr {F}$;

• if $A_i\in \mathscr {F}, A_i\cap A_j=\emptyset$ for $i, j=1,2,..., i\not = j$, then $\cup _{i=1}^\infty A_i\in \mathscr {F}$.

Then $\sigma (\mathscr {G})=\mathscr {F}$.

Proof

Since $\sigma (\mathscr {G})$ satisfies the three conditions for $\mathscr {F}$, we obviously have $\mathscr {F}\subset \sigma (\mathscr {G})$. For the opposite inclusion it remains to observe that $\mathscr {F}$ is a $\lambda$-system. (For a self-contained proof, see also [180], Proposition 1.4, p. 17.) $\square$

Definition 1.4

(Measurable space) If $\Omega$ is a set and $\mathscr {F}$ is a $\sigma$-field in $\Omega$, the pair $(\Omega , \mathscr {F})$ is called a measurable space .

Definition 1.5

(Probability measure, probability space) Consider a measurable space $(\Omega , \mathscr {F})$. A function $\mu :\mathscr {F}\rightarrow [0, +\infty )\cup \{+\infty \}$ is called a measure on $(\Omega , \mathscr {F})$ if $\mu (\emptyset ) = 0$, and whenever $A_i\in \mathscr {F}, A_i\cap A_j=\emptyset$ for $i, j=1,2,..., i\not = j$, then

$$ \mu \left( \bigcup _{i=1}^\infty A_i\right) =\sum _{i=1}^\infty \mu (A_i). $$

The triplet $(\Omega , \mathscr {F},\mu )$ is called a measure space . If $\mu (\Omega )<+\infty$ we say that $\mu$ is a bounded measure . If $\Omega =\bigcup _{n=1}^\infty A_n$, where $A_n\in \mathscr {F}, \mu (A_n)<+\infty , n=1,2,...$, we say that $\mu$ is a $\sigma$-finite measure. If $\mu (\Omega ) = 1$ we say that $\mu$ is a probability measure . We will use the symbol $\mathbb {P}$ to denote probability measures. The triplet $(\Omega , \mathscr {F},\mathbb {P})$ is called a probability space.

Thus a probability measure is a $\sigma$-additive function $\mathbb {P}:\mathscr {F}\rightarrow [0, 1]$ such that $\mathbb {P}(\Omega ) = 1$.

Given a measure space $(\Omega , \mathscr {F},\mu )$, we define $\mathcal {N}:=\{F\subset \Omega \; : \; \exists G\in \mathscr {F}, F\subset G,\mu (G)=0\}$ . The elements of $\mathcal {N}$ are called $\mu$-null sets . If $\mathcal {N}\subset \mathscr {F}$, the measure space $(\Omega , \mathscr {F},\mu )$ is said to be complete . The $\sigma$-field $\overline{\mathscr {F}}:=\sigma (\mathscr {F},\mathcal {N})$ is called the completion of $\mathscr {F}$ (with respect to $\mu$). It is easy to see that $\sigma (\mathscr {F},\mathcal {N})=\{A\cup B: A\in \mathscr {F}, B\in \mathcal {N}\}$. If $\mathscr {G}\subset \mathscr {F}$ is another $\sigma$-field then $\sigma (\mathscr {G},\mathcal {N})$ is called the augmentation of $\mathscr {G}$ by the null sets of $\mathscr {F}$. The augmentation of $\mathscr {G}$ may be different from its completion, as the latter is just the augmentation of $\mathscr {G}$ by the subsets of the sets of measure zero in $\mathscr {G}$. We also have $\sigma (\mathscr {G},\mathcal {N})= \{A\subset \Omega :A\Delta B\in \mathcal {N}\,\,\text {for some}\,\, B\in \mathscr {G}\}$.

Let $\mu ,\nu$ be two measures on a measurable space $(\Omega , \mathscr {F})$. We say that $\mu$ is absolutely continuous with respect to $\nu$ (we write $\mu<<\nu$) if for every $A\in \mathscr {F}$ such that $\nu (A)=0$ we have $\mu (A)=0$. If $\mu<<\nu$ and $\nu<<\mu$, we say that the measures $\mu$ and $\nu$ are equivalent (we write $\mu \sim \nu$). If there exists a set $A\in \mathscr {F}$ such that for every $B\in \mathscr {F}$ we have $\mu (B)=\mu (A\cap B)$, we say that $\mu$ is concentrated on the set A. If $\mu$ and $\nu$ are concentrated on disjoint sets we say that $\mu$ and $\nu$ are (mutually) singular and we write $\mu \perp \nu$.

Lemma 1.6

Let $\mu _1, \mu _2$ be two bounded measures on a measurable space $(\Omega , \mathscr {F})$, and let $\mathscr {G}$ be a $\pi$-system in $\Omega$ such that $\Omega \in \mathscr {G}$ and $\sigma (\mathscr {G})=\mathscr {F}$. Then $\mu _1=\mu _2$ if and only if $\mu _1(A)= \mu _2(A)$ for every $A\in \mathscr {G}$.

Proof

See [370], Lemma 1.17, p. 9. $\square$

Let $\Omega _t, t\in \mathcal {T}$ be a family of sets. We will denote the Cartesian product of the family $\Omega _t$ by ${\times }_{t\in \mathcal {T}}\Omega _t$. If $\mathcal {T}$ is finite ($\mathcal {T}=\{1,..., n\}$) or countable ($\mathcal {T}=\mathbb {N}$), we will also write $\Omega _1\times ...\times \Omega _n$, respectively $\Omega _1\times \Omega _2\times ...$. If each $\Omega _t$ is a topological space, we endow $\times _{t\in \mathcal {T}}\Omega _t$ with the product topology. If each $\Omega _t$ has a $\sigma$-field $\mathscr {F}_t$, we define the product $\sigma$-field $\otimes _{t\in \mathcal {T}}\mathscr {F}_t$ in $\times _{t\in \mathcal {T}}\Omega _t$ as the $\sigma$-field generated by the one-dimensional cylinder sets $A_t\times \left( \times _{s\not =t}\Omega _s \right)$. If $\mathcal {T}=\{1,..., n\}$ (respectively, $\mathcal {T}=\mathbb {N}$) we will just write $\otimes _{t\in \mathcal {T}}\mathscr {F}_t=\mathscr {F}_1\otimes ...\otimes \mathscr {F}_n$ (respectively, $\otimes _{t\in \mathcal {T}}\mathscr {F}_t=\mathscr {F}_1\otimes \mathscr {F}_2\otimes ...$).

If S is a topological space, the $\sigma$-field generated by the open sets of S is called the Borel $\sigma$-field . It will be denoted by $\mathcal {B}(S)$ . If S is a metric space, unless stated otherwise, its default $\sigma$-field will always be $\mathcal {B}(S)$. It is not difficult to see that if $S_1,S_2,...$ are separable metric spaces, then

$$ \mathcal {B}(S_1\times S_2\times ...)=\mathcal {B}(S_1)\otimes \mathcal {B}(S_2)\otimes .... $$

If $(S,\rho )$ is a metric space, $A\subset S$, and we consider $(A,\rho )$ as a metric space, then $\mathcal {B}(A) =A\cap \mathcal {B}(S)$. A complete separable metric space is called a Polish space . Also $\mathcal {B}(\overline{\mathbb {R}}^{\;+})= \sigma (\mathcal {B}(\mathbb {R}^+),\{+\infty \}), \mathcal {B}(\overline{\mathbb {R}})= \sigma (\mathcal {B}(\mathbb {R}),\{-\infty \},\{+\infty \})$.

A measurable space $(\Omega , \mathscr {F})$ is called countably determined (or $\mathscr {F}$ is called countably determined) if there is a countable set $\mathscr {F}_0\subset \mathscr {F}$ such that any two probability measures on $(\Omega , \mathscr {F})$ that agree on $\mathscr {F}_0$ must be the same. It follows from Lemma 1.6 that if $\mathscr {F}$ is countably generated then $\mathscr {F}$ is countably determined. If S is a Polish space then $\mathcal {B}(S)$ is countably generated.

If $(\Omega _i, \mathscr {F}_i,\mu _i), i=1,..., n$, are measure spaces, their product measure on $(\Omega _1\times ...\times \Omega _n,\mathscr {F}_1\otimes ...\otimes \mathscr {F}_n)$ is denoted by $\mu _1\otimes ...\otimes \mu _n$.

If S is a metric space, a bounded measure $\mu$ on $(S,\mathcal {B}(S))$ is called regular if

$$ \mu (A)=\sup \{\mu (C):C\subset A,C\,\,\text {closed}\}=\inf \{\mu (U):A\subset U, U\,\,\text {open}\}\quad \forall A\in \mathcal {B}(S). $$

Every bounded measure on $(S,\mathcal {B}(S))$ is regular (see [478], Chap. II, Theorem 1.2). A bounded measure $\mu$ on $(S,\mathcal {B}(S))$ is called tight if for every ${\varepsilon }>0$ there exists a compact set $K_{\varepsilon }\subset S$ such that $\mu (S\setminus K_{\varepsilon })<{\varepsilon }$. If S is a Polish space then every bounded measure on $(S,\mathcal {B}(S))$ is tight (see [478], Chap. II, Theorem 3.2).

We refer to [58, 61, 267, 370, 478] for more on the general theory of measure and probability.

1.1.2 Random Variables

Definition 1.7

(Random variable) A measurable mapX between two measurable spaces $(\Omega , \mathscr {F})$ and $({\tilde{\Omega }} , \mathscr {G})$ is a called a random variable . This means that X is a random variable if $X^{-1}(A) \in \mathscr {F}$ for every $A\in \mathscr {G}$. We write it shortly as $X^{-1}(\mathscr {G}) \subset \mathscr {F}$. Sometimes we will just say that X is $\mathscr {F}/ \mathscr {G}$-measurable.

If ${\tilde{\Omega }}=\mathbb {R}$ (resp. $\mathbb {R}^+$) and $\mathscr {G}$ is the Borel $\sigma$-field $\mathcal {B}(\mathbb {R})$ (resp. $\mathcal {B}(\mathbb {R}^+)$) then X is said to be a real random variable (resp. positive random variable).

If $\Omega , {\tilde{\Omega }}$ are topological spaces and $\mathscr {F}, \mathscr {G}$ are the Borel $\sigma$-fields then X is said to be Borel measurable .

If $(\Omega , \mathscr {F},\mu )$ is a measure space and $X, X_1:\Omega \rightarrow {\tilde{\Omega }}$, we say that $X_1$ is a version of X if $X=X_1$ $\mu$-a.e.

Given a random variable $X:(\Omega , \mathscr {F}) \rightarrow ({\tilde{\Omega }}, \mathscr {G})$ we denote by $\sigma (X)$ the smallest sub-$\sigma$-field of $\mathscr {F}$ that makes X measurable, i.e. $\sigma (X):=X^{-1}({\mathscr {G}})$. It is called the $\sigma$-field generated by X . Given a set of indices I and a family of random variables $X_i:(\Omega , \mathscr {F})\rightarrow ({\tilde{\Omega }} , \mathscr {G})$, $i\in I$, the $\sigma$-field $\sigma \left( X_i \, : \, i\in I \right)$ generated by $\left\{ X_i \right\} _{i\in I}$ is the smallest sub-$\sigma$-field of $\mathscr {F}$ that makes all the functions $X_i:\left( \Omega , \sigma \left( X_i \, : \, i\in I \right) \right) \rightarrow ({\tilde{\Omega }}, \mathscr {G})$ measurable, i.e. $\sigma \left( X_i \, : \, i\in I \right) = \sigma \left( X_i^{-1}(\mathscr {G}): i\in I\right)$.

Lemma 1.8

Let $(\Omega , \mathscr {F})$ be a measurable space. Then:

(i) If $({\tilde{\Omega }}, \mathscr {G})$ is a measurable space, $X:\Omega \rightarrow {\tilde{\Omega }}$, and $\mathscr {C}\subset \mathscr {G}$ is such that $\sigma (\mathscr {C})=\mathscr {G}$, then X is $\mathscr {F}/ \mathscr {G}$-measurable if and only if $X^{-1}(\mathscr {C})\subset \mathscr {F}$. Moreover, $\sigma (X)=\sigma (X^{-1}(\mathscr {C}))$.

(ii) If $X_n:\Omega \rightarrow \overline{\mathbb {R}}, n=1,2,...$, are random variables, then $\sup _n X_n, \inf _n X_n$, $\limsup _n X_n, \liminf _n X_n$ are random variables.

(iii) Let $X_n:\Omega \rightarrow S, n=1,2,...$, be random variables, where S is a metric space. Then:

• if S is complete then $\{\omega : X_n(\omega )\,\, \text {converges}\}\in \mathscr {F}$;

• if $X_n\rightarrow X$ on $\Omega$, then X is a random variable.

(iv) Let $(\Omega _i, \mathscr {F}_i), i=1,2$, be measurable spaces, and $X:\Omega _1\times \Omega _2\rightarrow \Omega$ be $(\mathscr {F}_1\otimes \mathscr {F}_2)/\mathscr {F}$-measurable. Then, for every $\omega _1\in \Omega _1$, $X_{\omega _1}(\cdot )=X(\omega _1,\cdot )$ is $\mathscr {F}_2/\mathscr {F}$-measurable, and, for every $\omega _2\in \Omega _2$, $X_{\omega _2}(\cdot )=X(\cdot ,\omega _2)$ is $\mathscr {F}_1/\mathscr {F}$-measurable.$\square$

Proof

See, for instance, [370], Lemmas 1.4, 1.9, 1.10, and [520], Theorem 7.5, p. 138. $\square$

Theorem 1.9

Let $(\Omega , \mathscr {F})$ and $({\tilde{\Omega }}, \mathscr {G})$ be two measurable spaces and (S, d) a Polish space. Let $X:(\Omega , \mathscr {F}) \rightarrow ({\tilde{\Omega }}, \mathscr {G})$ and $\phi :(\Omega , \mathscr {F}) \rightarrow (S, \mathcal {B}(S))$ be two random variables. Then $\phi$ is measurable as a map from $(\Omega ,\sigma (X))$ to $(S, \mathcal {B}(S))$ if and only if there exists a measurable map $\eta :({\tilde{\Omega }}, \mathscr {G}) \rightarrow (S, \mathcal {B}(S))$ such that $\phi = \eta \circ X$.

Proof

See [370], Lemma 1.13, p. 7, or [575] Theorem 1.7, p. 5. $\square$

We refer to [58, 267, 370, 520] for more on measurability and for the general theory of integration.

Definition 1.10

(Borel isomorphism) Let $(\Omega , \mathscr {F})$ and $({\tilde{\Omega }}, \mathscr {G})$ be two measurable spaces. A bijection f from $\Omega$ onto ${\tilde{\Omega }}$ is called a Borel isomorphism iff is $\mathscr {F}/ \mathscr {G}$-measurable and $f^{-1}$ is $\mathscr {G}/ \mathscr {F}$-measurable. We then say that $(\Omega , \mathscr {F})$ and $({\tilde{\Omega }}, \mathscr {G})$ are Borel isomorphic.

Definition 1.11

(Standard measurable space) A measurable space $(\Omega , \mathscr {F})$ is called standard if it is Borel isomorphic to one of the following spaces:

• (i) $\left( \{ 1, .. , n \}, \mathcal {B}(\{ 1, .. , n \}) \right)$,

• (ii) $\left( \mathbb {N}, \mathcal {B}(\mathbb {N})\right)$,

• (iii) $\left( \{0,1\}^\mathbb {N}, \mathcal {B}(\{0,1\}^\mathbb {N})\right)$,

where we have the discrete topologies in $\{ 1, .. , n \}$ and $\mathbb {N}$, and the product topology in $\{0,1\}^\mathbb {N}$.

The following theorem collects results that can be found in [478] (Chap. I, Theorems 2.8 and 2.12).

Theorem 1.12

If S is a Polish space, then $(S,\mathcal {B}(S))$ is standard. If a Borel subset of S is uncountable, then it is Borel isomorphic to $\{0,1\}^\mathbb {N}$. Two Borel subsets of S are Borel isomorphic if and only if they have the same cardinality. If $(\Omega , \mathscr {F})$ is standard and $A\in \mathscr {F}$, then $(A, \mathscr {F}\cap A)$ is standard.

In particular, we have the following result.

Theorem 1.13

If $(\Omega , \mathscr {F})$ is standard, then it is Borel isomorphic to a closed subset of [0, 1] (with its induced Borel sigma field).

Definition 1.14

(Simple random variable) Let$\left( \Omega ,\mathscr {F}\right)$ be a measurable space, and (S, d) be a metric space (endowed with the Borel $\sigma$-field induced by the distance). A random variable $X:\left( \Omega ,\mathscr {F}\right) \rightarrow (S,\mathcal {B}(S))$ is called simple (or a simple function) if it has a finite number of values.

Lemma 1.15

Let $f:\left( \Omega ,\mathscr {F}\right) \rightarrow S$ be a measurable function between a measurable space $\left( \Omega ,\mathscr {F}\right)$ and a separable metric space (S, d) (endowed with the Borel $\sigma$-field induced by the distance). Then there exists a sequence $f_n:\Omega \rightarrow S$ of simple, $\mathscr {F}/\mathcal {B}(S)$-measurable functions, such that $d \left( f(\omega ), f_n(\omega ) \right)$ is monotonically decreasing to 0 for every $\omega \in \Omega$.

Proof

See [180], Lemma 1.3, p. 16. $\square$

Lemma 1.16

Let S be a Polish space with metric d. Let $\left( \Omega ,\mathscr {F},\mathbb {P}\right)$ be a complete probability space and let $\mathscr {G}_1, \mathscr {G}_2\subset \mathscr {F}$ be two $\sigma$-fields with the following property: for every $A\in \mathscr {G}_2$ there exists a $B\in \mathscr {G}_1$ such that $\mathbb {P}(A\Delta B)=0$. Let $f:(\Omega ,\mathscr {G}_2)\rightarrow (S, \mathcal {B}(S))$ be a measurable function. Then there exists a function $g:(\Omega ,\mathscr {G}_1)\rightarrow (S, \mathcal {B}(S))$ such that $f=g$, $\mathbb {P}$-a.e., and simple functions $g_n:(\Omega ,\mathscr {G}_1)\rightarrow (S, \mathcal {B}(S))$ such that $d(f(\omega ), g_n(\omega ))$ monotonically decreases to 0, $\mathbb {P}$-a.e.

Proof

The proof follows the lines of the proof of Lemma 1.25, p. 13, in [370].

Step 1: Let us assume first that $f = x\mathbf{1}_{A}$ ($\mathbf{1}_{A}$ denotes the characteristic function of the set A) for some $A\in \mathscr {G}_2$ and $x\in S$. By hypothesis, we can find $B\in \mathscr {G}_1$ s.t. $\mathbb {P}(A\Delta B)=0$ and then the claim is proved if we choose $g_n \equiv g = x\mathbf{1}_B$. The same argument holds for a simple function f.

Step 2: For the case of a general f, thanks to Lemma 1.15 we can find a sequence of simple, $\mathscr {G}_2$-measurable functions $f_n$ such that $d(f(\omega ), f_n(\omega ))$ monotonically decreases to 0. By Step 1, we can find simple, $\mathscr {G}_1$-measurable functions $g_n$ such that $f_n= g_n$, $\mathbb {P}$-a.e. Thus the claim follows by taking $g(\omega ):= \lim g_n(\omega )$ if the limit exists and $g(\omega )=s$ (for some $s \in S$) otherwise. $\square$

Lemma 1.17

Let $\left( \Omega , \mathscr {F}\right)$ be a measurable space, and $V\subset E$ be two real separable Banach spaces such that the embedding of V into E is continuous. Then:

• (i) $\mathcal {B}(E)\cap V\subset \mathcal {B}(V)$ and $\mathcal {B}(V)\subset \mathcal {B}(E)$.

• (ii) If $X:\Omega \rightarrow V$ is $\mathscr {F}/\mathcal {B}(V)$-measurable, then it is $\mathscr {F}/\mathcal {B}(E)$-measurable.

• (iii) If $X:\Omega \rightarrow E$ is $\mathscr {F}/\mathcal {B}(E)$-measurable, then $X\cdot \mathbf{1}_{\{X\in V\}}$ is $\mathscr {F}/\mathcal {B}(V)$-measurable.

• (iv) $X:\Omega \rightarrow E$ is $\mathscr {F}/\mathcal {B}(E)$-measurable if and only if for every $f\in E^*$, $f\circ X$ is $\mathscr {F}/\mathcal {B}(\mathbb {R})$-measurable.

Proof

The embedding of V into E is continuous, so $\mathcal {B}(E)\cap V\subset \mathcal {B}(V)$. Since the embedding is also one-to-one, it follows from [478], Theorem 3.9, p. 21, that $\mathcal {B}(V)\subset \mathcal {B}(E)$, which completes the proof of (i). Parts (ii) and (iii) are direct consequences of (i). $f(\Omega )$ is separable because E is separable, so Part (iv) is a particular case of the Pettis theorem, see [488] Theorem 1.1. $\square$

Lemma 1.18

Let $(\Omega , \mathscr {F})$ be a measurable space and $(S_1,\rho _1),(S_2,\rho _2)$ be two metric spaces with $S_1$ separable. Let $f:\Omega \times S_1 \rightarrow S_2$ be such that

• (i) for each $x\in S_1$, the function $f(\cdot , x) :\Omega \rightarrow S_2$ is $\mathscr {F}/\mathcal {B}(S_2)$-measurable;

• (ii) for each $\omega \in \Omega$ the function $f(\omega , \cdot ) :S_1 \rightarrow S_2$ is continuous.

Then $f:\Omega \times S_1 \rightarrow S_2$ is $\mathscr {F}\otimes \mathcal {B}(S_1)/\mathcal {B}(S_2)$-measurable.

Proof

See Lemma 4.51, p. 153 of [8]. $\square$

Notation 1.19

If E is a Banach space we denote by $|\cdot |_E$ its norm. Given two Banach spaces E and F, we denote by ${\mathcal L}(E, F)$ the Banach space of all continuous linear operators from E to F. If $E=F$ we will usually write ${\mathcal L}(E)$ instead of ${\mathcal L}(E, F)$. If H is a Hilbert space we denote by $\langle \cdot ,\cdot \rangle$ its inner product. We will always identify H with its dual via Riesz representation theorem. If V, H are two real separable Hilbert spaces, we denote by $\mathcal {L}_2(V, H)$ the space of Hilbert–Schmidt operators from V to H (see Appendix B.3). The space $\mathcal {L}_2(V, H)$ is a real separable Hilbert space with the inner product $\langle \cdot ,\cdot \rangle _2$, see Proposition B.25. $\blacksquare$

Lemma 1.20

Let $\left( \Omega , \mathscr {F}\right)$ be a measurable space and V, H be real separable Hilbert spaces. Suppose that $F:\Omega \rightarrow \mathcal {L}_2(V, H)$ is a map such that for every $v\in V$, $F(\cdot )v$ is $\mathscr {F}/\mathcal {B}(H)$-measurable. Then F is $\mathscr {F}/\mathcal {B}(\mathcal {L}_2(V, H))$-measurable.

Proof

Since $\mathcal {L}_2(V, H)$ is separable, by Lemma 1.17-(iv) it is enough to show that for every $T\in \mathcal {L}_2(V, H)$

$$ \omega {\rightarrow } \langle F(\omega ), T\rangle _2=\sum _{k=1}^{+\infty }\langle F(\omega )e_k, Te_k\rangle $$

is $\mathscr {F}/\mathcal {B}(\mathbb {R})$-measurable, where $\{e_k\}$ is any orthonormal basis of V. But this is clear since for every $\omega$

$$ \langle F(\omega ), T\rangle _2=\lim _{n\rightarrow +\infty }F_n^T(\omega ), $$

where

$$ F_n^T(\omega )=\sum _{k=1}^{n} \langle F(\omega )e_k, Te_k\rangle $$

and $F_n^T(\omega )$ is $\mathscr {F}/\mathcal {B}(\mathbb {R})$-measurable because it is a finite sum of functions that are $\mathscr {F}/\mathcal {B}(\mathbb {R})$-measurable. $\square$

Let I be an interval in ${\mathbb {R}}$, E, F be two real Banach spaces, and let E be separable. If $f:I \times E \rightarrow F$ is Borel measurable then for every $t\in I$ the function $f(t,\cdot ):E \rightarrow F$ is Borel measurable (by Lemma 1.8-(iv)).

Assume now that, for all $t \in I$ and for some $m\ge 0$, $f(t,\cdot )\in B_m(E, F)$ (the space of Borel measurable functions with polynomial growth m, see Appendix A.2 for the precise definition). It is not true in general that the function

$$ I \rightarrow B_m(E, F), \qquad t {\rightarrow } f(t,\cdot ) $$

is Borel measurable. As a counterexample¹ one can take the function

$$ [0,1] \times L^2({\mathbb {R}}) \rightarrow L^2({\mathbb {R}}),\qquad (t, x){\rightarrow } S_t x, $$

where $(S_t)_{t \ge 0}$ is the semigroup of left translations. Indeed, the map

$$ [0,1] \rightarrow {\mathcal L}(L^2({\mathbb {R}})),\qquad t {\rightarrow } S_t $$

is not measurable (see e.g. [180], Sect. 1.2). Since ${\mathcal L}(L^2({\mathbb {R}})){\subset } B_1(L^2({\mathbb {R}}), L^2({\mathbb {R}}))$ and the norm in ${\mathcal L}(L^2({\mathbb {R}}))$ is equivalent to the one induced by $B_1(L^2({\mathbb {R}}), L^2({\mathbb {R}}))$, the claim follows in a straightforward way.

On the other hand, we have the following useful result.

Lemma 1.21

Let I and $\Lambda$ be two Polish spaces. Let $\mu$ be a measure defined on the Borel $\sigma$-field $\mathcal {B}(I)$ and denote by $\overline{\mathcal {B}(I)}$ the completion of $\mathcal {B}(I)$ with respect to $\mu$. Let $f:I \times \Lambda \rightarrow \mathbb {R}$ be Borel measurable and such that for every $t\in I$, $f(t,\cdot )$ is bounded from below (respectively, above). Then the function

$$\begin{aligned} \underline{f}:I \rightarrow {\mathbb {R}}, \qquad t {\rightarrow } \inf _{a\in \Lambda } f(t, a) \end{aligned}$$

(1.1)

(respectively, $\overline{f}:I \rightarrow {\mathbb {R}},$ $t {\rightarrow } \sup _{a\in \Lambda } f(t, a)$) is $\overline{\mathcal {B}(I)}/\mathcal {B}(\mathbb {R})$-measurable.²

In particular, if I is an interval in ${\mathbb {R}}$, E, F are two real Banach spaces with E separable, if $\rho :I \times E \rightarrow F$ is Borel measurable and, for all $t \in I$ and for some $m\ge 0$, $\rho (t,\cdot )\in B_m(E, F)$, then the function

$$\begin{aligned} \rho _1:I \rightarrow {\mathbb {R}}, \qquad t {\rightarrow } \Vert f(t,\cdot )\Vert _{B_m(E, F)} \end{aligned}$$

(1.2)

is Lebesgue measurable.

Proof

The first part is Example 7.4.2 in Volume 2 of [61] (recall that Polish spaces are Souslin spaces, see [61], Definition 6.6.1, and so $I\times \Lambda$ is a Souslin space).

For the second claim, observe that since f is Borel measurable, the function

$$ f: I\times E \rightarrow {\mathbb {R}}, \qquad f(t,x):= \frac{|\rho (t, x)|_{F}}{1+|x|_E^m} $$

is also Borel measurable (since it is the product of a continuous function with the composition of a continuous function and a Borel measurable function). The result thus follows from part one. $\square$

Definition 1.22

(Independence) Consider a probability space $(\Omega , \mathscr {F}, \mathbb {P})$. Let I be a set of indices, and $\mathscr {C}_{i}\subset \mathscr {F}$ for all ${i\in I}$. We say that the families $\mathscr {C}_{i}, i\in I$, are independent if, for every finite subset J of I and every choice of $A_i\in \mathscr {C}_{i}$, $({i\in J})$, we have

$$ \mathbb {P} \left( \bigcap _{i\in J} A_i \right) = \prod _{i\in J} \mathbb {P}(A_i). $$

If $\mathscr {C}_{i}\subset \mathscr {F}$ is, for all $i\in I$, a $\pi$-system (resp. $\sigma$-field), the definition above gives in particular the notion of independent $\pi$-systems (resp. $\sigma$-fields) . Random variables are said to be independent if they generate independent $\sigma$-fields. A random variable X is independent of some $\sigma$-field $\mathscr {G}$ if $\sigma (X)$ and $\mathscr {G}$ are independent $\sigma$-fields.

Lemma 1.23

Consider a probability space $(\Omega , \mathscr {F}, \mathbb {P})$. Let $\mathscr {C}_{i}\subset \mathscr {F}$ be a $\pi$-system for every $i\in I$. If $\mathscr {C}_{i}$, $i\in I$, are independent, then $\sigma \left( \mathscr {C}_{i} \right) , i\in I$, are independent.

Proof

See [370] Lemma 2.6, p. 27. $\square$

1.1.3 The Bochner Integral

Throughout this section $(\Omega , \mathscr {F},\mu )$ is a measure space where $\mu$ is $\sigma$-finite, and E is a separable Banach space with norm $|\cdot |_E$. We endow E with the Borel $\sigma$-field $\mathcal {B}(E)$.

Lemma 1.24

Let $X:(\Omega , \mathscr {F}) \rightarrow E$ be a random variable. Then the real-valued function $|X|_E$ is measurable.

Proof

See [180] Lemma 1.2, p. 16. $\square$

Let $p\ge 1$. We denote by $L^p(\Omega , \mathscr {F},\mu ; E)$ the quotient space of the set

$$ \tilde{L}^p(\Omega , \mathscr {F},\mu ; E):= \left\{ X :(\Omega , \mathscr {F}) \rightarrow (E, \mathcal {B}(E)) \text { measurable}: \int _\Omega \left| X(\omega ) \right| ^p_E {d} \mu (\omega ) <+\infty \right\} $$

with respect to the equivalence relation of equality $\mu$-a.e. $L^p(\Omega , \mathscr {F},\mu ; E)$ is a Banach space when endowed with the norm

$$|X|_{L^p(\Omega , \mathscr {F},\mu ; E)} = \left( \int _\Omega \left| X(\omega ) \right| ^p_E {d} \mu (\omega )\right) ^{1/p}$$

(see e.g. [191] Theorem 7.17 p. 104). We will often write $L^p(\Omega ,\mu ; E)$ or $L^p(\Omega ; E)$ for $L^p(\Omega , \mathscr {F},\mu ; E)$ and denote the norm by $|X|_{L^p}$ when the context is clear. If H is a separable Hilbert space, then $L^2(\Omega , \mathscr {F},\mu ; H)$ is a Hilbert space as well, equipped with the scalar product $\left\langle X, Y \right\rangle _{L^2(\Omega , \mathscr {F},\mu ; H)} = \int _\Omega \left\langle X(\omega ), Y(\omega ) \right\rangle _H {d} \mu (\omega )$.

The space $L^{\infty }(\Omega ,\mathscr {F},\mu ; E)$ is the quotient space of the space of bounded $\mathscr {F}/ \mathcal {B}(E)$-measurable functions with respect to the relation of being equal a.e. It is a Banach space equipped with the norm

$$|X|_{L^\infty (\Omega , \mathscr {F},\mu ; E)} = \mathrm{ess}\sup _\Omega \left| X(\omega ) \right| _E. $$

In the special case when $\Omega =I$ is an interval with endpoints a and b with $a<b$ (which may be $\pm \infty$), $\mathscr {F}$ is the Borel $\sigma$-field of I, and $\mu$ is the Lebesgue measure on I, we will simply write $L^p(I;E)$ or $L^p(a, b;E)$ for $L^p(I, \mathscr {F},\mu ; E)$. Finally, we denote by $L^{p}_{\mathrm {loc}}(I;E)$ the set of measurable functions $f:I \rightarrow E$ such that $\int _{K}^{}|f(s)|_{E}^{p} {d} s$ is finite for every compact subset K of I.

Lemma 1.25

If $\mathscr {F}$ is countably generated apart from null sets then $L^p(\Omega , \mathscr {F},\mu ; E)$ is a separable Banach space.

Proof

See [194], p. 92. $\square$

Definition 1.26

(Bochner integral) Let$X :(\Omega , \mathscr {F},\mu ) \rightarrow E$ be a simple random variable $X=\sum _{i=1}^{N} x_i \mathbf{1}_{A_i}$, where $x_i\in E$, $A_i\in \mathscr {F}, \mu (A_i)<+\infty$. The Bochner integral of X is defined as

$$ \int _\Omega X(\omega ) {d}\mu (\omega ) := \sum _{i=1}^{N} x_i \mu ({A_i}). $$

Let X be in $L^1(\Omega , \mathscr {F},\mu ; E)$. The Bochner integral of X is defined as

$$ \int _\Omega X(\omega ) {d}\mu (\omega ) := \lim _{n\rightarrow +\infty } \int _\Omega X_n(\omega ) {d}\mu (\omega ), $$

where $X_n :(\Omega , \mathscr {F},\mu ) \rightarrow E$ are simple random variables such that

$$\begin{aligned} \lim _{n\rightarrow +\infty } \int _\Omega |X(\omega ) - X_n(\omega )|_E {d}\mu (\omega ) =0. \end{aligned}$$

(1.3)

Remark 1.27

It follows easily from Lemma 1.15 that, for $X\in L^1(\Omega , \mathscr {F},\mu ; E)$, there always exists a sequence of simple random variables $X_n :(\Omega , \mathscr {F},\mu ) \rightarrow E$ as in Definition 1.26, satisfying (1.3). $\blacksquare$

Proposition 1.28

Let $X\in L^1(\Omega , \mathscr {F},\mu ; E)$. Then the Bochner integral of X is well defined and does not depend on the choice of the sequence. Moreover,

$$\begin{aligned} \left| \int _\Omega X(\omega )d\mu (\omega )\right| _E\le \int _\Omega |X(\omega )|_E d\mu (\omega ). \end{aligned}$$

(1.4)

Proof

See [180] Sect. 1.1 (in particular inequality (1.6), p. 19, and the part below Lemma 1.5). The proof there is done for a probability measure $\mu$, but the general case is identical. $\square$

Proposition 1.29

Assume that $(\Omega , \mathscr {F},\mu )$ is a complete measure space, E and F are separable Banach spaces and $A : D(A) \subset E \rightarrow F$ is a closed operator (see Definition B.3). If $X \in L^1(\Omega , \mathscr {F},\mu ; E)$ and $X \in D(A)$ a.s., then AX is an F-valued random variable, and X is a D(A)-valued random variable, where D(A) is endowed with the graph norm of A (see Definition B.3). If, moreover, $\int _{\Omega } |AX(\omega )|_F \, {d} \mu (\omega ) < +\infty$, then

$$ A \int _{\Omega } X(\omega ) {d} \mu (\omega ) = \int _{\Omega } AX(\omega ) {d} \mu (\omega ). $$

Proof

The facts that X is a D(A)-valued random variable and AX is an F-valued random variable follow from Lemma 1.17-(ii). For the last part, see the proof of Proposition 1.6, Chap. 1 of [180]. $\square$

Corollary 1.30

Assume that E and F are separable Banach spaces and $T :E \rightarrow F$ is a continuous linear operator. If $X \in L^1(\Omega , \mathscr {F},\mu ; E)$, then

$$ T \int _{\Omega } X(\omega ) {d} \mu (\omega ) = \int _{\Omega } TX(\omega ) {d} \mu (\omega ). $$

Proof

This is a particular case of Proposition 1.29. $\square$

Remark 1.31

In this subsection we assumed that the space E is separable. This was done for simplicity and since we will only need this case in the vast majority of the book. However, the Bochner integral of a random variable $X:(\Omega ,\mathscr {F},\mu )\rightarrow E$ can also be defined when E is non-separable, see Sect. II.2 of [190]. If E is non-separable the definition of measurability is different. The random variable X is called measurable if there exists a sequence of simple random variables $X_n :(\Omega , \mathscr {F},\mu ) \rightarrow E$ such that $\lim _{n\rightarrow +\infty }|X(\omega ) - X_n(\omega )|_E=0$ $\mu$-a.e. When E is separable this definition of measurability is equivalent to ours. Most of the results on the Bochner integral still hold in the non-separable case. In particular, Proposition 1.29 (hence also Corollary 1.30) still holds in the following form, which we will use later in Chap. 4 (see, for example, the proof of Corollary 4.14 and of Theorem 4.80).

Let $(\Omega , \mathscr {F},\mu )$ be a complete measure space, E and F be Banach spaces and $A : D(A) \subset E \rightarrow F$ be a closed operator. If $X \in L^1(\Omega , \mathscr {F},\mu ; E)$ and $AX\in L^1(\Omega , \mathscr {F},\mu ; F)$, then

$$ A \int _{\Omega } X(\omega ) {d} \mu (\omega ) = \int _{\Omega } AX(\omega ) {d} \mu (\omega ). $$

This is Theorem 6, p. 47 of [190]. $\blacksquare$

Theorem 1.32

Let $(\Omega _1, \mathscr {F}_1)$ and $(\Omega _2, \mathscr {F}_2)$ be two measurable spaces and $\mu _1$ (respectively $\mu _2$) be a $\sigma$-finite measure on $(\Omega _1,\mathscr {F}_1)$ (respectively on $(\Omega _2, \mathscr {F}_2)$). Then there exists a unique measure $\mu _1\otimes \mu _2$ on $\mathscr {F}_1 \otimes \mathscr {F}_2$ such that, for every $A\in \mathscr {F}_1$ and $B\in \mathscr {F}_2$ with finite measure,

$$ (\mu _1\otimes \mu _2 )(A\times B) = \mu _1(A) \mu _2(B). $$

The measure $\mu _1\otimes \mu _2$ is $\sigma$-finite.

Proof

See Theorem 8.2, p. 160 in Chap. VI, Sect. 8 of [397]. $\square$

Theorem 1.33

(Fubini’s Theorem) Let $(\Omega _1, \mathscr {F}_1)$ and $(\Omega _2, \mathscr {F}_2)$ be two measurable spaces and $\mu _1$ (respectively $\mu _2$) be a $\sigma$-finite measure on $(\Omega _1,\mathscr {F}_1)$ (respectively on $(\Omega _2, \mathscr {F}_2)$). Let E be a separable Banach space with norm $|\cdot |_E$.

• (i) Let X be in $L^1(\Omega _1\times \Omega _2, \mathscr {F}_1 \otimes \mathscr {F}_2,\mu _1\otimes \mu _2; E)$. Then, for $\mu _1$-almost every $\omega _1\in \Omega _1$, the function $X(\omega _1, \cdot )$ is in $L^1(\Omega _2, \mathscr {F}_2,\mu _2; E)$, and the function given by

  $$ \omega _1 {\rightarrow } \int _{\Omega _2} X(\omega _1, \omega _2) {d} \mu _2(\omega _2) $$

  for $\mu _1$-almost all $\omega _1$ (and defined arbitrarily for other $\omega _1$) is in $L^1(\Omega _1, \mathscr {F}_1,\mu _1; E)$. Moreover, we have

  $$ \int _{\Omega _1\times \Omega _2} X(\omega _1, \omega _2) {d} (\mu _1\otimes \mu _2)(\omega _1,\omega _2) = \int _{\Omega _1} \int _{\Omega _2} X(\omega _1, \omega _2) {d} \mu _1(\omega _1) {d} \mu _2(\omega _2). $$

• (ii) Let $X:\Omega _1\times \Omega _2 \rightarrow E$ be an $\mathscr {F}_1 \otimes \mathscr {F}_2$-measurable map. Assume that, for $\mu _1$-almost every $\omega _1\in \Omega _1$, the function $X(\omega _1,\cdot )$ is in $L^1(\Omega _2, \mathscr {F}_2,\mu _2; E)$ and that the map given by

  $$ \omega _1 {\rightarrow } \int _{\Omega _2} |X(\omega _1, \omega _2)| {d} \mu _2(\omega _2) $$

  for $\mu _1$-almost all $\omega _1$ (and defined arbitrarily for other $\omega _1$) is in $L^1(\Omega _1, \mathbb {R})$. Then X is in $L^1(\Omega _1\times \Omega _2, \mathscr {F}_1 \otimes \mathscr {F}_2,\mu _1\otimes \mu _2; E)$ and part (i) of the theorem applies.

Proof

See Theorems 8.4, p. 162, and 8.7, p. 165 in Chap. VI, Sect. 8 of [397]. $\square$

Theorem 1.34

Let E be a separable Banach space and $\mu$ be a bounded measure on $(E,\mathcal {B}(E))$. Then the set of uniformly continuous and bounded functions $UC_b(E)$ is dense in $L^p(E,\mathcal {B}(E),\mu )$ for $1\le p<+\infty$.

Proof

By Lemma 1.15 and the monotone convergence theorem it is enough to prove that every characteristic function $\mathbf{1}_A$ for some $A\in \mathcal {B}(E)$ can be approximated by functions in $UC_b(E)$. Since $\mu$ is regular, for every ${\varepsilon }>0$ we can find a closed set $C, C\subset A$, and an open set $U, A\subset U$, such that $\mu (U\setminus C)<{\varepsilon }^p$. Moreover, considering sets $U_n=\{x\in U: \mathrm{dist}(x:A)>1/n\}$ if necessary, we can assume that $\mathrm{dist}(C, U)>0$. Then the function

$$ f_{\varepsilon }(x):=\frac{\mathrm{dist}(x,U)}{\mathrm{dist}(x,A)+\mathrm{dist}(x, U)} $$

belongs to $UC_b(E)$ and $|\mathbf{1}_A-f_{\varepsilon }|_{L^p}<{\varepsilon }$. $\square$

1.1.4 Expectation, Covariance and Correlation

Let $(\Omega , \mathscr {F}, \mathbb {P})$ be a probability space and E be a separable Banach space with norm $|\cdot |_E$.

Definition 1.35

(Expectation) Given X in $L^1(\Omega , \mathscr {F},\mathbb {P};E)$, we denote by $\mathbb {E}[X]$ the (Bochner) integral $\int _\Omega X(\omega ) {d} \mathbb {P}(\omega )$. $\mathbb {E}[X]$ is said to be the expectation (or the mean) of X.

To define the covariance operator, we recall first that if $x\in E$, $y\in F$, where E, F are Hilbert spaces, the operator $x\otimes y:F\rightarrow E$ is defined by

$$ (x\otimes y)h=x\langle y, h\rangle _F. $$

Definition 1.36

(Covariance operator, correlation) Given a real, separable Hilbert space H and $X \in L^2(\Omega , \mathscr {F},\mathbb {P}; H)$, the covariance operator of X is defined by

$$ Cov(X):= \mathbb {E} \Big [ (X-\mathbb {E}[X]) \otimes (X-\mathbb {E}[X]) \Big ]. $$

For$X, Y \in L^2(\Omega , \mathscr {F},\mathbb {P}; H)$, the correlation of X and Y is the operator defined by

$$ Cor(X, Y):= \mathbb {E} \Big [ (X-\mathbb {E}[X]) \otimes (Y-\mathbb {E}[Y]) \Big ]. $$

Remark 1.37

For$X \in L^2(\Omega , \mathscr {F},\mathbb {P}; H)$, the operator Cov(X) is positive, symmetric and nuclear (see [180], p. 26). $\blacksquare$

1.1.5 Conditional Expectation and Conditional Probability

Theorem 1.38

Consider a separable Banach space E, a probability space $(\Omega , \mathscr {F},\mathbb {P})$ and a sub-$\sigma$-field $\mathscr {G}\subset \mathscr {F}$. There exists a unique contractive linear operator $\mathbb {E}[\cdot | \mathscr {G}] :L^1(\Omega , \mathscr {F},\mathbb {P}; E) \rightarrow L^1(\Omega , \mathscr {G},\mathbb {P};E)$ such that

$$ \int _A \mathbb {E}[\xi | \mathscr {G}](\omega ) {d} \mathbb {P}(\omega ) = \int _A \xi (\omega ) {d} \mathbb {P}(\omega )\qquad \text {for all } A\in \mathscr {G} \text { and }\xi \in L^1(\Omega , \mathscr {F},\mathbb {P};E). $$

If $E=H$ is a Hilbert space the restriction of $\mathbb {E}[\cdot | \mathscr {G}]$ to $L^2(\Omega , \mathscr {F},\mathbb {P};H)$ is the orthogonal projection $L^2(\Omega , \mathscr {F},\mathbb {P};H) \rightarrow L^2(\Omega , \mathscr {G},\mathbb {P};H)$.

Proof

See [180] Proposition 1.10, p. 26, and [458] Proposition V-2-5, pp. 102–103.    $\square$

Definition 1.39

(Conditional expectation) Given $X\in L^1(\Omega , \mathscr {F},\mathbb {P}; E)$ , the random variable $\mathbb {E}[X | \mathscr {G}] \in L^1(\Omega , \mathscr {G},\mathbb {P};E)$, defined by Theorem 1.38, is called the conditional expectation of X given $\mathscr {G}$.

Definition 1.40

Let $(\Omega , \mathscr {F},\mathbb {P})$ be a probability space and let E be a separable Banach space. A family $\mathcal {H}$ of integrable random variables $X\in L^1(\Omega , \mathscr {F},\mathbb {P}; E)$ is called uniformly integrable if

$$ \lim _{R\rightarrow \infty } \sup _{X\in \mathcal {H}} \int _{|X|_E\ge R} |X(\omega )|_E {d} \mathbb {P}(\omega ) =0. $$

The following proposition collects various properties of conditional expectation (see e.g. [487] Proposition 3.15, p. 25, see also [572] Sect. 9.7, p. 88, for similar properties for real-valued random variables).

Proposition 1.41

Let $(\Omega , \mathscr {F},\mathbb {P})$ be a probability space and let E be a separable Banach space. The conditional expectation has the following properties:

• (i) If $X\in L^1(\Omega , \mathscr {F},\mathbb {P}; E)$ is $\mathscr {G}$-measurable, then $\mathbb {E}[X|\mathscr {G}]=X$ $\mathbb {P}$-a.s.

• (ii) Given $X\in L^1(\Omega , \mathscr {F},\mathbb {P}; E)$ and two $\sigma$-fields $\mathscr {G}_1$ and $\mathscr {G}_2$ such that $\mathscr {G}_1\subset \mathscr {G}_2\subset \mathscr {F}$,

  $$ \mathbb {E} \Big [ \mathbb {E} \big [ X | \mathscr {G}_1 \big ] \big | \mathscr {G}_2 \Big ] = \mathbb {E} \Big [ \mathbb {E} \big [ X | \mathscr {G}_2 \big ] \big |\mathscr {G}_1 \Big ] = \mathbb {E} \big [ X | \mathscr {G}_1 \big ] \qquad \mathbb {P} \text {-a.s.} $$

• (iii) Let $X\in L^1(\Omega , \mathscr {F},\mathbb {P}; E)$. If X is independent of $\mathscr {G}$, then $\mathbb {E} \left[ X | \mathscr {G} \right] = \mathbb {E}[ X]$ $\mathbb {P}$-a.s. Moreover, X is independent of $\mathscr {G}$ if and only if, for any bounded, Borel measurable $f:E\rightarrow \mathbb {R}$, $\mathbb {E} \left[ f(X) | \mathscr {G} \right] = \mathbb {E} f(X)$ $\mathbb {P}$-a.s.

• (iv) If X is $\mathscr {G}$-measurable and $\zeta$ is a real-valued integrable random variable such that $\zeta X \in L^1(\Omega , \mathscr {F},\mathbb {P}; E)$, then

  $$ \mathbb {E} \Big [ \zeta X | \mathscr {G} \Big ] = X \mathbb {E} \Big [ \zeta | \mathscr {G} \Big ] \qquad \mathbb {P}\text {-a.s.} $$

• (v) If $X\in L^1(\Omega , \mathscr {F},\mathbb {P}; E)$ and $\zeta$ is an integrable, real-valued, $\mathscr {G}$-measurable random variable such that $\zeta X \in L^1(\Omega , \mathscr {F},\mathbb {P}; E)$, then

  $$ \mathbb {E} \Big [ \zeta X | \mathscr {G} \Big ] = \zeta \mathbb {E} \Big [ X | \mathscr {G} \Big ] \qquad \mathbb {P}\text {-a.s.} $$

• (vi) If $X\in L^1(\Omega , \mathscr {F},\mathbb {P}; E)$ and $f:\mathbb {R}\rightarrow \mathbb {R}$ is a convex function such that $\mathbb {E} \left[ |f(|X|_E)| \right] < +\infty$, then

  $$ f \left( \left| \mathbb {E} \Big [ X | \mathscr {G} \Big ] \right| _E \right) \le \mathbb {E} \Big [ f \left( \left| X \right| _E \right) | \mathscr {G} \Big ] \qquad \mathbb {P} \text { -a.s.} $$

• (vii) If $X, X_n\in L^1(\Omega , \mathscr {F},\mathbb {P}; E)$ for every $n\in \mathbb {N}$, the family $(X_n)_{n \in \mathbb {N}}$ is uniformly integrable and $X_n \xrightarrow {n\rightarrow \infty } X$, $\mathbb {P}$-a.s., then

  $$ \mathbb {E} \Big [ X_n | \mathscr {G} \Big ] \xrightarrow {n\rightarrow \infty } \mathbb {E} \Big [ X | \mathscr {G} \Big ] \qquad \mathbb {P} \text { -a.s.} $$

• (viii) Let $X\in L^1(\Omega , \mathscr {F},\mathbb {P}; E)$. Assume that $\mathscr {G}_n$ for $n\in \mathbb {N}$ is an increasing family of $\sigma$-fields such that $\mathscr {G} = \sigma \left( \mathscr {G}_n : n\in \mathbb {N} \right)$ is a sub-$\sigma$-field of $\mathscr {F}$. Then

  $$ \mathbb {E} \Big [ X | \mathscr {G}_n \Big ] \xrightarrow {n\rightarrow \infty } \mathbb {E} \Big [ X | \mathscr {G} \Big ] \qquad \mathbb {P} \text { -a.s.} $$

• (ix) Let Z be a separable Banach space and let $T\in \mathcal {L}(E, Z)$. Then

  $$ \mathbb {E}[TX|\mathscr {G}] = T\mathbb {E}[X|\mathscr {G}] \qquad \mathbb {P} \text { -a.s.} $$

Proposition 1.42

Let $(\Omega , \mathscr {F},\mathbb {P})$ be a probability space. Then:

• (i) If $X, Y\in L^1(\Omega , \mathscr {F},\mathbb {P}; \mathbb {R})$ and $X\ge Y$, then

  $$ \mathbb {E}[ X | \mathscr {G} ] \ge \mathbb {E} [ Y | \mathscr {G} ]. $$

• (ii) (Conditional Fatou Lemma) If $X_n\in L^1(\Omega , \mathscr {F},\mathbb {P}; \mathbb {R})$ and $X_n \ge 0$, then

  $$ \mathbb {E}[ \lim \inf _{n\rightarrow \infty } X_n | \mathscr {G} ] \le \lim \inf _{n\rightarrow \infty } \mathbb {E} [ X_n | \mathscr {G} ] \qquad \mathbb {P}\text {-a.s.} $$

Proof

See [572], Sect. 9.7, p. 88. $\square$

Proposition 1.43

Let $(E_{1},{\mathscr {E}}_{1})$ and $(E_{2},{\mathscr {E}}_{2})$ be two measurable spaces and $\psi :E_{1} \times E_{2}\rightarrow \mathbb {R}$ be a bounded measurable function. Let $X_1,X_2$ be two random variables in a probability space $(\Omega ,{\mathscr {F}},{\mathbb P})$ with values in $(E_{1},{\mathscr {E}}_{1})$ and $(E_{2},{\mathscr {E}}_{2})$ respectively, and let ${\mathscr {G}}\subset {\mathscr {F}}$ be a $\sigma$-field. If $X_1$ is ${\mathscr {G}}$-measurable and $X_2$ is independent of ${\mathscr {G}},$ then

$$\begin{aligned} {\mathbb E}[\psi (X_1,X_2)|{\mathscr {G}}]=\widehat{\psi }(X_1), \quad {\mathbb P}\text {-a.s.}, \end{aligned}$$

(1.5)

where

$$\begin{aligned} \widehat{\psi }(x_1)={\mathbb E}[\psi (x_1,X_2)], \;\quad x_1 \in E_{1}. \end{aligned}$$

(1.6)

Proof

See Proposition 1.12, p. 28 of [180]. $\square$

Let $(\Omega , \mathscr {F},\mathbb {P})$ be a probability space, and $\mathscr {G}$ be a sub-$\sigma$-field of $\mathscr {F}$. The conditional probability of $A\in \mathscr {F}$ given $\mathscr {G}$ is defined by

$$ \mathbb {P}(A|\mathscr {G})(\omega ) := \mathbb {E}[\mathbf{1}_A | \mathscr {G}](\omega ). $$

Definition 1.44

Let $(\Omega , \mathscr {F},\mathbb {P})$ be a probability space, and $\mathscr {G}$ be a sub-$\sigma$-field of $\mathscr {F}$. A function $p:\Omega \times \mathscr {F} \rightarrow [0,1]$ is called a regular conditional probability given $\mathscr {G}$ if it satisfies the following conditions:

• (i) for each $\omega \in \Omega$, $p(\omega , \cdot )$ is a probability measure on $(\Omega , \mathscr {F})$;

• (ii) for each $B\in \mathscr {F}$, the function $p(\cdot , B)$ is $\mathscr {G}$-measurable;

• (iii) for every $A\in \mathscr {F}$, $\mathbb {P}(A|\mathscr {G})(\omega )= p(\omega , A)$, $\mathbb {P}$-a.s.

It thus follows that, if $X\in L^1(\Omega , \mathscr {F},\mathbb {P}; E)$, where E is a separable Banach space, then

$$ \mathbb {E} [ X | \mathscr {G} ](\omega )= \int _\Omega X(\omega ')p(\omega , d\omega ')\quad \mathbb {P}\,\, a.s. $$

Theorem 1.45

Let $(\Omega , \mathscr {F},\mathbb {P})$ be a probability space, where $(\Omega , \mathscr {F})$ is a standard measurable space. Then, for every sub-$\sigma$-field $\mathscr {G}\subset \mathscr {F}$, there exists a regular conditional probability $p(\cdot ,\cdot )$ given $\mathscr {G}$. Moreover, if $p'(\cdot , \cdot )$ is another regular conditional probability given $\mathscr {G}$, then there exists a set $N\in \mathscr {G}, \mathbb {P}(N)=0$, such that, if $\omega \not \in N$ then $p(\omega ,A) = p'(\omega , A)$ for all $A\in \mathscr {F}$.

Moreover, if $\mathscr {H}$ is a countably determined sub-$\sigma$-field of $\mathscr {G}$, then there exists a $\mathbb {P}$-null set $N\in \mathscr {G}$ such that, if $\omega \not \in N$ then $p(\omega , A)=\mathbf{1}_{A}(\omega )$ for every $A\in \mathscr {H}$. In particular, if $(\Omega _1,\mathscr {F}_1)$ is a measurable space, $\mathscr {F}_1$ is countably determined, $\{ x \}\in \mathscr {F}_1$ for all $x \in \Omega _1$ and $\xi :(\Omega , \mathscr {F}) \rightarrow (\Omega _1,\mathscr {F}_1)$ is a $\mathscr {G}/\mathscr {F}_1$-random variable, then $p \left( \omega , \{ \omega ' \; : \; \xi (\omega )=\xi (\omega ') \} \right) = 1$ for $\mathbb {P}$-a.e. $\omega$.

Proof

See Theorem 8.1, p. 147 in [478], or Theorems 3.1, 3.2, and the corollary following them in [356] (see also [575] Proposition 1.9, p. 11). $\square$

Notation 1.46

If the regular conditional probability exists, we will often write $\mathbb {P}(\cdot |\mathscr {G})(\omega )$ or $\mathbb {P}_\omega$ for $p(\omega ,\cdot )$. $\blacksquare$

Definition 1.47

(Law of a random variable) Given a probability space $(\Omega , \mathscr {F}, \mathbb {P})$, a measurable space $(\Omega _1, \mathscr {F}_1)$, and a random variable $X:(\Omega , \mathscr {F}) \rightarrow (\Omega _1, \mathscr {F}_1)$, the probability measure on $(\Omega _1, \mathscr {F}_1)$ defined by

$$ \mathcal {L}_{\mathbb {P}}(X)(A) := \mathbb {P} \left( \left\{ \omega \in \Omega \; : \; X(\omega ) \in A \right\} \right) $$

is called the law (or distribution) ³ of X. We denote the law of X by $\mathcal {L}_{\mathbb {P}}(X)$.

Proposition 1.48

(Change of variables) Given a probability space $(\Omega , \mathscr {F}, \mathbb {P})$, a measurable space $(\Omega _1, \mathscr {F}_1)$, a random variable $X:(\Omega , \mathscr {F}) \rightarrow (\Omega _1, \mathscr {F}_1)$, and a bounded Borel function $\varphi :\Omega _1\rightarrow {\mathbb {R}}$ we have

$$ \int _\Omega \varphi (X (\omega ))d{\mathbb P}(\omega ) = \int _{\Omega _1} \varphi (\omega ')d\mathcal {L}_{\mathbb {P}}(X)(\omega '). $$

Definition 1.49

(Convergence of random variables) Consider a probability space $(\Omega , \mathscr {F},\mathbb {P})$ and a Polish space (S, d) endowed with the Borel $\sigma$-field. Let $X_n :\Omega \rightarrow S$ and $X :\Omega \rightarrow S$ be random variables. We say that:

• (i) $X_n$ converges to X $\mathbb {P}$-a.s. (and we write $X_n \rightarrow X$ $\mathbb {P}$-a.s.) if $\lim _{n\rightarrow \infty } d(X_n(\omega ), X(\omega ))=0$ $\mathbb {P}$-a.s.

• (ii) $X_n$ converges to X in probability if, for every ${\varepsilon } >0$, $\lim _{n\rightarrow +\infty } \mathbb {P} \left\{ \omega \in \Omega \; : \; d(X_n(\omega ), X(\omega ))>{\varepsilon } \right\} =0$.

• (iii) $X_n$ converges to X in law if, for every bounded and continuous $f:S\rightarrow \mathbb {R}$, $\int _S f(u) {d} \mathcal {L}_{\mathbb {P}}(X)(u) = \lim _{n\rightarrow \infty } \int _S f(u) {d} \mathcal {L}_{\mathbb {P}}(X_n)(u)$ (i.e. if $\mathbb {E}\left[ f(X) \right] =\lim _{n\rightarrow \infty } \mathbb {E}\left[ f(X_n) \right]$).

Lemma 1.50

Consider a probability space $(\Omega , \mathscr {F},\mathbb {P})$ and a Polish space (S, d) endowed with the Borel $\sigma$-field. Let $X_n :\Omega \rightarrow S$ and $X :\Omega \rightarrow S$ be random variables.

• (i) If $X_n$ converges to X $\mathbb {P}$-a.s. then $X_n$ converges to X in probability.

• (ii) If $X_n$ converges to X in probability then $X_n$ converges to X in law.

• (iii) If $X_n$ converges to X in probability then it contains a subsequence $X_{n_k}$ such that $X_{n_k}$ converges to X $\mathbb {P}$-a.s.

• (iv) (Egoroff’s theorem) If $X_n$ converges to X $\mathbb {P}$-a.s. then for every ${\varepsilon }>0$, there exists an ${\tilde{\Omega }}\in \mathscr {F}$ such that $\mathbb {P}(\Omega \setminus {\tilde{\Omega }})<{\varepsilon }$, and $X_n$ converges uniformly to X on ${\tilde{\Omega }}$.

• (v) Let $X, X_n\in L^p(\Omega , \mathscr {F},\mathbb {P}; E), n\in \mathbb {N}, p\ge 1$, and E be a separable Banach space. If $X_n$ converges to X in $L^p(\Omega , \mathscr {F},\mathbb {P}; E)$, then $X_n$ converges to X in probability.

Proof

For (i), (ii) and (iii) see, for instance, [370] Lemmas 4.2, p. 63 and 4.7, p. 66. Part (iv) can be found, for instance, in [73] Theorem 2, p. 170, Sect. 4.5.4. Property (v) is straightforward. $\square$

Lemma 1.51

Let $p>1$ and $X, X_n\in L^p(\Omega , \mathscr {F},\mathbb {P}; E), n\in \mathbb {N}$, for some separable Banach space E. Suppose that, for some $M>0$, $\mathbb {E} \left[ \left| X_n \right| _E^p \right] \le M$ for all $n\in \mathbb {N}$. If $X_n \rightarrow X$ in probability, then $\mathbb {E}\left[ \left| X-X_n \right| _E \right] \rightarrow 0$.

Proof

Since the sequence $(X_n)$ is bounded in $L^p(\Omega , \mathscr {F},\mathbb {P}; E)$, it is uniformly integrable (see e.g. [572], p. 127, Sect. 13.3). The claim follows, for example, from Theorem 13.7, p. 131 of [572]. $\square$

1.1.6 Gaussian Measures on Hilbert Spaces and the Fourier Transform

In this section we recall the notions of Gaussian measure and the Fourier transform for Hilbert space-valued random variables. For an extensive treatment of the subject we refer to [180], Chap. 2, [153], Chap. 1 or [154], Chap. 1.

For a real separable Hilbert space H we denote by $\mathcal {L}_1(H)$ the Banach space of the trace class operators on H, by $\mathcal {L}^+(H)$ the subspace (of $\mathcal {L}(H)$) of all bounded, linear, self-adjoint, positive operators, and we set $\mathcal {L}_{1}^+(H):= \mathcal {L}_{1}(H) \cap \mathcal {L}^+(H)$ (see Appendix B.3). We will denote by $M_1(H)$ the set of probability measures on $(H,\mathcal {B}(H))$.

Proposition 1.52

Consider a real, separable Hilbert space H with the Borel $\sigma$-field $\mathcal {B}(H)$ and a probability measure $\mathbb {P}$ on $(H, \mathcal {B}(H))$. If $\int _H |y| \, {d} \mathbb {P} (y) < +\infty$, then we can define

$$ m:=\int _H y \, {d} \mathbb {P} (y) \in H. $$

If $\int _H |y|^2 {d} \mathbb {P} (y) < +\infty$, then there exists a unique $Q\in \mathcal {L}_1^+(H)$ such that

$$ \left\langle Q x, y \right\rangle :=\int _H \left\langle x, h-m \right\rangle \left\langle y, h-m \right\rangle {d} \mathbb {P} (h). $$

Proof

See [153], p. 7. $\square$

Definition 1.53

(Mean and covariance of a measure on H) We call m and Q, defined by Proposition 1.52, respectively the mean and the covariance of $\mathbb {P}$. In other words, the mean (respectively covariance) of $\mathbb {P}$ is the mean (respectively covariance) of the identity random variable $I :(H,\mathcal {B}(H), \mathbb {P}) \rightarrow (H, \mathcal {B}(H))$.

Definition 1.54

(Fourier transform of a measure) LetH be a Hilbert space and $\mathcal {B}(H)$ be its Borel $\sigma$-field. Given a probability measure $\mathbb {P}$ on $(H,\mathcal {B}(H))$ we define, for $x\in H$,

$$ {\hat{\mathbb {P}}} (x) := \int _H e^{i \left\langle y, x \right\rangle } {d} \mathbb {P} (y). $$

We call ${\hat{\mathbb {P}}} :H \rightarrow \mathbb {C}$ the Fourier transform of ${\mathbb {P}}$.

Proposition 1.55

Let H be a real, separable Hilbert space, $\mathcal {B}(H)$ be its Borel $\sigma$-field, and $\mathbb {P}_1$ and $\mathbb {P}_2$ be two probability measures on $(H, \mathcal {B}(H))$. If ${\hat{\mathbb {P}}}_1 (x) = {\hat{\mathbb {P}}}_2(x)$ for all $x\in H$, then $\mathbb {P}_1=\mathbb {P}_2$.

Proof

See [153] Proposition 1.7, p. 6, or [180], Proposition 2.5, p. 35. $\square$

Theorem 1.56

Let $X_1,..., X_n$ be random variables in a real, separable Hilbert space H. The random variables are independent if and only if for every $y_1,..., y_n\in H$

$$\begin{aligned} \mathbb {E}\left[ e^{\left[ i \sum _{i=1}^n\left\langle X_i, y_i \right\rangle \right] }\right] =\prod _{i=1}^n\mathbb {E}\left[ e^{\left[ i \left\langle X_i, y_i \right\rangle \right] }\right] . \end{aligned}$$

(1.7)

Proof

Obviously if $X_1,..., X_n$ are independent then (1.7) holds. Also, Theorem 1.56 is well known if $H=\mathbb {R}^k$. Let now $k\in \mathbb {N}$ and $y_i^j\in H, i=1,..., n, j=1,..., k$, and consider random variables $X_i^k=(\langle X_i, y_i^1\rangle ,...,\langle X_i, y_i^k\rangle ), i=1,..., n$ in $\mathbb {R}^k$. Therefore, if (1.7) holds then $X_i^k, i=1,..., n$, are independent for every $k\in \mathbb {N}$ and $y_i^j\in H, j=1,..., k$. Since cylindrical sets of the form $\{x:(\langle x, y_i^1\rangle ,...,\langle x, y_i^k\rangle )\in A\in \mathcal {B}(\mathbb {R}^k)\}$ generate $\mathcal {B}(H)$ and are a $\pi$-system, the collection of sets $\{\omega :(\langle X_i, y_i^1\rangle ,...,\langle X_i, y_i^k\rangle )\in A\in \mathcal {B}(\mathbb {R}^k)\}$ over all $k\in \mathbb {N}$ and $y_i^j\in H, i=1,..., n, j=1,..., k, A\in \mathcal {B}(\mathbb {R}^k)$ is a $\pi$-system generating $\sigma (X_i)$. Thus, by Lemma 1.23, the sigma algebras $\sigma (X_1),...,\sigma (X_n)$ are independent. $\square$

Theorem 1.57

Let H be a real, separable Hilbert space, $\mathcal {B}(H)$ be its Borel $\sigma$-field, $a\in H$, and $Q\in \mathcal {L}_1^+(H)$. Then there exists a unique probability measure $\mathbb {P}$ on $(H, \mathcal {B}(H))$ such that

$$ {\hat{\mathbb {P}}} (x) = e^{i\left\langle a, x \right\rangle - \frac{1}{2} \left\langle Qx, x \right\rangle }. $$

The measure $\mathbb {P}$ has mean a and covariance Q.

Proof

See [153] Theorem 1.12, p. 12. $\square$

Definition 1.58

(Gaussian measure on H) LetH be a real, separable Hilbert space, $\mathcal {B}(H)$ be its Borel $\sigma$-field, $a\in H$, and $Q\in \mathcal {L}_1^+(H)$. The unique probability measure $\mathbb {P}$ identified by Theorem 1.57 is called the Gaussian measure with mean a and covariance Q, and is denoted by $\mathcal {N}(a, Q)$. When $a=0$ we will denote it by ${\mathcal N}_Q$ and call it a centered Gaussian measure.

We now provide two useful results about Gaussian measures.

Proposition 1.59

Let $Q\in {\mathcal L}_1^+(H)$. Then for all $y, z\in H$

$$\begin{aligned} \int _H \langle x,y\rangle \langle x,z\rangle {\mathcal N}_Q(dx) = \langle Qy, z\rangle . \end{aligned}$$

(1.8)

Define, for $y\in Q^{1/2}(H)$, ${\mathcal Q}_y \in L^2(H,{\mathcal N}_Q)$ as

$$\begin{aligned} {\mathcal Q}_y(x) :=\langle Q^{-1/2}y, x\rangle , \end{aligned}$$

(1.9)

where $Q^{-1/2}$ is the pseudoinverse of $Q^{1/2}$ (see Definition B.1). The map (called the “white noise function”, see e.g. [154] Sect. 2.5)

$$ y\in Q^{1/2}(H) \rightarrow {\mathcal Q}_y \in L^2(H,{\mathcal N}_Q) $$

can be extended to $H_0=\overline{Q^{1/2}(H)}=(\ker Q)^\perp$ and it satisfies

$$ \int _H {\mathcal Q}_y(x){\mathcal Q}_z(x) {\mathcal N}_Q(dx) = \langle y,z\rangle , \qquad y, z \in H_0. $$

Moreover, for all $m>0$ we have

$$\begin{aligned} \int _H |x|^{2m} {\mathcal N}_Q(dx) \le K(m)[\mathrm{Tr} (Q)]^{m} \end{aligned}$$

(1.10)

for some $K(m)>0$, independent of Q.

Proof

Formula (1.8) follows from Proposition 1.2.4 in [179].

The second statement is proved, when $\ker Q=\{0\}$, in [154] Sect. 2.5.2 (see also Sect. 1.2.4 of [179]). Since here we do not assume $\ker Q=\{0\}$, we provide a proof. First we observe that $\ker Q=\ker Q^{1/2}$ and that $Q^{1/2}(H)$ is dense in $(\ker Q)^\perp$ since $Q^{1/2}$ is self-adjoint. Moreover, by Definition B.1, the pseudoinverse of $Q^{1/2}$ is the operator $Q^{-1/2}:Q^{1/2}(H) \rightarrow (\ker Q)^\perp$, hence the map $y \rightarrow {\mathcal Q}_y=\langle Q^{-1/2}y, x\rangle$ is well defined for all $y\in Q^{1/2}(H)$. Furthermore, thanks to formula (1.8), we have, for $y_1,y_2 \in Q^{1/2}(H)$

$$ \int _H \langle Q^{-1/2}y_1,x\rangle \langle Q^{-1/2}y_2,x\rangle {\mathcal N}_Q(dx) =\langle Q(Q^{-1/2}y_1), Q^{-1/2}y_2\rangle =\langle y_1,y_2\rangle , $$

where we used that $Q^{1/2}Q^{-1/2}y=y$ for all $y \in Q^{1/2}(H)$. Hence, for $y_1,y_2 \in Q^{1/2}(H)$,

$$\begin{aligned} \int _H {\mathcal Q}_{y_1}(x){\mathcal Q}_{y_2}(x) {\mathcal N}_Q(dx) =\langle y_1,y_2\rangle . \end{aligned}$$

(1.11)

In view of the above the map $y \rightarrow {\mathcal Q}_y=\langle Q^{-1/2}y, x\rangle$ is an isometry and can be extended to $\overline{Q^{1/2}(H)}=(\ker Q)^\perp$ (endowed with the inner product inherited from H) and (1.11) extends to all $y_1,y_2 \in (\ker Q)^\perp$.

We remark that as pointed out in [154] Sect. 2.5.2, for a generic $y\in (\ker Q)^\perp$ the image ${\mathcal Q}_y$ is an element of $L^2(H,{\mathcal N}_Q)$, hence an equivalence class of random variables defined ${\mathcal N}_Q$-a.e.; in particular, writing ${\mathcal Q}_y(x)=\langle y, Q^{-1/2}x\rangle$, ${\mathcal N}_Q$-a.e., would be misleading since, as proved in [154] Proposition 2.22, ${\mathcal N}_Q(Q^{1/2}(H))=0$.

Concerning the third claim, by Proposition 2.19, p. 50, of [180], it holds for $m\in \mathbb N$. If $k-1<m<k$ for $k=1,2,...$, we use

$$ \int _H |x|^{2m} {\mathcal N}_Q(dx) \le \left[ \int _H |x|^{2k} {\mathcal N}_Q(dx)\right] ^{m/k}. $$

$\square$

Theorem 1.60

(Cameron–Martin formula) LetH be a real, separable Hilbert space. Let $a_1,a_2\in H$ and $Q\in \mathcal {L}_1^+(H)$. Then:

• (1) The Gaussian measures $\mathcal {N}(a_1,Q)$ and $\mathcal {N} (a_2,Q)$ are either singular or equivalent.

• (2) They are equivalent if and only if $a_1 - a_2 \in Q^{1/2}(H)$ and in this case

  $$ \frac{{d} \mathcal {N} (a_1, Q)}{{d} \mathcal {N} (a_2 , Q )}(x) = \exp \left( \langle Q^{-1/2} (a_1-a_2), Q^{-1/2} (x-a_2) \rangle -\frac{1}{2} \left| Q^{-1/2}(a_1-a_2)\right| ^2 \right) $$

  for $\mathcal {N} (a_2 , Q )$-a.e. $x\in H$.

Proof

See Theorem 2.23, p. 53 of [180]. $\square$

We now recall some results concerning compactness of a family of measures in $M_1(H)$ (see e.g. Sect. 2.1 in [180] or [219, 478] for more on this).

Definition 1.61

• (i) A sequence $(\mathbb P_n)$ in $M_1(H)$ is said to be weakly convergent to some $\mathbb P\in M_1(H)$ if, for every $\phi \in C_b(H)$,

  $$ \lim _{n \rightarrow + \infty } \int _H \phi (x)\mathbb P_n(dx) =\int _H \phi (x)\mathbb P(dx). $$

• (ii) A family $\Lambda {\subset } M_1(H)$ is said to be compact (respectively, relatively compact) if an arbitrary sequence $\mathbb P_n$ of elements of $\Lambda$ contains a subsequence $\mathbb P_{n_k}$ weakly convergent to a measure $\mathbb P\in \Lambda$ (respectively, to a measure $\mathbb P\in M_1(H)$).

• (iii) A family $\Lambda {\subset } M_1(H)$ is said to be tight if for any $\varepsilon > 0$ there exists a compact set $K_\varepsilon$ such that, for every $\mathbb P\in \Lambda$,

  $$ \mathbb P(K_\varepsilon )>1 -\varepsilon . $$

The following theorem (which also holds when H is a Polish space) is due to Prokhorov.

Theorem 1.62

Let H be a real separable Hilbert space. A family $\Lambda {\subset } M_1(H)$ is relatively compact if and only if it is tight.

Proof

See [180], the proof of Theorem 2.3. $\square$

The next theorem gives a useful sufficient condition for compactness.

Theorem 1.63

Let H be a real separable Hilbert space and let $\{e_i\}_{i\in \mathbb {N}}$ be an orthonormal basis in H. A family $\Lambda {\subset } M_1(H)$ is relatively compact if

$$ \lim _{N\rightarrow + \infty } \sup _{\mathbb P\in \Lambda } \int _H \sum _{i=N}^{+\infty }\langle x, e_i\rangle ^2 \mathbb P(dx)=0. $$

Proof

See [478], the proof of Theorem VI.2.2. $\square$

Concerning Gaussian measures, we have the following result (see Proposition 1.1.5 of [493]).

Proposition 1.64

Let ${\mathcal N}_{Q_n}$ ($n\in \mathbb {N}$) and ${\mathcal N}_Q$ be centered Gaussian measures on H. If $\lim _{n \rightarrow + \infty } \Vert Q_n-Q\Vert _{{\mathcal L}_1(H)} = 0$, then the measures ${\mathcal N}_{Q_n}$ converge weakly to ${\mathcal N}_{Q}$.

Proof

Observe that if $\{e_i\}_i$ is an orthonormal basis in H, it follows from (1.8) that for any $N\in \mathbb N$,

$$ \int _H \sum _{i=N}^{+\infty }\langle x, e_i\rangle ^2 {\mathcal N}_{Q_n}(dx)= \sum _{i=N}^{+\infty }\langle Q_ne_i, e_i\rangle . $$

Since $\lim _{n \rightarrow + \infty } \Vert Q_n-Q\Vert _{{\mathcal L}_1(H)} = 0$, the above formula implies in particular that Theorem 1.63 applies and thus the sequence $({\mathcal N}_{Q_n})$ is relatively compact.

Moreover, from Theorem 1.57 and Definition 1.58 it is immediate that, as $n\rightarrow + \infty$,

$$ \widehat{{\mathcal N}_{Q_n}}(x)=e^{-\frac{1}{2}\langle Q_n x, x\rangle } \quad \longrightarrow \quad e^{-\frac{1}{2}\langle Q x, x\rangle }=\widehat{{\mathcal N}_{Q}}(x), \qquad \forall x\in H. $$

Take now a subsequence ${\mathcal N}_{Q_{n_k}}$ weakly convergent to a probability measure $\mathbb P_0$. By Definition 1.54 we must have

$$ \widehat{{\mathcal N}_{Q_{n_k}}}(x) \rightarrow \widehat{\mathbb P_0}(x), \qquad \forall x \in H. $$

This implies that $\widehat{\mathbb P_0}=\widehat{{\mathcal N}_{Q}}$ and hence, by Proposition 1.55, that ${\mathbb P_0}={{\mathcal N}_{Q}}$. Since this is true for any convergent subsequence, the claim now follows by a standard contradiction argument. $\square$

We conclude with a useful result about uniformity of weak convergence. The result is also true if H is a Polish space, see [478], Theorem II.6.8.

Theorem 1.65

Let $\mathbb P_n$ be a sequence in $M_1(H)$ and $\mathbb P\in M_1(H)$. Then $\mathbb P_n$ is weakly convergent to $\mathbb P$ if and only if

$$ \lim _{n \rightarrow + \infty }\sup _{\phi \in {\mathcal C}_0} \left| \int _H \phi (x)\mathbb P_n(dx) -\int _H \phi (x)\mathbb P(dx) \right| =0 $$

for every family ${\mathcal C}_0 \subset C_b(H)$ which is equicontinuous at all points $x \in H$ and uniformly bounded, i.e., for some constant $M>0$, $|f(x)|\le M$ for all $x \in H$ and $f \in {\mathcal C}_0$.

Proof

See [478], the proof of Theorem II.6.8. $\square$

1.2 Stochastic Processes and Brownian Motion

1.2.1 Stochastic Processes

Definition 1.66

(Filtration, usual conditions) Let $t\ge 0$. A filtration$\left\{ \mathscr {F}^t_{s} \right\} _{s \ge t}$ in a complete probability space $(\Omega , \mathscr {F},\mathbb {P})$ is a family of $\sigma$-fields such that $\mathscr {F}^t_{s} \subset \mathscr {F}^t_{r}\subset \mathscr {F}$ whenever $t\le s\le r$.

• (i) We say that $\left\{ \mathscr {F}^t_{s} \right\} _{s \ge t}$ is right-continuous if, for all $s\ge t$, $\mathscr {F}^t_{s+}:=\bigcap _{r>s} \mathscr {F}^t_{r} = \mathscr {F}^t_{s}$.

• (ii) We say that $\left\{ \mathscr {F}^t_{s} \right\} _{s \ge t}$ is left-continuous if, for all $s> t$, $\mathscr {F}^t_{s-}:=\sigma \left( \bigcup _{r<s} \mathscr {F}^t_{r}\right) = \mathscr {F}^t_{s}$. We say that $\left\{ \mathscr {F}^t_{s} \right\} _{s \ge t}$ is continuous if it is both left and right-continuous.

• (iii) We say that $\left\{ \mathscr {F}^t_{s} \right\} _{s \ge t}$ satisfies the usual conditions if it is right-continuous and complete, i.e. if $\mathscr {F}^t_{s}$ contains all $\mathbb {P}$-null sets of $\mathscr {F}$ for every $s\ge t$.

We will often write $\mathscr {F}^t_{s}$ instead of $\left\{ \mathscr {F}^t_{s} \right\} _{s \ge t}$. We also set $\mathscr {F}^t_{+\infty }:=\sigma \left( \bigcup _{r<+\infty } \mathscr {F}^t_{r}\right)$.

Since we will mostly deal with filtrations satisfying the usual conditions we will assume from now on that this property holds unless explicitly stated otherwise. For this reason we include the usual conditions in the definition of a filtered probability space.

Definition 1.67

(Filtered probability space) Let $\mathscr {F}^t_{s}$ be a filtration satisfying the usual conditions on a complete probability space $(\Omega , \mathscr {F},\mathbb {P})$. The 4-tuple $\left( \Omega , \mathscr {F}, \mathscr {F}^t_{s}, \mathbb {P} \right)$ is called a filtered probability space .

Notation 1.68

We use the following convention in this section. When we write $s\in [t, T]$ we mean that $s\in [t, T]$ if $T\in \mathbb {R}$, and $s\in [t,+\infty )$ if $T=+\infty$. So [t, T] is understood to be $[t,+\infty )$ if $T=+\infty$. $\blacksquare$

Definition 1.69

(Stochastic process) Let $T\in (0,+\infty ]$, $t\in [0,T)$ and $(\Omega ,\mathscr {F})$ and $(\Omega _1,\mathscr {F}_1)$ be two measurable spaces. A family of random variables $X(\cdot )=\{ X(s) \}_{s\in [t, T]}$, $X(s) :\Omega \rightarrow \Omega _1$, is called a stochastic process in [t, T]. If $(\Omega _1,\mathscr {F}_1) = (\mathbb {R}, \mathcal {B}(\mathbb {R}))$ then $X(\cdot )$ is called a real stochastic process.

Definition 1.70

Let $\left( \Omega ,\mathscr {F}, \left\{ \mathscr {F}^t_{s}\right\} _{s \ge t},\mathbb {P} \right)$ be a filtered probability space and $(\Omega _1,\mathscr {F}_1)$ be a measurable space. A stochastic process $\{ X(s) \}_{s\in [t,T]} :[t, T] \times \Omega \rightarrow \Omega _1$ is said to be:

• (i) Measurable, if the map $(s,\omega ){\rightarrow } X(s)(\omega )$ is $\mathcal {B}([t, T]) \otimes \mathscr {F}/\mathscr {F}_1$-measurable.

• (ii) Adapted, if, for each $s\in [t, T]$, $X(s):\Omega \rightarrow \Omega _1$ is an $\mathscr {F}^t_{s}/\mathscr {F}_1$-measurable random variable.

• (iii) Progressively measurable, if for all $s\in (t, T]$, the restriction of $X(\cdot )$ to $[t, s]\times \Omega$ is $\mathcal {B}([t, s])\otimes \mathscr {F}^t_s/\mathscr {F}_1$-measurable.

• (iv) Predictable, if the map $(s,\omega ){\rightarrow } X(s)(\omega )$ is ${\mathcal {P}_{[t, T]}}/\mathscr {F}_1$-measurable, where $\mathcal {P}_{[t, T]}$ is the $\sigma$-field (the predictable $\sigma$-field) in $[t, T]\times \Omega$ generated by all sets of the form $(s,r]\times A, t\le s<r\le T, A\in \mathscr {F}^t_s$ and $\{t\}\times A, A\in \mathscr {F}^t_t$.

• (v) If E is a separable Banach space (endowed with its Borel $\sigma$-field), the process $\{ X(s) \}_{s\in [t,T]} :[t, T] \times \Omega \rightarrow E$ is called stochastically continuous at $s\in [t, T]$ if for every ${\varepsilon }, \delta >0$ there exists $\rho >0$ such that

  $$ \qquad \qquad \mathbb {P} \left( | X(r) - X(s) |\ge {\varepsilon } \right) \le \delta , \qquad \text {for all }r\in (s-\rho , s+\rho ) \cap [t, T]. $$

• (vi) If (S, d) is a metric space (endowed with its Borel $\sigma$-field), the process $\{ X(s) \}_{s\in [t,T]} :[t, T] \times \Omega \rightarrow S$ is called continuous (respectively, right-continuous, left-continuous), if for $\mathbb {P}$-a.e. $\omega \in \Omega$, the function $s {\rightarrow } X(s)(\omega )$ is continuous (respectively, right-continuous, left-continuous).

• (vii) If E is a separable Banach space (endowed with its Borel $\sigma$-field), the process $\{ X(s) \}_{s\in [t,T]} :[t, T] \times \Omega \rightarrow E$ is called integrable (respectively square- integrable) if $\mathbb {E}[|X(s)|] < +\infty$ (respectively $\mathbb {E}[|X(s)|^2] < +\infty$) for all $s\in [t, T]$. The process is called uniformly integrable if it is integrable and the family $\{ X(s) \}_{s\in [t, T]}$ is uniformly integrable (see Definition 1.40).

• (viii) If E is a separable Banach space (endowed with the Borel $\sigma$-field induced by the norm), the process $\{ X(s) \}_{s\in [t,T]} :[t, T] \times \Omega \rightarrow E$ is said to be mean square continuous if $\mathbb {E}[|X(s)|^2] < +\infty$ for all $s\in [t, T]$ and $\lim _{r\rightarrow s} \mathbb {E}[|X(r) - X(s)|^2] =0$ for all $s\in [t, T]$.

It is easy to see that if a process is mean square continuous then it is stochastically continuous.

The concepts of adapted, progressively measurable, and predictable processes can be defined for any filtration $\mathscr {G}^t_s$. To emphasize the filtration used, we will refer to the processes as $\mathscr {G}^t_s$-adapted, $\mathscr {G}^t_s$-progressively measurable, and $\mathscr {G}^t_s$-predictable.

Progressive measurability can also be defined using the concept of progressively measurable sets, see e.g. [447], p. 4, or [219], p. 71. We say that a set $A\subset [t, T]\times \Omega$ is $\mathscr {F}^t_s$-progressively measurable if the function $\mathbf{1}_A$ is a progressively measurable process. Equivalently this means that $A\cap ([t,s]\times \Omega )\in \mathcal {B}([t, s])\otimes \mathscr {F}^t_s$ for every $s\in [t, T]$. It can be proved that the $\mathscr {F}^t_s$-progressively measurable sets form a $\sigma$-field and that a process $X(\cdot )$ is progressively measurable if and only if it is measurable with respect to the $\sigma$-field of $\mathscr {F}^t_s$-progressively measurable sets.

Definition 1.71

(Stochastic equivalence, modification) Let $(\Omega ,\mathscr {F},\mathbb {P})$ be a probability space, and $(\Omega _1,\mathscr {F}_1)$ be a measurable space. Processes $X(\cdot ),Y(\cdot ):[t, T]\times \Omega \rightarrow \Omega _1$ are called stochastically equivalent if for all $s\in [t, T]$, $\mathbb {P}(X(s)=Y(s))=1$. In this case, $Y(\cdot )$ is said to be a modification or version of $X(\cdot )$. The processes $X(\cdot )$ and $Y(\cdot )$ are called indistinguishable if $\mathbb {P}(X(s)=Y(s):\forall s\in [t, T])=1$. We will also say that $Y(\cdot )$ is an indistinguishable version of $X(\cdot )$.

Lemma 1.72

Let $\left( \Omega ,\mathscr {F}, \left\{ \mathscr {F}^t_{s}\right\} _{s \ge t},\mathbb {P} \right)$ be a filtered probability space and let $\left\{ X(s)\right\} _{s \ge t}$ be a process with values in a Polish space (S, d), endowed with the Borel $\sigma$-field induced by the distance.

• (i) If $X(\cdot )$ is $\mathcal {B}([t, T])\otimes \mathscr {F}/\mathcal {B}(S)$-measurable and $\mathscr {F}^t_{s}$-adapted, then $X(\cdot )$ has an $\mathscr {F}^t_{s}$-progressively measurable modification.

• (ii) If $X(\cdot )$ is $\mathscr {F}^t_{s}$-adapted and $X(\cdot )$ is left- (or right-) continuous for every $\omega$, then $X(\cdot )$ itself is $\mathscr {F}^t_{s}$-progressively measurable.

Proof

Part (i): Since S is Borel isomorphic to a Borel subset A of $\mathbb {R}$, without loss of generality we can consider $X(\cdot )$ to be an $\mathbb {R}$-valued process with values in A. By [449], Theorem T46, p. 68, $X(\cdot )$ has an $\mathbb {R}$-valued, $\mathscr {F}^t_{s}$-progressively measurable modification $\tilde{X}(\cdot )$. Let $a\in A$. We define a process $Y(\cdot )$ by $Y(s) := \tilde{X}(s)\mathbf{1}_{\tilde{X}(s)\in A}+a\mathbf{1}_{\tilde{X}(s)\in (\mathbb {R}\setminus A)}$. The process $Y(\cdot )$ is $\mathscr {F}^t_{s}$-progressively measurable. Moreover, if $\tilde{X}(s)=X(s)$, then $Y(s)=X(s)$, so $Y(\cdot )$ is a modification of $X(\cdot )$. Part (ii): See [449], Theorem T47, p. 70, or [372], Proposition 1.13, p. 5. $\square$

Lemma 1.73

Let $\left( \Omega ,\mathscr {F}, \mathbb {P} \right)$ be a complete probability space and let $\left\{ X(s)\right\} _{s \ge t}$ be a stochastic process with values in a separable Banach space E endowed with the Borel $\sigma$-field. If $X(\cdot )$ is stochastically continuous then it has a measurable modification.

Proof

See [180], Proposition 3.2. $\square$

Lemma 1.74

Let $\left( \Omega ,\mathscr {F}, \left\{ \mathscr {F}^t_{s}\right\} _{s \ge t},\mathbb {P} \right)$ be a filtered probability space and let $\left\{ X(s)\right\} _{s \ge t}$ be an adapted process with values in a separable Banach space E endowed with the Borel $\sigma$-field. If $X(\cdot )$ is stochastically continuous then it has an $\mathscr {F}^t_{s}$-progressively measurable modification.

Proof

See [180], Proposition 3.6. It is also a corollary of Lemmas 1.72-(i) and 1.73.    $\square$

1.2.2 Martingales

Notation 1.75

Unless specified otherwise, any Banach space E and any metric space (S, d) will be understood to be endowed with the Borel $\sigma$-field induced respectively by the norm and by the distance.$\blacksquare$

Definition 1.76

(Martingale) Let$\left( \Omega ,\mathscr {F},\mathscr {F}^t_{s},\mathbb {P} \right)$ be a filtered probability space, and let $M(\cdot )$ be an $\mathscr {F}^t_s$-adapted and integrable process with values in a separable Banach space E. Then $M(\cdot )$ is said to be a martingale if, for all $r,s\in [t, T], s\le r$,

$$ \mathbb {E} \left[ M(r) | \mathscr {F}^t_s \right] = M(s) \qquad \mathbb {P}-a.s. $$

If $E=\mathbb {R}$, we say that M(s) is a submartingale (respectively, supermartingale) if

$$ \mathbb {E} \left[ M(r) | \mathscr {F}_s^t \right] \ge M(s), \quad (\text {respectively}, \,\, \mathbb {E} \left[ M(r) | \mathscr {F}_s^t \right] \le M(s))\,\,\,\mathbb {P}-a.s. $$

Theorem 1.77

(Doob’s maximal inequalities) Let $T>0$, $\left( \Omega ,\mathscr {F},\mathscr {F}^t_{s},\mathbb {P} \right)$ be a filtered probability space, and H be a separable Hilbert space. Let $M(\cdot )$ be a right-continuous H-valued martingale such that $M(s) \in L^p \left( \Omega ,\mathscr {F},\mathbb {P};H \right)$ for all $s\in [t, T]$. Then:

• (i) If $p\ge 1$, $\mathbb {P} \left( \sup _{s\in [t, T]} |M(s)| >\lambda \right) \le \frac{1}{\lambda ^p} \mathbb {E} \left[ |M(T)| ^p \right]$, for all $\lambda >0$.

• (ii) If $p > 1$, $\mathbb {E} \left[ \sup _{s\in [t, T]} |M(s)|^p \right] \le \left( \frac{p}{p-1}\right) ^p \mathbb {E} \left[ |M(T)| ^p \right]$.

Proof

We observe that, if $M(\cdot )$ is a right-continuous H-valued martingale such that $M(s) \in L^p \left( \Omega ,\mathscr {F},\mathbb {P};H \right)$, $p\ge 1$, for all $s\in [t, T]$, then by Proposition 1.41-(vi), $|M(\cdot )|^p$ is a right-continuous $\mathbb {R}$-valued submartingale with $|M(s)| \in L^p \left( \Omega ,\mathscr {F},\mathbb {P};\mathbb {R} \right)$ for all $s\in [t, T]$. The claims now easily follow from [372] Theorem 3.8 (i) and (iii), pp. 13–14. $\square$

In particular, we see that a right-continuous E-valued martingale $M(\cdot )$ is square-integrable if and only if $\mathbb {E} |M(T)|^2 <+\infty$.

Notation 1.78

(Square-integrable martingales) Let $T\in (0,+\infty )$, $t\in [0,T)$, let $\left( \Omega ,\mathscr {F},\mathscr {F}^t_{s},\mathbb {P} \right)$ be a filtered probability space, and E be a separable Banach space. The class of all continuous square-integrable martingales $M:[t, T] \times \Omega \rightarrow E$ is denoted by $\mathcal {M}^2_{t, T}(E)$. $\blacksquare$

If H is a separable Hilbert space then $\mathcal {M}^2_{t, T}(H)$ endowed with the scalar product

$$ \left\langle M, N \right\rangle _{\mathcal {M}^2_{t, T}} := \mathbb {E} \left[ \left\langle M(T), N(T) \right\rangle \right] . $$

is a Hilbert space (see [294], p. 22).

Theorem 1.79

(Angle bracket process, Quadratic variation process) Let $T>0$ , $t\in [0,T)$, H be a separable Hilbert space, and $\left( \Omega ,\mathscr {F},\mathscr {F}^t_{s},\mathbb {P} \right)$ be a filtered probability space. For every $M \in \mathcal {M}^2_{t, T}(H)$ there exists a unique (real) increasing, adapted, continuous process starting from 0 at t, called the angle bracket process, and denoted by $\left\langle M \right\rangle _t$, such that $|M_s|^2 - \left\langle M\right\rangle _s$ is a continuous martingale. Moreover, there exists a unique $\mathcal {L}_1^+(H)$-valued continuous adapted process starting from 0 at t, called the quadratic variation of M, and denoted by $\left\langle \left\langle M \right\rangle \right\rangle _s$, such that, for all $x, y \in H$, the process

$$ \left\langle M_s, x \right\rangle \left\langle M_s, y \right\rangle - \Big \langle \left\langle \left\langle M\right\rangle \right\rangle _s(x) , y \Big \rangle , \qquad s\in [t, T] $$

is a continuous martingale. Moreover, $\left\langle M\right\rangle _s=\mathrm{Tr}( \left\langle \left\langle M\right\rangle \right\rangle _s)$.

Proof

See [294], Definition 2.9 and Lemma 2.1, p. 22. $\square$

Theorem 1.80

(Burkholder–Davis–Gundy inequality) Let $T>0, t\in [0,T)$, H be a separable Hilbert space, and $\left( \Omega ,\mathscr {F},\mathscr {F}^t_{s},\mathbb {P} \right)$ be a filtered probability space. For every $p>0$ there exists a $c_p>0$ such that, for every $M \in \mathcal {M}^2_{t, T}(H)$ with $M(0)=0$,

$$ c_p^{-1} \mathbb {E} \left[ \left\langle M \right\rangle _T^{p/2}\right] \le \mathbb {E} \left[ \sup _{s\in [t, T]} |M(s)|^p\right] \le c_p \mathbb {E} \left[ \left\langle M \right\rangle _T^{p/2}\right] . $$

Proof

See [487], Theorem 3.49, p. 37. $\square$

1.2.3 Stopping Times

Definition 1.81

(Stopping time) Consider a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and a filtration $\left\{ \mathscr {F}^t_s\right\} _{s \ge t}$ on $\Omega$. A random variable $\tau :(\Omega , \mathscr {F}) \rightarrow [t,+\infty ]$ is said to be an $\mathscr {F}^t_s$-stopping time if, for all $s\ge t$,

$$ \left\{ \tau \le s \right\} := \left\{ \omega \in \Omega \; : \; \tau (\omega ) \le s \right\} \in \mathscr {F}^t_s. $$

Given a stopping time $\tau$ we denote by $\mathscr {F}_\tau$ the sub-$\sigma$-field of $\mathscr {F}$ defined by

$$ \mathscr {F}_\tau := \Big \{ A\in \mathscr {F} \; : \; A \cap \left\{ \tau \le s \right\} \in \mathscr {F}^t_s \;\; \text {for all }s\ge t \Big \}. $$

Proposition 1.82

Let $(\Omega , \mathscr {F}, \mathscr {F}^t_s,\mathbb {P})$ be a filtered probability space.

• (i) If $\tau$ and $\sigma$ are $\mathscr {F}^t_s$-stopping times, so are $\tau \wedge \sigma$, $\tau \vee \sigma$ and $\tau + \sigma$.

• (ii) If $\sigma _n$ (for $n=1,2...$) are $\mathscr {F}^t_s$-stopping times, then

  $$ \sup _n\sigma _n,\,\, \inf _n\sigma _n, \,\,\limsup _n\sigma _n, \,\,\liminf _n\sigma _n $$

  are $\mathscr {F}^t_s$-stopping times.

• (iii) For any $\mathscr {F}^t_s$-stopping time $\tau$ there exists a decreasing sequence of discrete-valued $\mathscr {F}^t_s$-stopping times $\tau _n$, such that $\lim _{n\rightarrow \infty } \tau _n = \tau$.

• (iv) Let (S, d) be a metric space (endowed with the Borel $\sigma$-field induced by the distance), and $X:[t,+\infty ) \times \Omega \rightarrow S$ be a continuous and $\mathscr {F}^t_s$-adapted process. Let $A\subset S$ be an open or a closed set. Then the hitting time

  $$ \tau _A := \inf \{ s \ge t \; : \; X(s) \in A \} $$

  is a stopping time. (It is understood that $\inf \{\emptyset \}=+\infty$.)

Proof

(i) and (ii) see [372], Lemmas 2.9 and 2.11, p. 7. (iii) see [370], Lemma 7.4, p. 122. (iv) see [575], Example 3.3, p. 24, or [452], Proposition 1.3.2, p. 12 (there $S=\mathbb {R}^n$, but the proofs are the same). $\square$

Proposition 1.83

Let $\left( \Omega , \mathscr {F}, \left\{ \mathscr {F}^t_s\right\} _{s \ge t}, \mathbb {P} \right)$ be a filtered probability space, $(\Omega _1, \mathscr {F}_1)$ be a measurable space, $X:[t,+\infty ) \times \Omega \rightarrow \Omega _1$ be an $\mathscr {F}^t_s$-progressively measurable process, and $\tau$ be an $\mathscr {F}^t_s$-stopping time. Then the random variable $X(\tau )$, (where $X(\tau )(\omega ):=X(\tau (\omega ),\omega )$), is $\mathscr {F}_\tau$-measurable and the process defined, for any $s\in [t, +\infty )$, by $X(s\wedge \tau )$ is $\mathscr {F}^t_s$-progressively measurable.

Proof

See [452], Proposition 1.3.5, p. 13, or [575], Proposition 3.5, p. 25. $\square$

Theorem 1.84

(Doob’s optional sampling theorem) Let$\left( \Omega , \mathscr {F}, \left\{ \mathscr {F}^t_s \right\} _{s \ge t}, \mathbb {P} \right)$ be a filtered probability space, $X:[t,+\infty ) \times \Omega \rightarrow \mathbb {R}$ be a right-continuous $\mathscr {F}^t_s$-submartingale, and $\tau ,\sigma$ be two $\mathscr {F}^t_s$-stopping times with $\tau$ bounded. Then $X_\tau$ is integrable and

$$ \mathbb {E}[X_\tau |\mathscr {F}^t_\sigma ]\ge X_{\tau \wedge \sigma },\quad \mathbb {P}\,\, a.s. $$

If $X^+$ (the positive part of the process) is uniformly integrable then the statement extends to unbounded $\tau$.

Proof

See [370], Theorem 7.29, p. 135. $\square$

Definition 1.85

(Local martingale) Let $\left( \Omega ,\mathscr {F}, \left\{ \mathscr {F}^t_{s} \right\} _{s \ge t},\mathbb {P} \right)$ be a filtered probability space. An $\left\{ \mathscr {F}^t_{s} \right\} _{s \ge t}$-adapted process $\left\{ X(s)\right\} _{s \ge t}$ with values in a separable Banach space E is said to be a local martingale if there exists an increasing sequence of stopping times $\left( \tau _n \right) _{n\in \mathbb {N}}$ with $\mathbb {P}({\tau _n} \uparrow +\infty ) =1$, such that the process $\{ X(s \wedge \tau _n) \}_{s\ge t}$ is a martingale for every $n\in \mathbb {N}$.

1.2.4 Q-Wiener Processes

Definition 1.86

(Real Brownian motion) Given $t\in \mathbb {R}$, a real stochastic process $\beta :[t,+\infty )\times \Omega \rightarrow \mathbb {R}$ on a complete probability space $\left( \Omega , \mathscr {F}, \mathbb {P} \right)$ is a standard (one-dimensional) real Brownian motion on $[t,+\infty )$ starting at 0, if

• (1) $\beta$ is continuous and $\beta (t)=0$;

• (2) for all $t\le t_1<t_2<...<t_n$ the random variables $\beta (t_1)$, $\beta (t_2)-\beta (t_1)$, ..., $\beta (t_n)-\beta (t_{n-1})$ are independent;

• (3) for all $t\le t_1 \le t_2$, $\beta (t_2) - \beta (t_1)$ has a Gaussian distribution with mean 0 and covariance $t_2-t_1$.

Consider a real, separable Hilbert space $\Xi$ and $Q \in \mathcal {L}^+(\Xi )$. Define $\Xi _0:=Q^{1/2} (\Xi )$ and let $Q^{-1/2}$ be the pseudo-inverse of $Q^{1/2}$ (see Definition B.1). $\Xi _0$ is a separable Hilbert space when endowed with the inner product $\left\langle x, y\right\rangle _{\Xi _0} := \left\langle {Q}^{-1/2} x, {Q}^{-1/2} y \right\rangle _\Xi$ . Let $\Xi _1$ be an arbitrary real, separable Hilbert space such that $\Xi \subset \Xi _1$ with continuous embedding and $\Xi _0 \subset \Xi _1$ with Hilbert–Schmidt embedding $J:\Xi _0 \hookrightarrow \Xi _1$ (see Appendix B.3 on Hilbert–Schmidt operators). The operator $Q_1:=J J^*$ belongs to $\mathcal {L}_1^+(\Xi _1)$ and $\Xi _0$ is identical with the space $Q_1^{\frac{1}{2}}(\Xi _1)$ (see [180] Proposition 4.7, p. 85).

Theorem 1.87

Consider the setting described above. Let $\left\{ g_{k} \right\} _{k \in \mathbb {N}}$ be an orthonormal basis of $\Xi _0$ and $(\beta _{k})_{k \in \mathbb {N}}$ be a sequence of mutually independent, standard one-dimensional Brownian motions $\beta _k:[t,+\infty ) \times \Omega \rightarrow \mathbb {R}$ on $[t,+\infty )$ starting at 0. Then for every $s\in [t, +\infty )$ the series

$$\begin{aligned} W_Q(s):=\sum _{k=1}^{ \infty } g_{k} \beta _{k}(s) \end{aligned}$$

(1.12)

is convergent in $L^2(\Omega , \mathscr {F}, \mathbb {P}; \Xi _1)$.

Proof

See [180] Propositions 4.3, p. 82, and 4.7, p. 85. $\square$

Definition 1.88

(Q-Wiener process) The process $W_Q$ defined by (1.12) is called a Q-Wiener process on $[t,+\infty )$ starting at 0.

Remark 1.89

We will use the notation $W_Q$ to denote a Q-Wiener process. If Q is trace-class, $\Xi _1=\Xi$ is a canonical choice and it will be understood that $W_Q$ is a $\Xi$-valued process. If Q is not trace-class, writing $W_Q$ and calling it a Q-Wiener process is a slight abuse of notation as it would be more precise to write $W_{Q_1}$ and call it a $Q_1$-Wiener process with values in $\Xi _1$. However, even though the construction we have described is not canonical if $\mathrm{Tr}(Q)=+\infty$, and the choice of $\Xi _1$ is not unique, the class of the integrable processes is independent of the choice of $\Xi _1$ (see [180] Sect. 4.1 and in particular Proposition 4.7). Moreover (see [180] Sect. 4.1.2), for arbitrary $a \in \Xi$ the stochastic process

$$\begin{aligned} <a, W(s)>:=\sum _{k=1}^{ \infty }\langle a, g_{k}\rangle \beta _{k}(s), \quad s \ge t, \end{aligned}$$

is a real-valued Wiener process and

$$\begin{aligned} \mathbb {E}<a, W(s_1)> <b, W(s_2)>=((s_1-t)\wedge (s_2-t)) \langle Qa,b\rangle , \; a, b \in \Xi . \end{aligned}$$

For these reasons, even when $\mathrm{Tr}(Q)=+\infty$, we will still use the notation $W_Q$. When Q is the identity on $\Xi$ we will call it a cylindrical Wiener process in $\Xi$. $\blacksquare$

Proposition 1.90

Let $\Xi$ be a real, separable Hilbert space, $Q \in \mathcal {L}^+(\Xi )$ and let $\Xi _0$, $\Xi _1$ and J be as described above. Let $\left( \Omega , \mathscr {F}, \mathbb {P} \right)$ be a complete probability space and $B:[t,+\infty ) \times \Omega \rightarrow \Xi _1$ be a stochastic process. Denote by $\mathscr {F}_{s}^{t, 0}$ the filtration generated by B, i.e.

$$ \mathscr {F}_{s}^{t, 0}=\sigma (B(r) \; : \; t\le r\le s), $$

and $\mathscr {F}_{s}^{t}:=\sigma (\mathscr {F}_{s}^{t, 0},\mathcal {N})$ , where $\mathcal {N}$ is the class of the $\mathbb {P}$-null sets. Then B is a Q-Wiener process on $[t,+\infty )$ starting at 0 if and only if:

• (1) $B(t)=0$.

• (2) B has continuous trajectories.

• (3) For all $t\le t_1 \le t_2$ the random variable $B(t_2) - B(t_1)$ is independent of $\mathscr {F}^t_{t_1}$.

• (4) $\mathcal {L}_{\mathbb {P}} \left( B(t_2) - B(t_1) \right) = \mathcal {N}(0, \, (t_2 - t_1) Q_1)$, where $Q_1= J J^*$.

Proof

The “only if” part follows from [180], Proposition 4.7, p. 85 (observe that in [180] a Wiener process is in fact defined using the four properties (1)–(4)). The “if” part is proved in [180] Proposition 4.3-(ii), p. 81 (if $\mathrm{Tr}(Q) = +\infty$ we apply the proposition in the space $\Xi _1$). $\square$

The existence of a process satisfying conditions (1)–(4) above can also be proved using the Kolmogorov extension theorem (see [180], Proposition 4.4).

Remark 1.91

If $W_Q(s)=\sum _{k=1}^\infty g_k\beta _k(s)$ for some orthonormal basis $\{g_k\}_{k\in \mathbb {N}}$ of $\Xi _0$, it is easy to see that regardless of the choice of $\Xi _1$, $\mathscr {F}_{s}^{t, 0}=\sigma ( \beta _k(r):t\le r\le s, k\in \mathbb {N})$. Thus the filtration generated by $W_Q$ does not depend on the choice of $\Xi _1$. $\blacksquare$

Definition 1.92

(Translated $\mathscr {G}^t_s$-Q-Wiener process) Let $0\le t<T\le +\infty$. Let $\Xi$ be a real, separable Hilbert space, $Q \in \mathcal {L}^+(\Xi )$ and let $\Xi _0$, $\Xi _1$ and J be as described above. Let $\left( \Omega , \mathscr {F}, \mathscr {G}^t_s, \mathbb {P} \right)$ be a filtered probability space. We say that a stochastic process $B:[t, T] \times \Omega \rightarrow \Xi _1$ is a translated $\mathscr {G}^t_s$-Q-Wiener process on [t, T] if:

• (1) B has continuous trajectories.

• (2) B is adapted to $\mathscr {G}^t_s$.

• (3) For all $t\le t_1<t_2\le T$, $B(t_2) - B(t_1)$ is independent of $\mathscr {G}^t_{t_1}$.

• (4) $\mathcal {L}_{\mathbb {P}} \left( B(t_2) - B(t_1) \right) = \mathcal {N}(0, \, (t_2 - t_1) Q_1)$, where $Q_1= J J^*$.

If we also have $B(t)=0$ then we call B a $\mathscr {G}^t_s$-Q -Wiener process on [t, T].

We remark that if B is a translated $\mathscr {G}^t_s$-Q-Wiener process, then it is also a translated $\mathscr {F}^{t}_s$-Q-Wiener process, where $\mathscr {F}^{t}_s$ is the augmented filtration generated by B. Moreover, if $W_Q$ is a Q-Wiener process as in Definition 1.88 then it is also a $\mathscr {F}^{t}_s$-Q-Wiener process, where $\mathscr {F}^{t}_s$ is the augmented filtration generated by B.

Lemma 1.93

Let $0\le t<T\le +\infty$. Let $\Xi$ be a real, separable Hilbert space, $Q \in \mathcal {L}^+(\Xi )$ and let $\Xi _0$ and $\Xi _1$ be as described above. Let $\left( \Omega , \mathscr {F}, \mathbb {P} \right)$ be a complete probability space. Let $B:[t , T]\times \Omega \rightarrow \Xi _1$ be a continuous stochastic process such that $B(t)=0$. Then B is a Q-Wiener process on [t, T] if and only if, for all $a\in \Xi _1$, $t\le t_1\le t_2\le T$, we have

$$\begin{aligned} \mathbb {E} \left[ e^{i \left\langle a, B(t_2) - B(t_1) \right\rangle _{\Xi _1}} | \mathscr {F}_{t_1}^t \right] = e^{-\frac{\left\langle Q_1 a, a \right\rangle _{\Xi _1}}{2}(t_2-t_1)}. \end{aligned}$$

(1.13)

Proof

(The proof uses the same arguments as in the finite-dimensional case, see Proposition 1.2.7 of [452].)

The “only if” part: if B is a Q-Wiener process then, by Proposition 1.90-(4), Theorem 1.57 and Definition 1.58,

$$ \mathbb {E} \left[ e^{i \left\langle a, B(t_2) - B(t_1) \right\rangle _{\Xi _1}} \right] = e^{-\frac{\left\langle Q_1 a, a \right\rangle _{\Xi _1}}{2}(t_2-t_1)}. $$

Moreover, since $B(t_2) - B(t_1)$ is independent of $\mathscr {F}_{t_1}^t$,

$$ \mathbb {E} \left[ e^{i \left\langle a, B(t_2) - B(t_1) \right\rangle _{\Xi _1}} \right] = \mathbb {E} \left[ e^{i \left\langle a, B(t_2) - B(t_1) \right\rangle _{\Xi _1}} | \mathscr {F}_{t_1}^t \right] . $$

The “if” part: We have to prove the four conditions in Proposition 1.90: (1) and (2) are already in the assumptions of the lemma. Condition (4) follows easily from (1.13), Theorem 1.57 and Definition 1.58. To prove condition (3), i.e. that $Y:=B(t_2) - B(t_1)$ is independent of $\mathscr {F}_{t_1}^t$, observe that, for all $Z :\Omega \rightarrow \Xi _1$ which are $\mathscr {F}_{t_1}^t$-measurable, one has, for all $a, b\in \Xi _1$,

$$\begin{aligned}&\mathbb {E} \left[ e^{i \left\langle a, Y \right\rangle _{\Xi _1}} e^{i \left\langle b, Z \right\rangle _{\Xi _1}} \right] = \mathbb {E} \left[ \mathbb {E} \left[ e^{i \left\langle a, Y \right\rangle _{\Xi _1}} | \mathscr {F}_{t_1}^t \right] e^{i \left\langle b, Z \right\rangle _{\Xi _1}} \right] \\&\qquad \qquad \qquad = e^{-\frac{\left\langle Q_1 a, a \right\rangle _{\Xi _1}}{2}(t_2-t_1)} \; \mathbb {E} \left[ e^{i \left\langle b, Z \right\rangle _{\Xi _1}} \right] = \mathbb {E} \left[ e^{i \left\langle a, Y \right\rangle _{\Xi _1}} \right] \mathbb {E} \left[ e^{i \left\langle b, Z \right\rangle _{\Xi _1}} \right] . \end{aligned}$$

Since the above holds for all $Z :\Omega \rightarrow \Xi _1$ which are $\mathscr {F}_{t_1}^t$-measurable, and for all $a, b\in \Xi _1$, we conclude that Y is independent of $\mathscr {F}_{t_1}^t$ by Theorem 1.56. $\square$

Lemma 1.94

Let $\mathscr {F}_{s}^{t, 0}$ and $\mathscr {F}_{s}^{t}$ be the filtrations defined in Proposition 1.90 for a Q-Wiener process $W_Q$. Then $\mathscr {F}_{s}^{t}$ is right-continuous. Moreover, for all $T > t$, $\mathscr {F}_{T}^{t, 0}$, and consequently $\mathscr {F}_{T}^{t}$, are countably generated up to sets of measure zero. If the trajectories of $W_Q$ are everywhere continuous then

$$\begin{aligned} \mathscr {F}_{T}^{t, 0}=\mathscr {F}_{T-}^{t, 0}=\sigma \left( W_Q(s_i):i=1,2,...\right) , \end{aligned}$$

(1.14)

where $(s_i), i=1,2,...$ is any dense sequence in [t, T), and hence the filtration $\mathscr {F}_{s}^{t, 0}$ is countably generated and left-continuous.

Proof

The proof follows arguments from [513] and [372] (Sect. 2.7-A). Consider $\tau >s$ and ${\varepsilon }>0$. Since $W_Q(\tau +{\varepsilon })-W_Q(s+{\varepsilon })$ is independent of $\mathscr {F}_{s+}^{t, 0}$, for every $A\in \mathscr {F}_{s+}^{t, 0}$ and $f\in C_b(\Xi _1)$

$$ \mathbb {E}\left( \mathbf{1}_A f(W_Q(\tau +{\varepsilon })-W_Q(s+{\varepsilon }))\right) = \mathbb {P}(A)\mathbb {E}f(W_Q(\tau +{\varepsilon })-W_Q(s+{\varepsilon })). $$

Letting ${\varepsilon }\rightarrow 0$ we thus have by the dominated convergence theorem that

$$\begin{aligned} \mathbb {E}\left( \mathbf{1}_A f(W_Q(\tau )-W_Q(s))\right) = \mathbb {P}(A)\mathbb {E}f(W_Q(\tau )-W_Q(s)). \end{aligned}$$

(1.15)

Now if $B=\overline{B}\subset \Xi _1$ then there exist functions $f_n\in C_b(\Xi _1), 0\le f_n\le 1$, such that $f_n(x)\rightarrow \mathbf{1}_B(x)$ as $n\rightarrow +\infty$ for every $x\in \Xi _1$. Therefore (1.15) implies that

$$ \mathbb {P}(A\cap \{W_Q(\tau )-W_Q(s)\in B\}) =\mathbb {P}(A)\mathbb {P}(\{W_Q(\tau )-W_Q(s)\in B\}) $$

and since the sets $\{\{W_Q(\tau )-W_Q(s)\in B\}:B=\overline{B}\subset \Xi _1\}$ are a $\pi$-system generating $\sigma (W_Q(\tau )-W_Q(s))$, it follows from Lemma 1.23 that $\mathscr {F}_{s+}^{t, 0}$ and $\sigma (W_Q(\tau )-W_Q(s))$ are independent.

Now let $s=\tau _0<\tau _1<...<\tau _k\le T$. We have $\sigma (W_Q(\tau _i)-W_Q(s):i=1,..., k)=\sigma (W_Q(\tau _i)-W_Q(\tau _{i-1}):i=1,..., k)$. Let now $A\in \mathscr {F}_{s+}^{t, 0}$ and $B_i\in \sigma (W_Q(\tau _i)-W_Q(\tau _{i-1})), i=1,..., k$. Since $B_i$ is independent of $A\cap B_1\cap ...\cap B_{i-1}\in \mathscr {F}_{\tau _{i-1}}^{t, 0}, i=1,..., k$ and $B_1,..., B_k$ are independent

$$\begin{aligned}&\mathbb {P}(A\cap B_1\cap ...\cap B_k)=\mathbb {P}(A\cap B_1\cap ...\cap B_{k-1})\mathbb {P}(B_{k})=... \\&\quad \quad =\mathbb {P}(A\cap B_1)\prod _{i=2}^k\mathbb {P}(B_i)=\mathbb {P}(A)\prod _{i=1}^k\mathbb {P}(B_i) =\mathbb {P}(A)\mathbb {P}(B_1\cap ...\cap B_k). \end{aligned}$$

Therefore $\bigcup \sigma (W_Q(\tau _i)-W_Q(s):i=1,..., k)$ (where the union is taken over all partitions $s=\tau _0<\tau _1<...<\tau _k\le T$) is a $\pi$-system independent of $\mathscr {F}_{s+}^{t, 0}$ and thus $\mathscr {G}_s=\sigma (W_Q(\tau )-W_Q(s):s\le \tau \le T)$ is independent of $\mathscr {F}_{s+}^{t, 0}$.

Since $\mathscr {F}_{T}^{t, 0}=\sigma (\mathscr {F}_{s}^{t, 0},\mathscr {G}_s)$, the family $\{A_s\cap B_s:A_s\in \mathscr {F}_{s}^{t, 0}, B_s\in \mathscr {G}_s\}$ is a $\pi$-system generating $\mathscr {F}_{T}^{t, 0}$. Let now $A\in \mathscr {F}_{s+}^{t, 0}$ and let $\xi$ be a version of $\mathbf{1}_A-\mathbb {E}(\mathbf{1}_A|\mathscr {F}_{s}^{t, 0})$. Since $\xi$ is $\mathscr {F}_{s+}^{t, 0}$-measurable, it is independent of $\mathscr {G}_s$, so if $A_s\in \mathscr {F}_{s}^{t, 0}, B_s\in \mathscr {G}_s$ then

$$\begin{aligned}&\mathbb {E}\left( \xi \mathbf{1}_{A_s\cap B_s} \right) =\mathbb {E}\left( \xi \mathbf{1}_{A_s}{} \mathbf{1}_{B_s} \right) =\mathbb {P}(B_s)\mathbb {E}\left( \xi \mathbf{1}_{A_s} \right) \\&=\mathbb {P}(B_s)\int _{A_s}\xi d\mathbb {P}=\mathbb {P}(B_s)\left[ \int _{A_s}{} \mathbf{1}_A d\mathbb {P}-\int _{A_s}\mathbb {E}(\mathbf{1}_A|\mathscr {F}_{s}^{t, 0})d\mathbb {P}\right] =0 \end{aligned}$$

by the definition of conditional expectation. This implies that $\int _D\xi d\mathbb {P}=0$ for every $D\in \mathscr {F}_{T}^t$ and thus $\xi =0, \mathbb {P}$-a.e. Therefore $\mathbf{1}_A=\mathbb {E}(\mathbf{1}_A|\mathscr {F}_{s}^{t, 0}), \mathbb {P}$-a.e., i.e. if $\tilde{A}=\mathbb {E}(\mathbf{1}_A|\mathscr {F}_{s}^{t, 0})^{-1}(1)$ then $\tilde{A}\in \mathscr {F}_{s}^{t, 0}$ and $\mathbb {P}(A\Delta \tilde{A})=0$. This shows that $\mathscr {F}_{s+}^{t, 0}\subset \mathscr {F}_{s}^{t}$.

Now let $A\in \mathscr {F}_{s+}^{t}$, which means that for every $n\ge 1,$ $A\in \mathscr {F}_{s+1/n}^{t}$ and there exists a $B_n\in \mathscr {F}_{s+1/n}^{t, 0}$ such that $A\Delta B_n\in \mathcal {N}$. Set

$$ B=\bigcap _{m=1}^{+\infty }\bigcup _{n=m}^{+\infty }B_n. $$

Then $B\in \mathscr {F}_{s+}^{t, 0}\subset \mathscr {F}_{s}^{t}$ and

$$ B\setminus A\subset \left( \bigcup _{n=1}^{+\infty }B_n\right) \setminus A =\bigcup _{n=1}^{+\infty }(B_n\setminus A)\in \mathcal {N}. $$

Moreover,

$$\begin{aligned}&A\setminus B=A\cap \left( \bigcap _{m=1}^{+\infty }\bigcup _{n=m}^{+\infty }B_n\right) ^c =A\cap \left( \bigcup _{m=1}^{+\infty }\bigcap _{n=m}^{+\infty }B_n^c\right) \\&\quad = \bigcup _{m=1}^{+\infty }\bigcap _{n=m}^{+\infty }(A\cap B_n^c)\subset \bigcup _{m=1}^{+\infty }(A\cap B_m^c)= \bigcup _{m=1}^{+\infty }(A\setminus B_m)\in \mathcal {N}. \end{aligned}$$

Thus $A\Delta B\in \mathcal {N}$, which implies that $A\in \mathscr {F}_{s}^{t}$, which completes the proof of the right continuity.

To show that $\mathscr {F}_{T}^{t, 0}$ is countably generated up to sets of measure zero we take a dense sequence $(s_i), i=1,2,...$, in [t, T). Since $\mathcal {B}(\Xi _1)$ is countably generated (for instance by open balls with rational radii centered at points of a countable dense set), each $\sigma (W_Q(s_i))$ is countably generated and so $\sigma (W_Q(s_i):i\ge 1)$ is countably generated. It remains to show that for every $s\in (t,T], \sigma (W_Q(s))\subset \sigma (\mathcal {N}, W_Q(s_i):s_i<s)$. Let $\Omega _0\subset \Omega ,\mathbb {P}(\Omega _0)=1$ be such that $W_Q$ has continuous trajectories on [t, T] for $\omega \in \Omega _0$. Let A be an open subset of $\Xi _1$ and set $A_n=\{x\in A: \mathrm{dist}(x, A^c)>1/n\}, n=1,2,...$. Then $A_n$ is open, $\overline{A}_n\subset A_{n+1}$, and $\bigcup _{n=1}^{+\infty }A_n=A$. Let $s_{i_k}$ be a sequence of $s_{i}$ such that $s_{i_k}<s$ and $s_{i_k}\rightarrow s$ as $k\rightarrow +\infty$. Then, using the continuity of the trajectories of $W_Q$, it is easy to see that

$$ \Omega _0\cap W_Q(s)^{-1}(A)=\Omega _0\cap \bigcup _{n=1}^{+\infty }\bigcap _{k=n}^{+\infty }W_Q(s_{i_k})^{-1}(A_n) \in \sigma (\mathcal {N}, W_Q(s_i):s_i<s). $$

Therefore $W_Q(s)^{-1}(A) \in \sigma (\mathcal {N}, W_Q(s_i):s_i<s)$ and since the sets $\{W_Q(s)^{-1}(A):A\,\,\,\text{ is } \text{ an } \text{ open } \text{ subset } \text{ of }\,\,\, \Xi _1\}$ generate $\sigma (W_Q(s))$, the result follows. If $\Omega _0=\Omega$ then we have above

$$ W_Q(s)^{-1}(A)=\bigcup _{n=1}^{+\infty }\bigcap _{k=n}^{+\infty }W_Q(s_{i_k})^{-1}(A_n) \in \sigma (W_Q(s_i):s_i<s). $$

The argument that $\sigma (W_Q(t))\subset \sigma (W_Q(s_i):i=1,2,...)$ is similar (or we can just assume that $s_1=t$). This yields (1.14). $\square$

In fact the above argument shows that if S is a Polish space, $T>t$, and $X:[t, T]\times \Omega \rightarrow S$ is a stochastic process with everywhere continuous trajectories, then the filtration generated by X, $\mathscr {F}_{s}^{X}:=\sigma (X(\tau ):t\le \tau \le s)$ is countably generated and left-continuous.

1.2.5 Simple and Elementary Processes

Definition 1.95

($\mathscr {F}^t_s$-simple process) Let E be a Banach space (endowed with the Borel $\sigma$-field) and let $(\Omega , \mathscr {F}, \left\{ \mathscr {F}^t_s \right\} _{s\in [t, T]}, \mathbb {P})$ be a filtered probability space. A process $X:[t, T] \times (\Omega , \mathscr {F}, \mathbb {P}) \rightarrow E$ is called $\mathscr {F}^t_s$-simple if:

• (i) Case $T=+\infty$: there exists a sequence of real numbers $(t_n )_{n \in {\mathbb N}}$ with $t=t_0<t_1< ...<t_n <...$ and $\lim _{n\rightarrow \infty } t_n = +\infty$, a constant $C<+\infty$, and a sequence of random variables $\xi _n:\Omega \rightarrow E$ with $\sup _{n \ge 0} |\xi _n(\omega )|_E \le C$ for every $\omega \in \Omega$, such that $\xi _n$ is $\mathscr {F}^t_{t_n}$-measurable for every $n\ge 0$, and

  $$ X(s)(\omega ) = \left\{ \begin{array}{l} \xi _0(\omega ) \qquad \text {if }s=t\\ \xi _i(\omega ) \qquad \text {if }s\in (t_i, t_{i+1}]. \end{array} \right. $$

• (ii) Case $T<+\infty$: there exist $t= t_0<t_1< ... <t_N=T$, a constant $C<+\infty$, and random variables $\xi _n:\Omega \rightarrow E$ for $n=0,..., N-1$ with $\sup _{0\le n \le N-1} |\xi _n(\omega )|_E \le C$ for every $\omega \in \Omega$, such that $\xi _n$ is $\mathscr {F}^t_{t_n}$-measurable, and

  $$ X(s)(\omega ) = \left\{ \begin{array}{l} \xi _0(\omega ) \qquad \text {if }s=t\\ \xi _i(\omega ) \qquad \text {if }s\in (t_i, t_{i+1}]. \end{array} \right. $$

Definition 1.96

($\mathscr {F}^t_s$-elementary process) Let $T\in (0,+\infty )$ , $t\in [0,T)$. Let (S, d) be a complete metric space (endowed with the Borel $\sigma$-field), and $(\Omega , \mathscr {F}, \left\{ \mathscr {F}^t_s \right\} _{s\in [t, T]}, \mathbb {P})$ be a filtered probability space. We say that a process $X:[t, T] \times (\Omega , \mathscr {F}, \mathbb {P}) \rightarrow S$ is $\mathscr {F}^t_s$-elementary if there exist S-valued random variables $\xi _0, \xi _1, .. , \xi _{N-1}$, and a sequence $t=t_0< t_1< .. < t_N=T$, such that

• (1) $\xi _i$ has a finite numbers of values for every $i\in \{ 0, .. N-1 \}$.

• (2) $\xi _i$ is $\mathscr {F}^t_{t_i}$-measurable for every $i\in \{ 0, .. N-1 \}$.

• (3) $X(s)(\omega ) = \xi _i(\omega )$ for $s\in (t_i, t_{i+1}]$ for $i\in \{ 0, .. N-1 \}$, and $X(t)=\xi _0$.

Finally, we say that a process $X:[t,+\infty ) \times (\Omega , \mathscr {F}, \mathbb {P}) \rightarrow S$ is $\mathscr {F}^t_s$-elementary if there exists $T_1>t$ such that the restriction of X to $[t, T_1]$ is $\mathscr {F}^t_s$-elementary and $X(s)=0$ for $s>T_1$.

It is immediate from the definitions that simple and elementary processes are progressively measurable and predictable.

Remark 1.97

In Definitions 1.14, 1.95 and 1.96 we introduced the concepts of a simple random variable, $\mathscr {F}^t_s$-simple process, and $\mathscr {F}^t_s$-elementary process. The reader should be aware that in the literature the use of these terms varies and the same word is often used by different authors to mean different things. $\blacksquare$

Lemma 1.98

Let E be a separable Banach space endowed with the Borel $\sigma$-field, $(\Omega , \mathscr {F}, \mathscr {F}^t_s, \mathbb {P})$ be a filtered probability space and $X:[t, T] \times \Omega \rightarrow E$ be a bounded, measurable, $\mathscr {F}^t_s$-adapted process, where $T\in [t,+\infty )\cup \{+\infty \}$. There exists a sequence $X^m(\cdot )$ of $\mathscr {F}^t_s$-elementary E-valued processes on [t, T] such that, for every $1\le p<+\infty$ and $R>t$,

$$\begin{aligned} \lim _{m\rightarrow +\infty } \mathbb {E} \int _t^{R\wedge T} \left| X^m(s) - X(s) \right| ^p_E {d} s =0. \end{aligned}$$

(1.16)

The same claim holds if, instead of the Banach space, we consider E to be an interval $[a, b]\subset \mathbb {R}$ or a countable closed subset of [a, b]. In these cases the norm $|\cdot |_E$ in (1.16) is replaced by $|\cdot |_{\mathbb {R}}$.

Proof

It is enough to prove the result for a single $p\ge 1$. To obtain a sequence of $\mathscr {F}^t_s$-simple processes $X^m(\cdot )$ with the required properties, the proof follows exactly the proof of Lemma 3.2.4, p. 132, in [372] with obvious technical modifications as we now have to deal with Bochner integrals in E. We then use Lemma 1.16 to approximate the random variables $\xi _i$ defining $X^m(\cdot )$ by simple random variables to obtain $\mathscr {F}^t_s$-elementary approximating processes.

If E is a countable closed subset of [a, b], we first produce [a, b]-valued $\mathscr {F}^t_s$-elementary approximating processes $X^m(\cdot )$. We then construct an E-valued $\mathscr {F}^t_s$-elementary process $Y^m(\cdot )$ from $X^m(\cdot )$ as follows. Let $X^m(s) = \xi _i$ for $s\in (t_i, t_{i+1}]$ for $i\in \{ 0, .. N-1 \}$, and $X(t)=\xi _0$. Let ${\tilde{\xi }}_i$ be defined in the following way. If $\xi _i(\omega )\in E$, we set $\tilde{\xi }_i(\omega )=\xi _i(\omega )$. If $\xi _i(\omega )\not \in E$, we set $\tilde{\xi }_i(\omega )=\arg \min _{x\in E }|\xi (\omega )-x|$ if $\arg \min _{x\in E }|\xi (\omega )-x|$ is a singleton. If $\arg \min _{x\in E }|\xi (\omega )-x|$ has two points $x_1<x_2$, we set $\tilde{\xi }_i(\omega )=x_1$. Obviously $\tilde{\xi }_i$ is a simple, $\mathscr {F}^t_{t_i}$-measurable process. We now define $Y^m(s) = \tilde{\xi }_i$ for $s\in (t_i, t_{i+1}]$ for $i\in \{ 0, .. N-1 \}$, and $X(t)=\tilde{\xi }_0$. Then, since $X(\cdot )$ has values in E, it is easy to see that $|Y^m(s)-X(s)|\le 2|X^m(s)-X(s)|$ for any $s\in [t, +\infty )$ and $\omega \in \Omega$. Therefore the result follows. $\square$

Lemma 1.99

Let $\mathscr {F}_{s}^{t, 0}$ and $\mathscr {F}_{s}^{t}$ be as in Proposition 1.90, $T\in [t,+\infty )\cup \{+\infty \}$, and let $a(\cdot ):[t, T] \times \Omega \rightarrow S$ be an $\mathscr {F}_{s}^{t}$-progressively measurable process, where (S, d) is a Polish space endowed with the Borel $\sigma$-field. Then there exists an $\mathscr {F}_{s}^{t, 0}$-progressively measurable and $\mathscr {F}_{s}^{t, 0}$-predictable process $a_1(\cdot ):[t, T]\times \Omega \rightarrow S$, such that $a(\cdot )=a_{1}(\cdot )$, ${d} t \otimes \mathbb {P}$-a.e. on $[t, T]\times \Omega$.

Proof

In light of Theorems 1.12 and 1.13 we can assume that $S = [0,1]$ or S is a countable closed subset of [0, 1]. Using Lemma 1.98, we can find approximating $\mathscr {F}_s^t$-elementary processes $a^n(\cdot )$ on [t, T] of the form

$$ a^n(t)(\omega ) = \left\{ \begin{array}{l} \xi _0^n(\omega ) \qquad \text {if }s=t\\ \xi _i^n(\omega ) \qquad \text {if }s\in (t_i, t_{i+1}] \end{array} \right. $$

such that

$$ \sup _{R\ge t} \lim _{n\rightarrow \infty } \mathbb {E} \int _t^{R\wedge T} \left| a(s) - a^n (s) \right| ^2_\mathbb {R} {d} s =0. $$

Using Lemma 1.16, we can change every $\xi _i^n$ on a null-set to obtain a sequence of $\mathscr {F}_s^{t, 0}$-elementary processes $a_1^n(\cdot )$ that still satisfy

$$ \sup _{R\ge t} \lim _{n\rightarrow \infty } \mathbb {E} \int _t^{R\wedge T} \left| a(s) - a_1^n (s) \right| ^2_\mathbb {R} {d} s =0. $$

Obviously the processes $a_1^n(\cdot )$ are $\mathscr {F}_{s}^{t, 0}$-progressively measurable. We can now extract a subsequence (still denoted by $a_1^n(\cdot )$) such that $a_1^n(\cdot ) \rightarrow a(\cdot )$ ${d} t \otimes \mathbb {P}$-a.e. on $[t, T]\times \Omega$, and define $a_1(\cdot ) := \lim \inf _{n\rightarrow + \infty } a_1^n(\cdot )$. The process $a_1(\cdot )$ is $\mathscr {F}_{s}^{t, 0}$-progressively measurable, $\mathscr {F}_{s}^{t, 0}$-predictable, and $a(\cdot )=a_{1}(\cdot )$, ${d} t \otimes \mathbb {P}$-a.e. on $[t, T]\times \Omega$. $\square$

1.3 The Stochastic Integral

Let $T\in (0,+\infty )$, and $t\in [0,T)$. Throughout the whole section $\Xi$ and H will be two real, separable Hilbert spaces, Q will be an operator in $\mathcal {L}^+(\Xi )$, $\left( \Omega , \mathscr {F},\right. \left. \left\{ \mathscr {F}^t_s \right\} _{s\in [t, T]},\mathbb {P} \right)$ will be a filtered probability space, and $W_Q$ will be a translated $\mathscr {F}^t_s$-Q-Wiener process on $\Omega$ on [0, T]. The following concept will be used in Chap. 2.

Definition 1.100

A 5-tuple $\mu :=\left( \Omega , \mathscr {F}, \left\{ \mathscr {F}^t_s \right\} _{s\in [t, T]}, \mathbb {P}, W_Q \right)$ described above is called a generalized reference probability space.

A process $X(\cdot )$ will always be assumed to be defined on $\Omega$, and the expressions “adapted” and “progressively measurable” will always refer to the filtration $\mathscr {F}^t_s$.

1.3.1 Definition of the Stochastic Integral

In this section we will assume that $\mathrm{Tr}(Q)<+\infty$. If $\mathrm{Tr}(Q)=+\infty$, the construction of the stochastic integral is the same, we just have to consider $W_Q$ as a $\Xi _1$-valued Wiener process with nuclear covariance $Q_1$ (see Sect. 1.2.4). This way $W_Q$ is not uniquely determined but $Q_{1}^{1/2}(\Xi _{1})=\Xi _0=Q^{1/2}(\Xi ), |x|_{\Xi _0}=|Q_1^{-1/2}x|_{\Xi _1}$ for all possible extensions $\Xi _{1}$ and the class of integrands and the value of the integrals are independent of the choice of the space $\Xi _{1}$ (see [180], Proposition 4.7 and Sect. 4.1.2).

We recall that we denote by $\mathcal {L}_2(\Xi _0,H)$ the space of Hilbert–Schmidt operators from $\Xi _0$ to H (see Appendix B.3). It is equipped with its Borel $\sigma$-field $\mathcal {B}(\mathcal {L}_2(\Xi _0,H))$. $\mathcal {L}_2(\Xi _0,H)$ is a real, separable Hilbert space (see Proposition B.25), and $\mathcal {L}(\Xi , H)$ is dense in $\mathcal {L}_2(\Xi _0,H)$ (see e.g. [294], pp. 24–25).

Definition 1.101

(The space $\mathcal {N}_Q^{p}(t, T;H)$) Given$p\ge 1$, we denote by $\mathcal {N}_Q^{p}(t, T;H)$ the space of all $\mathcal {L}_2(\Xi _0,H)$-valued, progressively measurable processes $X(\cdot )$, such that

$$ |X(\cdot )|_{\mathcal {N}_Q^{p}(t, T;H)}:= \left( \mathbb {E}\int _{t}^{T} \Vert X(s)\Vert ^{p}_{\mathcal {L}_2(\Xi _0,H)} {d} s \right) ^{1/p} < \infty . $$

$\mathcal {N}_Q^{p}(t, T;H)$ is a Banach space if it is endowed with the norm $|\cdot |_{\mathcal {N}_Q^{p}(t, T;H)}$.

We remark that, as always, two processes in $\mathcal {N}_Q^{p}(t, T;H)$ are identified if they are equal $\mathbb {P}\otimes dt$-a.e.

Remark 1.102

In several classical references (see e.g. [180] or [491]), the theory of stochastic integration is developed for predictable processes instead of progressively measurable ones like in our case. However, it follows for instance from Lemma 1.99, that for every ${\mathcal {L}_2(\Xi _0,H)}$-valued progressively measurable process X there exists a predictable process $X_1$ which is $\mathbb {P}\otimes dt$-a.e. equal to X. Thus, since we are working with stochastic integrals with respect to Wiener processes (which are continuous), the two concepts coincide. $\blacksquare$

For an $\mathcal {L}(\Xi , H)$-valued, $\mathscr {F}^t_s$-simple process $\Phi$ on [t, T], $\Phi (s) = \Phi _0 \mathbf{1}_{\{t\}}(s) + \sum _{i=0}^{i=N-1} \mathbf{1}_{(t_i, t_{i+1}]}(s) \Phi _i$, the stochastic integral with respect to $W_Q$ is defined by

$$ \int _t^T \Phi (s) {d} W_Q(s) := \sum _{i=0}^{N-1} \Phi _i(W_Q(t_{i+1}) - W_Q(t_i)) \in L^2(\Omega ;H). $$

Note that if we take $\Phi$ to be $\mathcal {L}_2(\Xi _0,H)$-valued, we cannot guarantee that the expression above is well defined, since $\mathcal {L}_2(\Xi _0,H)$ contains genuinely unbounded operators in $\Xi$ (see e.g. [294], p. 25, Exercise 2.7).

We now extend the stochastic integral to all processes in $\mathcal {N}_Q^{2}(t, T;H)$ by the following theorem.

Theorem 1.103

(Itô isometry) For every $\mathcal {L}(\Xi , H)$-valued, $\mathscr {F}^t_s$-simple process $\Phi$ we have

$$ \mathbb {E}\left| \int _t^T \Phi (s) {d} W_Q(s)\right| _H^2= \mathbb {E}\int _{t}^{T} \Vert \Phi (s)\Vert ^{2}_{\mathcal {L}_2(\Xi _0,H)} {d} s. $$

Thus the stochastic integral is an isometry between the set of $\mathcal {L}(\Xi , H)$-valued, $\mathscr {F}^t_s$-simple processes in $\mathcal {N}_Q^{2}(t, T;H)$ and its image in $L^2(\Omega ;H)$. Moreover, since $\mathcal {L}(\Xi , H)$-valued, $\mathscr {F}^t_s$-simple (and in fact elementary) processes are dense in $\mathcal {N}_Q^{2}(t, T;H)$, it can be uniquely extended to all processes in $\mathcal {N}_Q^{2}(t, T;H)$. We denote this unique extension by

$$ \int _t^T \Phi (s) {d} W_Q(s) $$

and call it the stochastic integral of $\Phi$ with respect to $W_Q$.

Proof

See [294], Propositions 2.1, 2.2, and Definition 2.10. See also [180], Proposition 4.22 in the context of predictable processes. $\square$

Proposition 1.104

For $\Phi \in \mathcal {N}_Q^{2}(t, T;H)$, consider the process

$$ \left\{ \begin{array}{l} I(\Phi ) :[t, T] \times \Omega \rightarrow H\\ I(\Phi )(r) := \int _t^r \Phi (s) {d} W_Q(s):=\int _t^T \Phi (s)\mathbf{1}_{[t, r]} {d} W_Q(s). \end{array} \right. $$

$I(\Phi )$ is a continuous square-integrable martingale and $I:\mathcal {N}_Q^{2}(t, T;H)\rightarrow \mathcal {M}^2_{t, T}(H)$ is an isometry. Moreover,

$$ \left\langle \left\langle I(\Phi )\right\rangle \right\rangle _s=\int _t^s\left( \Phi (s)Q^{\frac{1}{2}}\right) \left( \Phi (s)Q^{\frac{1}{2}}\right) ^*ds, $$

$$ \left\langle I(\Phi )\right\rangle _s=\int _t^s\Vert \Phi (s)\Vert _{\mathcal {L}_2(\Xi _0,H)}^2 ds. $$

Proof

See [294] Theorem 2.3, p. 34. $\square$

The definition of stochastic integral can be further extended to all $\mathcal {L}_2(\Xi _0,H)$-valued progressively measurable processes $\Phi (\cdot )$ such that

$$\begin{aligned} \mathbb {P} \left( \int _t^T \Vert \Phi (s)\Vert _{\mathcal {L}_2(\Xi _0,H)}^2 {d} s <+\infty \right) = 1. \end{aligned}$$

(1.17)

Lemma 1.105

Let $\{\Phi (s)\}_{s\in [t, T]}$ be an $\mathcal {L}_2(\Xi _0,H)$-valued progressively measurable process satisfying (1.17). Then there exists a sequence $\Phi _n$ of $\mathcal {L}(\Xi , H)$-valued $\mathscr {F}^t_s$-simple processes such that

$$\begin{aligned} \lim _{n\rightarrow \infty } \int _t^T \Vert \Phi (s) - \Phi _n(s) \Vert ^2_{\mathcal {L}_2(\Xi _0,H)} {d} s =0 \qquad \mathbb {P}-a.s. \end{aligned}$$

(1.18)

Moreover, there exists an H-valued random variable, denoted by $\mathcal {I}$, such that

$$ \lim _{n\rightarrow \infty } \int _t^T \Phi _n(s) {d} W_Q(s) = \mathcal {I} \qquad \text {in probability}. $$

$\mathcal {I}$ does not depend on the choice of approximating sequence, more precisely, given $\Phi _n^1$ and $\Phi _n^2$ satisfying (1.18), if $\mathcal {I}_1:= \lim _{n\rightarrow \infty } \int _t^T \Phi ^1_n(s) {d} W_Q(s)$ and $\mathcal {I}_2:= \lim _{n\rightarrow \infty } \int _t^T \Phi ^2_n(s) {d} W_Q(s)$, then $\mathcal {I}_1=\mathcal {I}_2$ $\mathbb {P}-a.s.$

Proof

See [294], Lemmas 2.3, p. 39, and 2.6, p. 41. $\square$

The process $\mathcal {I}$ defined by Lemma 1.105 is called the stochastic integral of $\Phi$ with respect to $W_Q$, and is denoted by $\int _t^T \Phi (s) {d} W_Q(s)$. We also set $\int _t^r \Phi (s) {d} W_Q(s):=\int _t^T \Phi (s)\mathbf{1}_{[t, r]} {d} W_Q(s)$.

Proposition 1.106

Let $\{\Phi (s)\}_{s\in [t, T]}$ be an $\mathcal {L}_2(\Xi _0,H)$-valued progressively measurable process satisfying (1.17). Then the process

$$ \left\{ \begin{array}{l} I(\Phi ) :[t, T] \times \Omega \rightarrow H\\ I(\Phi )(r) := \int _t^r \Phi (s) {d} W_Q(s) \end{array} \right. $$

is a continuous local martingale.

Proof

See [294], pp. 42–44. $\square$

Finally, we may extend the definition of stochastic integral to all processes (not necessarily progressively measurable) that are $dt\otimes \mathbb {P}$-equivalent to progressively measurable processes satisfying (1.17) in the sense of the following definition (see also [372], p. 130).

Definition 1.107

We say that two processes $\Phi _1$ and $\Phi _2$ are $dt\otimes \mathbb {P}$-equivalent if $\Phi _1=\Phi _2$, $dt\otimes \mathbb {P}$-a.e. If $\Phi$ belongs to the equivalence class of a progressively measurable process $\Phi _1$ satisfying (1.17),⁴ we set

$$ \int _t^T \Phi (s) {d} W_Q(s):=\int _t^T \Phi _1(s){d} W_Q(s). $$

This definition is obviously independent of the choice of a representative process $\Phi _1$. Thus a representative process defines the stochastic integral for the whole equivalence class.

Example 1.108

Every $\mathcal {L}_2(\Xi _0,H)$-valued, $\mathscr {F}^t_s$-adapted, and $\overline{\mathcal {B}([t, T])\otimes \mathscr {F}}$-measurable process $\Phi$ satisfying (1.17) is stochastically integrable, where $\overline{\mathcal {B}([t, T])\otimes \mathscr {F}}$ is the completion of $\mathcal {B}([t, T])\otimes \mathscr {F}$ with respect to $dt\otimes \mathbb {P}$. To see this we need to find a progressively measurable process $\Phi _1$ which is equivalent to $\Phi$. First, let $\Phi _2$ be a $\mathcal {B}([t, T])\otimes \mathscr {F}$-measurable process equivalent to $\Phi$ (which exists by Lemma 1.16). Then, for a.e. $s \in [t, T]$, we have $\Phi _2(s,\cdot )=\Phi (s,\cdot )$ $\mathbb P$-a.s. and, since every $\mathscr {F}^t_s$ is complete, also $\Phi _2(s,\cdot )$ is $\mathscr {F}^t_s$-measurable for a.e. s. Thus there exists an $A\in \mathcal {B}([t, T])$ of full measure such that $\Phi _2(s,\cdot )$ is $\mathscr {F}^t_s$-measurable for $s\in A$. We then define $\Phi _3=\Phi _2\mathbf{1}_A$. $\Phi _3$ is $\mathcal {B}([t, T])\otimes \mathscr {F}$-measurable and $\mathscr {F}^t_s$-adapted, thanks to Lemma 1.72 it has a progressively measurable modification $\Phi _1$ which is clearly equivalent to $\Phi$. $\blacksquare$

Theorem 1.109

Let $(E, \mathscr {G} , \mu )$ be a measure space with bounded measure. Let $\Phi :[t, T] \times \Omega \times E\rightarrow \mathcal {L}_2(\Xi _0,H)$ be $(\mathcal {B}([t, T])\otimes \mathscr {F}_T^t \otimes \mathscr {G})/\mathcal {B}(\mathcal {L}_2(\Xi _0,H))$-measurable. Suppose that, for any $x\in E$, $\{\Phi (s, \cdot , x)\}_{s\in [t, T]}$ is progressively measurable and

$$ \int _E | \Phi (\cdot , \cdot , x) |_{\mathcal {N}_Q^{2}(t, T;H)} {d} \mu (x) < +\infty . $$

Then:

• (i) $\displaystyle \int _t^T \Phi (s, \cdot , \cdot ) {d} W_Q(s)$ has an $\mathscr {F}_T^t\otimes \mathscr {G} /\mathcal {B}(H)$-measurable version.

• (ii) $\displaystyle \int _E \Phi (\cdot , \cdot , x) {d} \mu (x)$ is progressively measurable.

• (iii) The following equality holds $\mathbb {P}$-a.s.:

  $$ \int _E \int _t^T \Phi (s, \cdot , x) {d} W_Q(s) {d} \mu (x) = \int _t^T \int _E \Phi (s, \cdot , x) {d} \mu (x) {d} W_Q(s). $$

Proof

See Theorem 2.8, Sect. 2.2.6, p. 57 of [294] and Theorem 4.33, Sect. 4.5, p. 110 of [180]. $\square$

1.3.2 Basic Properties and Estimates

Lemma 1.110

Let $T>0$ and $t\in [0,T)$. Assume that $\Phi$ is in $\mathcal {N}_Q^{2}(t, T;H)$ and that $\tau$ is an $\mathscr {F}^t_{s}$-stopping time such that $\mathbb {P}(\tau \le T) =1$. Then $\mathbb {P}$-a.s.

$$ \int _t^T \mathbf{1}_{[t,\tau ]} (r) \Phi (r) {d} W_Q(r) = \int _t^\tau \Phi (r) {d} W_Q(r). $$

Proof

See [294], Lemma 2.7, p. 43 (also [180], Lemma 4.24, p. 99). $\square$

As a consequence of Theorem 1.80 and Proposition 1.104 we obtain the following theorem (see also e.g. [177], Theorem 5.2.4, p. 58).

Theorem 1.111

(Burkholder–Davis–Gundy inequality for stochastic integrals) Let$T>0$ and $t\in [0,T)$. For every $p\ge 2$, there exists a constant $c_p$ such that, for every $\Phi$ in $\mathcal {N}_Q^{p}(t, T;H)$,

$$\begin{aligned}&\mathbb {E} \left[ \sup _{s\in [t, T]} \left| \int _t^s\Phi (r) {d} W_Q(r) \right| ^p \right] \le c_p \mathbb {E} \left[ \int _t^T \Vert \Phi (r)\Vert ^2_{\mathcal {L}_2(\Xi _0, H)} {d} r \right] ^{p/2} \\&\qquad \qquad \qquad \qquad \qquad \qquad \quad \le c_p (T-t)^{\frac{p}{2}-1}\mathbb {E} \left[ \int _t^T \Vert \Phi (r)\Vert ^p_{\mathcal {L}_2(\Xi _0, H)} {d} r \right] . \end{aligned}$$

Proposition 1.112

Let $T>0$ and $t\in [0,T)$. Let A be the generator of a $C_0$-semigroup $\{e^{rA},\; r\ge 0\}$ on H such that $\Vert e^{rA} \Vert \le M e^{\alpha r}$ for every $r\ge 0$ for some $\alpha \in \mathbb {R}$, $M>0$. Let $p> 2$ and $\Phi \in \mathcal {N}^p_Q(t, T;H)$. Let $A_n$ be the Yosida approximation of A. Then the stochastic convolution process

$$\begin{aligned} \Psi (s):=\int _t^s e^{(s-r)A} \Phi (r) {d} W_Q(r), \qquad s\in [t, T], \end{aligned}$$

(1.19)

has a continuous modification,

$$\begin{aligned} \mathbb {E} \left[ \sup _{s\in [t, T]} \left| \int _t^s e^{(s-r)A} \Phi (r) {d} W_Q(r) \right| ^p \right] \le C \mathbb {E} \left[ \int _t^T \Vert \Phi (r)\Vert ^p_{\mathcal {L}_2(\Xi _0, H)} {d} r \right] , \end{aligned}$$

(1.20)

where the constants c and C depend only on $T-t$, p, M, $\alpha$, and

$$\begin{aligned} \lim _{n\rightarrow \infty } \mathbb {E} \left[ \sup _{s\in [t, T]} \left| \int _t^s \left( e^{(s-r)A_n} - e^{(s-r)A} \right) \Phi (r) {d} W_Q(r) \right| ^{p} \right] =0. \end{aligned}$$

(1.21)

If, moreover, A generates a $C_0$-pseudo-contraction semigroup (i.e. $M=1$ above, see Appendix B.4) then the claims are also true for $p=2$.

Proof

See [294], Lemma 3.3, p. 87. The claims for p=2 can be proved by repeating the arguments of the proof of Proposition 3.3 of [543], which uses the Unitary Dilation Theorem. $\square$

Proposition 1.113

Let A be the generator of a $C_0$-semigroup on H, $T>0$, and $t\in [0,T)$. Assume that $\Phi : [t, T]\times \Omega \rightarrow \mathcal {L}_2(\Xi _0, H)$ is a progressively measurable process such that $\Phi (s)\in \mathcal {L}_2(\Xi _0, D(A))$ $\mathbb {P}$-a.s., for a.e. $s \in [t, T]$. Assume that

$$ \mathbb {P}\left( \int _t^T \Vert \Phi (s) \Vert ^2_{\mathcal {L}_2(\Xi _0, D(A))} ds <+\infty \right) =1. $$

Then

$$\begin{aligned} \mathbb {P}\left( \int _t^T \Phi (s) dW_Q(s) \in D(A) \right) =1 \end{aligned}$$

(1.22)

and

$$\begin{aligned} A\int _t^T \Phi (s) dW_Q(s)=\int _t^T A\Phi (s) dW_Q(s), \qquad \mathbb {P}-a.s. \end{aligned}$$

(1.23)

Proof

We can assume without loss of generality that $Q\in \mathcal {L}^+_1(\Xi )$. The proof follows the proof of Proposition 3.1 (p. 76) of [294], however we present it here to clarify a measurability issue. Indeed, we first need to show that $\Phi$ is an $\mathcal {L}_2(\Xi _0, D(A))$-valued, progressively measurable process. To do this we take $\Psi _n=J_n \Phi$, where $J_n=n(nI-A)^{-1}$ (see Definition B.40). Since $J_n\in \mathcal {L}(H, D(A))$, $\Psi _n$ is an $\mathcal {L}_2(\Xi _0, D(A))$-valued, progressively measurable process. Moreover, it is easy to see that if, for some $s\in [t, T]$ and $\omega \in \Omega$, $\Phi (s)(\omega )\in \mathcal {L}_2(\Xi _0, D(A))$, then $\Psi _n(s)(\omega )\rightarrow \Phi (s)(\omega )$ in $\mathcal {L}_2(\Xi _0, D(A))$. Therefore, defining $V:=\{(s,\omega ):\Psi _n(s)(\omega )\,\,\text {converges in}\,\,\mathcal {L}_2(\Xi _0, D(A))\}$, it follows from Lemma 1.8-(iii) that $\Phi$ is equivalent to a progressively measurable process $\lim _{n\rightarrow +\infty }\mathbf{1}_V\Psi _n$. The proof is now done in two steps.

Step 1: The claim is true for $\mathscr {F}_s^t$-simple $\mathcal {L}(\Xi , D(A))$-valued processes.

Step 2: If $\Phi$ is a $\mathcal {L}_2(\Xi _0, D(A))$-valued progressively measurable process satisfying the hypotheses of this proposition, we take a sequence of $\mathscr {F}_s^t$-simple $\mathcal {L}(\Xi , D(A))$-valued processes $\Phi _n$ approximating $\Phi$ in the sense of (1.18) so that

$$ \lim _{n\rightarrow + \infty } \int _t^T \Vert \Phi (s) - \Phi _n(s) \Vert ^2_{\mathcal {L}_2(\Xi _0, D(A))} ds =0 \qquad \mathbb {P}-a.s. $$

In particular we have

$$ \int _t^T \Phi _n(s) dW_Q(s) \xrightarrow {n\rightarrow \infty } \int _t^T \Phi (s) dW_Q(s), $$

$$ A \int _t^T \Phi _n(s) dW_Q(s) = \int _t^T A \Phi _n(s) dW_Q(s) \xrightarrow {n\rightarrow \infty } \int _t^T A \Phi (s) dW_Q(s) $$

in probability, so the claim follows since A is a closed operator. $\square$

In the rest of this section we explain how the factorization method is used to prove continuity of trajectories of stochastic convolution processes.

Lemma 1.114

(Factorization Lemma) Let $T>0$, $t\in [0,T)$, and $0<\alpha <1$. Let A be the generator of a $C_0$-semigroup $\{e^{rA},\; r\ge 0\}$ on H. Consider a linear, densely defined, closed operator $A_1:D(A_1){\subset } H \rightarrow H$ such that, for any $r>0$, $e^{rA}H{\subset } D(A_1)$, $A_1e^{rA}$ is bounded and $A_1e^{rA}=e^{rA}A_1$ on $D(A_1)$. Let $\Phi : [t, T]\times \Omega \rightarrow \mathcal {L}_2(\Xi _0, H)$ be progressively measurable and such that for every $s\in [t, T]$

$$ \mathbb {E} \int _t^s \left\| A_1e^{(s-r)A} \Phi (r) \right\| ^2_{\mathcal {L}_2(\Xi _0, H)} {d} r<+\infty . $$

Assume that, for all $s\in [t, T]$,

$$\begin{aligned} \int _t^s (s-r)^{\alpha -1} \left( \int _t^r (r-h)^{-2\alpha } \mathbb {E} \left[ \left\| A_1e^{(r-h)A}\Phi (h) \right\| ^2_{\mathcal {L}_2(\Xi _0, H)} \right] {d} h \right) ^{1/2} {d} r < +\infty . \end{aligned}$$

(1.24)

Then

$$ \int _t^s A_1e^{(s-r)A} \Phi (r) {d} W_Q(r) = \frac{\sin (\alpha \pi )}{\pi } \int _t^s (s-r)^{\alpha -1} e^{(s-r)A} Y^{\Phi }_\alpha (r) {d} r \qquad \mathbb {P}-a.s. $$

for all $s\in [t, T]$, where $Y^{\Phi }_\alpha (\cdot )$ is a $\mathcal {B}([t, T])\otimes \mathscr {F}_T^t /\mathcal {B}(H)$-measurable process which is ${d} t\otimes \mathbb {P}$-equivalent to

$$ \int _t^r (r-h)^{-\alpha } A_1e^{(r-h)A} \Phi (h) {d} W_Q(h). $$

Proof

The statement is similar to [177], Theorem 5.2.5, p. 58, Sect. 5.2.1. We give the proof for completeness.

We use the identity

$$\begin{aligned} \int _{\sigma }^{t}(t-s)^{\alpha -1}(s-\sigma )^{- \alpha }{d} s = \frac{\pi }{\sin (\pi \alpha ) }, \qquad \text {for all } \sigma \le s \le t, \; 0<\alpha <1 \end{aligned}$$

(which can be proved by a simple direct computation). Define

$$ X(r, h)=\mathbf{1}_{[t, r]}(h)(r-h)^{-\alpha }A_1e^{(r-h)A}\Phi (h). $$

Since (1.24) implies

$$ \int _t^T \left( \mathbb {E} \int _t^T \left\| X(r, h) \right\| ^2_{\mathcal {L}_2(\Xi _0, H)} {d} h \right) ^{1/2} {d} r < +\infty , $$

by the stochastic Fubini Theorem 1.109 (see also Theorem 4.33, p. 110 of [180] or Theorem 2.8, p. 57 of [294]) there exists a $\mathcal {B}([t, T])\otimes \mathscr {F}_T^t /\mathcal {B}(H)$-measurable process $Y^{\Phi }_\alpha :[t, T]\times \Omega \rightarrow H$ such that

$$ \int _t^T X(r, h) dW_Q(h)=\int _t^r(r-h)^{-\alpha }A_1e^{(r-h)A}\Phi (h){d} W_Q(h) = Y^{\Phi }_\alpha (r), \quad {d} t\otimes \mathbb {P}\text {-a.e.} $$

Then for every $s\in [t, T]$ the process $Z^{\Phi , s}_\alpha (\cdot )$, defined for any $r\in [t, s]$ by $Z^{\Phi , s}_\alpha (r)=(s-r)^{\alpha -1}e^{(s-r)A}Y^{\Phi }_\alpha (r)$, is jointly measurable and ${d} t\otimes \mathbb {P}$-equivalent to

$$ (s-r)^{\alpha -1} e^{(s-r)A} \int _t^r (r-h)^{- \alpha }A_1e^{(r-h)A} \Phi (h) {d} W_Q(h) $$

on $[t, s]\times \Omega$. Thus fixing any $s\in [t, T]$ and applying the stochastic Fubini Theorem on $[t,s]\times [t, s]\times \Omega$ (whose assumptions are satisfied by (1.24)) and noticing that we can use the process $Z^{\Phi , s}_\alpha (\cdot )$ in place of a process provided by the stochastic Fubini Theorem (since it will give $\mathbb {P}$-a.e. the same integrals) we obtain for $\mathbb {P}$-a.e. $\omega$

$$\begin{aligned}&\frac{\pi }{\sin (\pi \alpha ) } \int _t^s A_1e^{(s-h)A} \Phi (h) {d} W_Q(h) \\&\qquad = \int _t^s\int _t^s \mathbf{1}_{[h, s]}(r)(s-r)^{\alpha -1} e^{(s-r)A} (r-h)^{- \alpha } A_1e^{(r-h)A} \Phi (h) {d} r {d} W_Q(h) \\&\qquad \qquad \qquad \qquad \qquad \quad \qquad \qquad \qquad = \int _t^s (s-r)^{\alpha -1} e^{(s-r)A} Y^{\Phi }_\alpha (r) {d} r. \end{aligned}$$

$\square$

Lemma 1.115

Let A be the generator of a $C_0$-semigroup $\{e^{rA},\; r\ge 0\}$ on H, $T>0$, $t\in [0,T)$ and $f\in L^p(t, T; H)$, $p\ge 1$. Then:

• (i) If either $1/p < \alpha \le 1$, or $p=\alpha =1$, then the function

  $$ G_{\alpha } f (s) := \int _t^s (s-r)^{\alpha -1} e^{(s-r)A} f(r) {d} r $$

  is in C([t, T], H).

• (ii) If the semigroup $e^{tA}$ is analytic, $\lambda \in \mathbb {R}$ is such that $(\lambda I - A)^{-1} \in \mathcal {L}(H)$, $\beta >0$ and $\alpha >\beta +1/p$, then the function

  $$ G_{\alpha ,\beta } f (s) := \int _t^s (s-r)^{\alpha -1} (\lambda I- A)^{\beta }e^{(s-r)A} f(r) {d} r $$

  is in C([t, T], H).

Proof

Part (i): Let $1/p < \alpha \le 1$. Let $t\le s_1\le s_2 \le T$ and put $h=s_2-s_1$. We have

$$\begin{aligned}&\left| \int _t^{s_2} (s_2-r)^{\alpha -1} e^{(s_2-r)A} f(r) {d} r - \int _t^{s_1} (s_1-r)^{\alpha -1} e^{(s_1-r)A} f(r) {d} r \right| \\&\qquad \qquad \qquad \le I_1 + I_2 := \int _{t}^{t+h} \left| (s_2-r)^{\alpha -1} e^{(s_2-r)A} f(r) \right| {d} r \\&\qquad \quad +\left| \int _{t+h}^{s_2} (s_2-r)^{\alpha -1} e^{(s_2-r)A}f(r){d} r - \int _{t}^{s_1}(s_1-r)^{\alpha -1} e^{(s_1 -r)A} f(r) {d} r\right| . \end{aligned}$$

Set $q:= \frac{p}{p-1}$ and let $R>0$ be such that $\left\| e^{sA} \right\| \le R$ for all $s\in [0,T]$. Then

$$ I_1 \le R\left( \int _{0}^{h} (h-r)^{q(\alpha -1)} {d} r \right) ^{1/q} \left( \int _{t}^{T} | f(r)|^p {d} r \right) ^{1/p}\rightarrow 0\,\,\,\text {as}\,\, h\rightarrow 0 $$

since $0\ge q(\alpha -1) > -1$. As regards $I_2$, after a change of variables we have

$$\begin{aligned}&\quad I_2\le \int _{t}^{s_1}(s_1-r)^{\alpha -1} e^{(s_1 -r)A} |f(r+h)-f(r)| {d} r \\&\le R\left( \int _{t}^{T} (T-r)^{q(\alpha -1)} {d} r \right) ^{1/q} \left( \int _{t}^{T-h} |f(r+h)- f(r)|^p {d} r \right) ^{1/p}\rightarrow 0\,\,\,\text {as}\,\, h\rightarrow 0. \end{aligned}$$

The proof in the case $p=\alpha =1$ is straightforward.

Part (ii) follows from Proposition A.1.1 in Appendix A, p. 307 of [177]. $\square$

Proposition 1.116

Let $T>0$ and $t\in [0,T)$. Let A, $A_1$, $\Phi$ satisfy the assumptions of Lemma 1.114 except (1.24). Assume that there exist $0<\alpha <1$, $C>0$ and $p>\frac{1}{\alpha }, p\ge 2$ such that

$$\begin{aligned} \int _t^T \mathbb {E} \left( \int _t^r \Vert (r-h)^{-\alpha }A_1 e^{(r-h)A} \Phi (h)\Vert ^2_{\mathcal {L}_2(\Xi _0, H)} {d} h \right) ^{p/2} {d} r <C. \end{aligned}$$

(1.25)

Then

$$\begin{aligned} \Psi (s):=\int _t^s A_1e^{(s-r)A} \Phi (r) {d} W_Q(r), \qquad s\in [t, T], \end{aligned}$$

has a continuous modification.

Proof

We follow the scheme of the proof of Theorem 5.2.6 in [177] (p. 59, Sect. 5.2.1). We give some details because our claim is slightly more general.

Observe that using Hölder’s and Jensen’s inequalities we obtain

$$ \begin{aligned} \int _t^s (s-r)^{\alpha -1} \left( \int _t^r (r-h)^{-2\alpha } \mathbb {E} \left[ \left\| A_1e^{(r-h)A}\Phi (h) \right\| ^2_{\mathcal {L}_2(\Xi _0, H)} \right] {d} h \right) ^{1/2} {d} r \\ \le \left( \int _t^s (s-r)^{\frac{(\alpha -1)p}{p-1}}\right) ^{\frac{p-1}{p}} \left( \int _t^s\mathbb {E} \left( \int _t^r (r-h)^{-2\alpha } \left\| A_1e^{(r-h)A} \Phi (h) \right\| ^2_{\mathcal {L}_2(\Xi _0, H)} {d} h \right) ^{p/2} \right) ^{\frac{1}{p}} \\ <+\infty , \end{aligned} $$

where we used (1.25) and that $\frac{(1-\alpha )p}{p-1}<1$, which follows from $p>1/\alpha$. Therefore the hypotheses of Lemma 1.114 are satisfied and thus we have

$$ \int _t^s A_1e^{(s-r)A} \Phi (r) {d} W_Q(r) = \frac{\sin (\alpha \pi )}{\pi } \int _t^s (s-r)^{\alpha -1} e^{(s-r)A} Y^{\Phi }_\alpha (r) {d} r \qquad \mathbb {P}-a.s. $$

for all $s\in [t, T]$, where $Y^{\Phi }_\alpha (\cdot )$ is defined in Lemma 1.114. The claim will follow from Lemma 1.115-(i) applied to a.e. trajectory. Thus we need to know that the process $Y^{\Phi }_\alpha (\cdot )$ has p-integrable trajectories a.s. This is guaranteed if

$$ \mathbb {E} \int _t^T \left| Y^{\Phi }_\alpha (s) \right| ^p {d} s <+\infty . $$

However, from Theorem 1.111, we have

$$\begin{aligned} \int _t^T \mathbb {E} \left( \left[ \left| Y^{\Phi }_\alpha (s) \right| ^p \right] \right) {d} s \le c_p \int _t^T \mathbb {E} \left( \int _t^s \Vert (s-r)^{-\alpha }A_1 e^{(s-r)A} \Phi (r)\Vert ^2_{\mathcal {L}_2(\Xi _0, H)} {d} r \right) ^{p/2} {d} s, \end{aligned}$$

(1.26)

which is bounded thanks to (1.25). $\square$

The factorization method can also be used to show the continuity of deterministic convolution integrals. The following lemma deals with a case which arises in Sects. 1.5.2 and 1.5.3.

Lemma 1.117

Let $T>0$, $t\in [0,T)$, and $0<\alpha <1$. Let A be the generator of a $C_0$-semigroup $\{e^{rA},\; r\ge 0\}$ on H. Let $\phi$ be a function defined on [t, T] such that, for every $s\in (0,T-t]$, $e^{sA}\phi :[t, T]\rightarrow H$ is well defined, measurable and

$$\begin{aligned} |e^{sA} \phi (r)|\le s^{-\beta } g(r)\quad \text {for}\,\,r\in [t, T], \end{aligned}$$

(1.27)

where $0\le \beta <1,g\in L^q(t, T;H), q>\frac{1}{1-\beta }$. Then the function

$$ \psi (s)=\int _t^s e^{(s-r)A} \phi (r) {d} r $$

belongs to C([t, T], H).

Proof

Let $0<\alpha$ be such that $\alpha +\beta <1$ and $q>\frac{1}{1-(\alpha +\beta )}$. We have, by the Fubini Theorem 1.33,

$$ \int _t^s e^{(s-r)A} \phi (r) {d} r =\frac{ \sin (\pi \alpha )}{\pi }\int _t^s (s-r)^{\alpha -1} e^{(s-r)A} Y(r) {d} r, $$

where

$$ Y(r)=\int _t^r (r-h)^{-\alpha } e^{(r-h)A} \phi (h) {d} h. $$

It remains to notice that, using (1.27) and Hölder’s inequality, we have for $t\le r\le T$

$$ |Y(r)|\le \int _t^r (r-h)^{-(\alpha +\beta )} g(h) {d} h\le C_T |g|_{L^q(t, T;H)}. $$

Thus the result follows from Lemma 1.115-(i). $\square$

1.4 Stochastic Differential Equations

In this section we consider $T>0$ and take H, $\Xi$, Q, and a generalized reference probability space $\mu =(\Omega , \mathscr {F}, \{\mathscr {F}_s\}_{s\in [0,T]}, \mathbb {P}, W_Q)$ as in Sect. 1.3 (with $t=0$). A is the infinitesimal generator of a $C_0$-semigroup on H, and $\Lambda$ is a Polish space. We will look at stochastic differential equations (SDEs) on the interval [0, T], however all results would be the same if, instead of [0, T], we took an interval [t, T], for $0\le t<T$.

1.4.1 Mild and Strong Solutions

Let $b:[0,T] \times H\times \Omega \rightarrow H$ and $\sigma :[0,T]\times H \times \Omega \rightarrow \mathcal {L}_2(\Xi _0, H)$. We consider the following general stochastic differential equation (SDE)

$$\begin{aligned} \left\{ \begin{array}{l} {d} X(s) =(AX(s)+ b(s,X(s))) {d} s + \sigma (s, X(s)) {d} W_Q(s) \qquad s\in (0,T]\\ X(0)=\xi , \end{array} \right. \end{aligned}$$

(1.28)

where $\xi$ is an H-valued $\mathscr {F}_0$-measurable random variable. To simplify the notation we dropped the $\omega$ variable in (1.28) and we use this convention throughout the section.

Definition 1.118

(Strong solution of (1.28)) An H-valued progressively measurable process $X(\cdot )$ is called a strong solution of (1.28) if:

• (i) For ${d} t \otimes \mathbb {P}$-a.e. $(s,\omega )\in [0,T]\times \Omega$, $X(s)(\omega )\in D(A)$.

• (ii) $\displaystyle \mathbb {P} \left( \int _0^T\left( |X(s)| + |AX(s)| + |b(s, X(s))|\right) {d} s <+\infty \right) = 1$ and

  $$ \mathbb {P} \left( \int _0^T \Vert \sigma (s, X(s))\Vert _{\mathcal {L}_2(\Xi _0, H)}^2 {d} s <+\infty \right) = 1. $$

• (iii) For every $t\in [0,T]$

  $$ X(t) =\xi + \int _0^t AX(s) + b(s, X(s)) {d} s + \int _0^t \sigma (s, X(s)) {d} W_Q(s) \;\; \mathbb {P}\text {-a.e.} $$

Definition 1.119

(Mild solution of (1.28)) An H-valued progressively measurable process $X(\cdot )$ is called a mild solution of (1.28) if:

• (i) For every $t\in [0,T]$

  $$ \mathbb {P} \left( \int _0^t \left( |X(s)| + |e^{(t-s)A} b(s, X(s))| \right) {d} s <+\infty \right) = 1 $$

  and

  $$ \mathbb {P} \left( \int _0^t \Vert e^{(t-s)A} \sigma (s, X(s))\Vert _{\mathcal {L}_2(\Xi _0, H)}^2 {d} s <+\infty \right) = 1. $$

• (ii) For every $t\in [0,T]$

  $$ X(t) = e^{tA}\xi + \int _0^t e^{(t-s)A} b(s, X(s)) {d} s + \int _0^t e^{(t-s)A} \sigma (s, X(s)) {d} W_Q(s) \;\; \mathbb {P}\text {-a.e.} $$

In order for the above definitions to be meaningful, all the processes involved must be well defined and have proper measurability properties so that the integrals that appear in the definitions make sense. We do not want to analyze here the required measurability properties in the most generality. Instead, we discuss one case which will frequently appear in applications to optimal control in Remark 1.123 below. Moreover, note that if $A_n$ is the Yosida approximation of A, since by Lemma 1.17-(i) $D(A)\in \mathcal {B}(H)$, it follows that the processes $\mathbf{1}_{X(\cdot )\in D(A)}A_nX(\cdot )$ are progressively measurable and they converge as $n\rightarrow +\infty$ to $\mathbf{1}_{X(\cdot )\in D(A)}AX(\cdot )$ for every $(s,\omega )$. Thus the process $AX(\cdot )$ (understood as $\mathbf{1}_{X(\cdot )\in D(A)}AX(\cdot )$) is progressively measurable.

Remark 1.120

In the definition of a mild solution we assumed that $b:[0,T] \times H\times \Omega \rightarrow H$ and $\sigma :[0,T]\times H \times \Omega \rightarrow \mathcal {L}_2(\Xi _0, H)$. However, Definition 1.119 may still make sense even if b and $\sigma$ do not have values in H and $\mathcal {L}_2(\Xi _0, H)$, provided that the terms $e^{(t-s)A} b(s, X(s))$ and $e^{(t-s)A} \sigma (s, X(s))$ have values in these spaces when they are interpreted properly (see, for instance, Sect. 1.5.1 and also Remark 1.123). Therefore in the future when we are dealing with such cases, we will not repeat the definition of a mild solution, instead we will just explain how to interpret the above terms. $\blacksquare$

Definition 1.121

(Weak mild solution of (1.28)) Assume that in (1.28) we have $b:[0,T]\times H\rightarrow H$ and $\sigma :[0,T]\times H\rightarrow {\mathcal L}_2(\Xi _0,H)$. A weak mild solution of (1.28) is defined to be any 6-tuple $(\Omega , \mathscr {F}, \mathscr {F}_s, W_Q, \mathbb {P}, X(\cdot ))$, where $(\Omega , \mathscr {F}, \mathscr {F}_s, \mathbb {P})$ is a filtered probability space, $W_Q$ is a translated $\mathscr {F}_s$-Q-Wiener process on $\Omega$, and $X(\cdot )$ is a mild solution for (1.28) in the generalized reference probability space $(\Omega , \mathscr {F}, \mathscr {F}_s, W_Q, \mathbb {P})$.

Notation 1.122

In the existing literature, different authors often give different names to the same notion of solution, and the same name does not always correspond to the same definition. For instance, the weak mild solution introduced above is often called a weak solution and in [180] Chap. 8 it is called a martingale solution. $\blacksquare$

Remark 1.123

Let $\Lambda$ be a Polish space. Suppose that $\sigma :[0,T]\times H\times \Lambda \rightarrow \mathcal {L}(\Xi _0, H)$ is such that for every $u\in \Xi _0$, the map $(t,x,a)\rightarrow \sigma (t,x, a)u$ is $\mathcal {B}([0,T])\otimes \mathcal {B}(H)\otimes \mathcal {B}(\Lambda )/\mathcal {B}(H)$-measurable, and $e^{sA}\sigma (t,x, a)\in \mathcal {L}_2(\Xi _0, H)$ for every (t, x, a) and $s>0$. It then follows from Lemma 1.20 that, after possibly redefining it at $s=0$, the map $(s,t,x,a)\rightarrow e^{sA}\sigma (t,x, a)$ is $\mathcal {B}([0,T])\otimes \mathcal {B}([0,T])\otimes \mathcal {B}(H)\otimes \mathcal {B}(\Lambda )/\mathcal {B}( \mathcal {L}_2(\Xi _0, H))$-measurable. Now, if $X(\cdot ):[0,T]\times \Omega \rightarrow H, a(\cdot ):[0,T]\times \Omega \rightarrow \Lambda$ are $\mathscr {F}_s$-progressively measurable, then for every $t\in [0,T]$,

$$ (s,\omega )\rightarrow e^{(t-s)A}\sigma (s,X(s), a(s)) $$

is an $\mathcal {L}_2(\Xi _0, H)$-valued $\mathscr {F}_s$-progressively measurable process on $[0,t]\times \Omega$. If this process is in $\mathcal {N}_Q^{2}(0,t;H)$ for every t then the process

$$ Z(t)=\int _0^te^{(t-s)A}\sigma (s,X(s), a(s)){d} W_Q(s), \qquad t\in [0,T] $$

is an H-valued $\mathscr {F}_t$-adapted process. One way to argue that $Z(\cdot )$ has a progressively measurable modification is the following.

Suppose that there is a constant $K\ge 0$ such that

$$ \mathbb {E}|Z(t)|\le K\quad \text {for all}\,\, t\in [0,T] $$

and that for all $0\le t\le h\le T$

$$ \mathbb {E}\int _t^h\left\| e^{(h-s)A}\sigma (s,X(s), a(s))\right\| _{ \mathcal {L}_2(\Xi _0, H)}^2{d} s\le \rho (h-t) $$

for some modulus $\rho$. We have for $0\le t\le h\le T$

$$ Z(h)-Z(t)=\left( e^{(h-t)A}-I\right) Z(t)+\int _t^he^{(h-s)A}\sigma (s,X(s), a(s)){d} W_Q(s). $$

Let $\{e_n\}$ be an orthonormal basis of H. Then

$$ \langle Z(h)-Z(t),e_n\rangle =\left\langle Z(t),e^{(h-t)A}e_n-e_n\right\rangle +\left\langle \int _t^he^{(h-s)A}\sigma (s,X(s),a(s)){d} W_Q(s), e_n \right\rangle $$

and hence

$$ \mathbb {E}\left| \langle Z(h)-Z(t), e_n\rangle \right| \le K|e^{(h-t)A^*}e_n-e_n|+\sqrt{\rho (h-t)}\le \rho _n(h-t) $$

for some modulus $\rho _n$. Therefore it is easy to see that the process $\langle Z(t), e_n\rangle$ is stochastically continuous and thus, by Lemma 1.74, it has a progressively measurable modification which we denote by $Z_n(\cdot )$. The process $\tilde{Z}(\cdot )$ defined, for $t\in [0,T]$, by

$$ \tilde{Z}(t)=\left\{ \begin{array}{l} {\sum }_{n=1}^{+\infty }Z_n(t)e_n\quad \text {if the limit exists}, \\ 0\quad \text {otherwise} \end{array} \right. $$

is a progressively measurable modification of $Z(\cdot )$. $\blacksquare$

1.4.2 Existence and Uniqueness of Solutions

Definition 1.124

(The space $M^{p}_{\mu }(t, T;E)$) In this definition $T\in (0,+\infty )\cup \{+\infty \}$. Let $p\ge 1$ and $0\le t <T$. Given a Banach space E, we denote by $M^{p}_{\mu }(t, T;E)$ the space of all E-valued progressively measurable processes $X(\cdot )$ such that

$$\begin{aligned} |X(\cdot )|_{M^{p}_{\mu }(t, T;E)}:=\left( \mathbb {E}\left( \int _{t}^{T} |X(s)|^{p} {d} s \right) \right) ^{1/p} < +\infty . \end{aligned}$$

(1.29)

$M^{p}_{\mu }(t, T;E)$ is a Banach space endowed with the norm $|\cdot |_{M^{p}_{\mu }(t, T;E)}$.

Note that in the notation $M^{p}_{\mu }(t, T;E)$ we emphasize the dependence on the generalized reference probability space $\mu$. Processes in $M^{p}_{\mu }(t, T;E)$ are identified if they are equal $\mathbb {P}\otimes dt$-a.e.

Let $a:[0,T] \times \Omega \rightarrow \Lambda$ be an $\mathscr {F}_s$-progressively measurable process (a control process), where $\Lambda$ is, as before, a Polish space. We consider the controlled SDE

$$\begin{aligned} \left\{ \begin{array}{l} {d} X(s) = \left( A X(s) + b(s, X(s), a(s)) \right) {d} s + \sigma (s, X(s), a(s)) {d} W_Q(s)\\ X(0)=\xi . \end{array} \right. \end{aligned}$$

(1.30)

This equation falls into the category of equations (1.28) with $b(s,x,\omega ):=b(s,x, a(s,\omega ))$ and $\sigma (s,x,\omega ):=\sigma (s,x, a(s,\omega ))$. Thus strong, mild and weak mild solutions of (1.30) are defined using the definitions for Eq. (1.28).

Hypothesis 1.125

The operator A is the generator of a strongly continuous semigroup $e^{sA}$ on H. The function $b:[0,T] \times H \times \Lambda \rightarrow H$ is $\mathcal {B}([0,T])\otimes \mathcal {B}(H)\otimes \mathcal {B}(\Lambda )/\mathcal {B}(H)$-measurable, $\sigma :[0,T]\times H \times \Lambda \rightarrow \mathcal {L}_2(\Xi _0, H)$ is $\mathcal {B}([0,T])\otimes \mathcal {B}(H) \otimes \mathcal {B}(\Lambda )/ \mathcal {B}(\mathcal {L}_2(\Xi _0, H))$-measurable, and there exists a constant $C>0$ such that

$$\begin{aligned}&|b(s,x,a) - b(s,y,a)| \le C |x-y| \qquad&\forall x, y \in H, s\in [0,T], a\in \Lambda , \end{aligned}$$

(1.31)

$$\begin{aligned}&\Vert \sigma (s,x,a) - \sigma (s,y, a)\Vert _{\mathcal {L}_2(\Xi _0, H)} \le C |x-y|&\forall x, y \in H, s\in [0,T], a\in \Lambda , \end{aligned}$$

(1.32)

$$\begin{aligned}&|b(s,x, a)| \le C (1+|x|)&\forall x\in H, s\in [0,T], a\in \Lambda ,\end{aligned}$$

(1.33)

$$\begin{aligned}&\Vert \sigma (s,x, a)\Vert _{\mathcal {L}_2(\Xi _0, H)} \le C (1+|x|)&\forall x \in H, s\in [0,T], a\in \Lambda . \end{aligned}$$

(1.34)

Definition 1.126

(The space ${\mathcal {H}_p^\mu (t, T;E)}$) Let $p\ge 1$ and $0\le t<T$. Given a Banach space E, we denote by $\mathcal {H}^\mu _p(t, T;E)$ the set of all progressively measurable processes $X:[t, T] \times \Omega \rightarrow E$ such that

$$\begin{aligned} |X(\cdot )|_{\mathcal {H}_p^\mu (t,T;E)} := \left( \sup _{s\in [t, T]} \mathbb {E} |X(s)|^p \right) ^{1/p} < +\infty . \end{aligned}$$

(1.35)

It is a Banach space with the norm $|\cdot |_{\mathcal {H}_p^\mu (t, T;E)}$.

Processes in $\mathcal {H}_p^\mu (t, T;E)$ are identified if they are equal $\mathbb {P}\otimes dt$-a.e. Therefore the sup in the definition of $\mathcal {H}_p^\mu (t, T;E)$ must be understood as esssup. However, we will keep the notation sup here and in all subsequent uses of this space. If the generalized reference probability space $\mu$ is clear we will just write $M^{p}(t, T;E)$ and ${\mathcal {H}_p(t, T;E)}$ for simplicity.

Mild solutions in $\mathcal {H}_p^\mu (0,T;E)$ (or $M^{p}_\mu (0,T;E)$) of various versions of (1.30) will be obtained as fixed points in these spaces of some maps. We point out that this will not imply that every representative of the equivalence class is a mild solution. Since a mild solution $X(\cdot )$ satisfies the integral equality in Definition 1.119-(ii) for every $t\in [0,T]$, X(t) is prescribed by the right-hand side of this equality, which does not depend on the choice of a representative of the equivalence class. Thus there is a unique (up to a modification) representative of the equivalence class which is a mild solution. We will then always be able to evaluate $\mathbb {E} |X(t)|^p$ for the mild solution $X(\cdot )$ for every $t\in [0,T]$ (and in fact compute the $\mathcal {H}_p^\mu (0,T;E)$ norm of this representative by taking the sup over all $t\in [0,T]$ instead of the esssup).

Theorem 1.127

Let $\xi \in L^p(\Omega ,\mathscr {F}_0, \mathbb {P})$ for some $p\ge 2$, and let A, b and $\sigma$ satisfy Hypothesis 1.125. Let $a(\cdot ):[0,T] \rightarrow \Lambda$ be an $\mathscr {F}_s$-progressively measurable process. Then the SDE (1.30) has a unique, up to a modification, mild solution $X(\cdot )\in \mathcal {H}_p(0,T;H)$. The solution is in fact unique among all processes such that $\mathbb {P} \left( \int _0^T |X(s)|^2 {d} s <+\infty \right) =1$, in particular among the processes in $M^2_\mu (0,T;H)$. $X(\cdot )$ has a continuous modification. Given two continuous versions $X_1(\cdot )$, $X_2(\cdot )$ of the solution, there exists a ${\tilde{\Omega }} \subset \Omega$ with $\mathbb {P}({\tilde{\Omega }}) =1$ s.t. $X_1(s) = X_2(s)$ for all $s\in [0,T]$ and $\omega \in {\tilde{\Omega }}$, i.e. they are indistinguishable.

Proof

The proof can be found, for instance, in [180], Theorem 7.2, p. 188 or [294], Theorems 3.3, p. 97, and 3.5, p. 105. For the last claim, we can take

$$ {\tilde{\Omega }}:= \bigcap _{s\in \mathbb {Q}\cap [0,T]} \left\{ \omega \in \Omega \; : \; X_1(s)(\omega )=X_2(s)(\omega ) \right\} . $$

Since $X_1(\cdot )$ is a modification of $X_2(\cdot )$, we have $\mathbb {P}({\tilde{\Omega }})=1$, and since $X_1(\cdot )$ and $X_2(\cdot )$ are continuous, it follows that $X_1(s)(\omega )=X_2(s)(\omega )$ for all $s\in [0,T]$, $\omega \in {\tilde{\Omega }}$. $\square$

We will denote the solution of (1.30) by $X(\cdot ;\xi , a(\cdot ))$ if we want to emphasize the dependence on the initial datum and the control.

Corollary 1.128

Let $\xi \in L^p(\Omega ,\mathscr {F}_0, \mathbb {P})$ for some $p\ge 2$, let A, b and $\sigma$ satisfy Hypothesis 1.125. If $a_1(\cdot ), a_2(\cdot ) :[0,T] \times \Omega \rightarrow \Lambda$ are two progressively measurable processes such that $a_{1}(\cdot )=a_{2}(\cdot )$, ${d} t \otimes \mathbb {P}$-a.e. on $[0,T]\times \Omega$, then, $\mathbb {P}-a.e.$,

$$ X\left( s;\xi , a_{1}(\cdot )\right) =X\left( s;\xi , a_{2}(\cdot )\right) \,\, \text {for all }s\in [0,T]. $$

Proof

Define $X_i (\cdot ) := X\left( \cdot ;\xi , a_{i}(\cdot )\right)$. Using Theorem 1.103, Jensen’s inequality, and $\sup _{s\in [0,T]}\Vert e^{sA} \Vert \le C$ for some $C \ge 0$, it follows that, for suitable positive $C_1$ and $C_2$:

$$\begin{aligned}&\mathbb {E} \left[ |X_1(s) - X_2(s) |^2 \right] \le C_1 \bigg ( \int _0^s \mathbb {E} | b(r, X_1(r), a_1(r)) - b(r, X_2(r), a_2(r)) |^2 {d} r \\&\qquad \quad + \int _0^s \mathbb {E} \Vert \sigma (r, X_1(r), a_1(r)) - \sigma (r, X_2(r), a_2(r)) \Vert ^2_{\mathcal {L}_2(\Xi _0,H)} {d} r \bigg )\\&\qquad \qquad \qquad \qquad \qquad \qquad \le C_2 \int _0^s \mathbb {E} | X_1(r) - X_2(r) |^2 {d} r,\qquad s \in [0,T], \end{aligned}$$

and the claim follows by using Gronwall’s lemma and the continuity of the trajectories.    $\square$

Remark 1.129

Above we assumed that the $\sigma$ always takes values in ${\mathcal L}_2(\Xi _0,H)$. Existence and uniqueness results for SDEs with more general $\sigma$ can be found, for instance, in [294] Theorem 3.15, p. 143, or in [180] Theorem 7.5, p. 197. To treat some specific examples we will also prove more general results in Sect. 1.5. $\blacksquare$

1.4.3 Properties of Solutions

Theorem 1.130

Let $\xi \in L^p(\Omega ,\mathscr {F}_0, \mathbb {P})$ for some $p\ge 2$, $a:[0,T]\times \Omega \rightarrow \Lambda$ be $\mathscr {F}_s$-progressively measurable, and let A, b and $\sigma$ satisfy Hypothesis 1.125.

• (i) Let $X(\cdot )=X(\cdot ;\xi , a(\cdot ))$ be the unique mild solution of (1.30) (provided by Theorem 1.127). Then, for any $s\in [0,T]$,

  $$\begin{aligned} \sup _{s\in [0,T]} \mathbb {E} \left[ |X(s)|^{p} \right] \le C_p(T) (1+ \mathbb {E}|\xi |^{p})\quad \text {if}\,\, p\ge 2, \end{aligned}$$

  (1.36)
  $$\begin{aligned} \mathbb {E} \left[ \sup _{s\in [0,T]} |X(s)|^{p} \right] \le C_p(T) (1+ \mathbb {E}|\xi |^{p})\quad \text {if}\,\, p> 2, \end{aligned}$$

  (1.37)

  and

  $$\begin{aligned} \mathbb {E} \left[ \sup _{r\in [0,s]} |X(r)-\xi |^{p} \right] \le \omega _\xi (s)\quad \text {if}\,\, p> 2, \end{aligned}$$

  (1.38)

  where $C_p(T)$ is a constant depending on p, T, C (from Hypothesis 1.125) and $M,\alpha$ (where $\Vert e^{rA}\Vert \le Me^{r\alpha }$ for $r \ge 0$), and $\omega _\xi$ is a modulus depending on the same constants and on $\xi$ (in particular they are independent of the process $a(\cdot )$ and of the generalized reference probability space).

• (ii) If $\xi ,\eta \in L^p(\Omega ,\mathscr {F}_0, \mathbb {P})$ for $p>2$, and $X(\cdot )=X(\cdot ;\xi ,a(\cdot )), Y(\cdot )=Y(\cdot ;\eta , a(\cdot ))$ are the solutions of (1.30), then, for all $s\in [0,T]$,

  $$\begin{aligned} \mathbb {E} \left[ \sup _{s\in [0,T]} |X(s)-Y(s)|^{2} \right] \le C_T \left( \mathbb {E} \left[ |\xi -\eta |^{p} \right] \right) ^{\frac{2}{p}}, \end{aligned}$$

  (1.39)

  where $C_T$ depends only on p, T, C, M, $\alpha$.

Proof

Part (i): For (1.36) and (1.37) we refer, for instance, to [180] Theorem 9.1, p. 235, or [294], Lemma 3.6, p. 102, and Corollary 3.3, p. 104. Regarding (1.38), we have that there is a constant $c_1$ depending only on p and $\sup _{t\in [0,T]} \Vert e^{tA} \Vert$, such that

$$\begin{aligned}&\mathbb {E} \left[ \sup _{r\in [0,s]} \left| X(r) -\xi \right| ^{p} \right] \le c \bigg (\mathbb {E}\left[ \sup _{r\in [0,s]} \left| e^{rA}\xi -\xi \right| ^p \right] \\&\qquad \qquad \quad \qquad + \mathbb {E} \left[ \sup _{r\in [0,s]} \left( \int _0^r | b(u, X(u), a(u))| {d} u \right) ^p \right] \\&\qquad \qquad \qquad \qquad \quad + \mathbb {E} \left[ \sup _{r\in [0,s]} \left| \int _0^r e^{(r-u)A} \sigma (u, X(u), a(u)) {d} W_Q(u) \right| ^p \right] \bigg ). \end{aligned}$$

Using Hypothesis 1.125, (1.37), Hölder’s inequality, and Proposition 1.112, we see that

$$ \mathbb {E} \left[ \sup _{r\in [0,s]} \left| X(r) -\xi \right| ^{p} \right] \le c_2 \bigg ( \mathbb {E} \left[ \sup _{r\in [0,s]} \left| e^{rA}\xi -\xi \right| ^p \right] + \int _0^s \left( 1 + \mathbb {E} |\xi |^p \right) {d} r \bigg ). $$

Since $\sup _{r\in [0,s]} \left| e^{rA}\xi -\xi \right| ^p \xrightarrow {s\rightarrow 0^+} 0$ a.e., and $\sup _{r\in [0,s]} \left| e^{rA}\xi -\xi \right| ^p\le C_1|\xi |^p$, the result follows by the Lebesgue dominated convergence theorem.

Part (ii): See [180] Theorem 9.1, p. 235. $\square$

Theorem 1.131

Let$\xi \in L^p(\Omega ,\mathscr {F}_0, \mathbb {P})$ for some $p> 2$, and let A, b and $\sigma$ satisfy Hypothesis 1.125. Let $a:[0,T]\times \Omega \rightarrow \Lambda$ be a progressively measurable process. Let $X(\cdot )$ be the unique mild solution of (1.30). Consider the approximating equations

$$\begin{aligned} \left\{ \begin{array}{l} {d} X^n(s) = \left( A_n X^n(s) + b(s, X^n(s), a(s)) \right) {d} s + \sigma (s, X^n(s), a(s)) {d} W_Q(s)\\ X^n(0)=\xi , \end{array} \right. \end{aligned}$$

(1.40)

where $A_n$ is the Yosida approximation of A. Let $X_n(\cdot )$ be the solution of (1.40). Then

$$\begin{aligned} \lim _{n\rightarrow \infty } \mathbb {E} \left[ \sup _{s\in [0,T]} |X^n(s) - X(s)|^{p} \right] =0. \end{aligned}$$

(1.41)

Proof

See [180] Proposition 7.4, p. 196, or [294], Proposition 3.2, p. 101. $\square$

The next proposition is a simpler version of Theorem 1.131 which will be useful in the proofs of the results of Sect. 1.7.

Proposition 1.132

Let $\xi \in L^p(\Omega ,\mathscr {F}_0, \mathbb {P})$, $f\in M^p_\mu (0,T;H)$, and $\Phi \in \mathcal {N}_Q^p(0,T;H)$ for some $p\ge 2$. Let $X(\cdot )$ be the mild solution of

$$\begin{aligned} \left\{ \begin{array}{l} {d} X(s) = \left( A X(s) + f(s)\right) {d} s + \Phi (s) {d} W_Q(s)\\ X(0)=\xi \end{array} \right. \end{aligned}$$

(1.42)

and $X^n(\cdot )$ be the solution of

$$\begin{aligned} \left\{ \begin{array}{l} {d} X^n(s) = \left( A_n X^n(s) + f(s)\right) {d} s + \Phi (s) {d} W_Q(s)\\ X^n(0)=\xi , \end{array} \right. \end{aligned}$$

(1.43)

where A generates a $C_0$-semigroup and $A_n$ is the Yosida approximation of A. Then, if $p>2$,

$$\begin{aligned} \lim _{n\rightarrow \infty } \mathbb {E} \left[ \sup _{s\in [0,T]} |X^n(s) - X(s)|^{p} \right] =0. \end{aligned}$$

(1.44)

Moreover, for $p\ge 2$, there exists an $M>0$, independent of n, such that

$$\begin{aligned} \sup _{s\in [0,T]} \mathbb {E} \left[ |X^n(s)|^p \right] \le M, \qquad \sup _{s\in [0,T]} \mathbb {E} \left[ |X(s)|^p \right] \le M. \end{aligned}$$

(1.45)

Proof

Observe first that the mild solution of (1.42) is well defined thanks to the assumptions on $\xi$, f and $\Phi$, and

$$ X(s) = e^{sA} \xi + \int _0^s e^{(s-r)A} f(r) {d} r + \int _0^s e^{(s-r)A} \Phi (r) {d} W_Q(r), \qquad s\in [0,T]. $$

The same is true for the mild solution of (1.43) (which is also a strong solution).

To prove (1.44), we write, for $s\in [0,T]$,

$$\begin{aligned}&X^n(s) - X(s) = \left( e^{sA_n} - e^{sA} \right) \xi + \int _0^s \left( e^{(s-r)A_n} - e^{(s-r)A} \right) f(r) {d} r \\&\qquad \qquad + \int _0^s \left( e^{(s-r)A_n} - e^{(s-r)A} \right) \Phi (r) {d} W_Q(r) =: I^n_1(s) + I^n_2(s) + I^n_3(s). \end{aligned}$$

It is enough to show that $\lim _{n\rightarrow \infty } \mathbb {E} \left[ \sup _{s\in [0,T]} |I^n_i(s)|^{p} \right] =0$ for $i\in \{1,2,3 \}$. For $i=3$ this follows from (1.21). To prove it for $i=2$, we observe that (B.15) implies that if

$$ \psi _n(r):=\sup _{s\in [r, T]}\left| \left( e^{(s-r)A_n} - e^{(s-r)A} \right) f(r)\right| , $$

then $\psi _n(r) \xrightarrow {n\rightarrow \infty } 0$ a.e. on $\Omega$. Moreover, thanks to (B.14), there exists a $C_1$ such that, for all $t\in [0,T]$ and all n, $\left\| e^{tA_n} \right\| \le C_1$, so $\psi _n(r) \le 2C_1 |f(r)|$ for all n. Since $\int _t^T |f(r)| {d} r < +\infty$ for almost every $\omega \in \Omega$, by the Lebesgue dominated convergence theorem we have

$$\begin{aligned}&\sup _{s\in [0,T]} \left| \int _0^s \left| \left( e^{(s-r)A_n} - e^{(s-r)A} \right) f(r) \right| {d} r \right| ^p\\&\qquad \qquad \qquad \qquad \quad \le \sup _{s\in [0,T]} \left| \int _0^s \psi _n(r) {d} r \right| ^p \le \left| \int _0^T \psi _n(r) {d} r \right| ^p \xrightarrow {n\rightarrow \infty } 0 \end{aligned}$$

for a.e. $\omega \in \Omega$. Now observe that

$$\begin{aligned}&\sup _{s\in [0,T]} \left| \int _0^s \left| \left( e^{(s-r)A_n} - e^{(s-r)A} \right) f(r) \right| {d} r \right| ^p\\&\qquad \qquad \qquad \qquad \le \sup _{s\in [0,T]} \int _0^s (2 C_1)^p \left| f(r) \right| ^p {d} r \le \int _0^T (2 C_1)^p \left| f(r) \right| ^p {d} r, \end{aligned}$$

and the last expression is integrable (on $\Omega$), since $f\in M^p_\mu (0,T;H)$. Therefore we can apply the Lebesgue dominated convergence theorem, obtaining $\lim _{n\rightarrow \infty } \mathbb {E} \left[ \sup _{s\in [0,T]} |I^n_2(s)|^{p} \right] =0$. The claim for $i=1$ follows again from (B.15) and the Lebesgue dominated convergence theorem.

Estimates (1.45) are easy consequences of (B.14) and the assumptions on $\xi , f,\Phi$.    $\square$

1.4.4 Uniqueness in Law

Definition 1.133

(Finite-dimensional distributions) Let $T>0$ and $t\in [0,T)$. Consider a measurable space $(\Omega , \mathscr {F})$, two probability spaces $(\Omega _i, \mathscr {F}_i, \mathbb {P}_i)$ for $i=1,2$, and two processes $\left\{ X_i(s) \right\} _{s\in [t, T]} :(\Omega _i, \mathscr {F}_i, \mathbb {P}_i) \rightarrow (\Omega , \mathscr {F})$. We say that $X_1(\cdot )$ and $X_2(\cdot )$ have the same finite-dimensional distributions on $D\subset [t, T]$ if for any $t\le t_1< t_2< ... <t_n \le T, t_i \in D$ and $A\in \underbrace{\mathscr {F}\otimes \mathscr {F}\otimes ... \otimes \mathscr {F}}_{n \text { times}}$, we have

$$ \mathbb {P}_1 \left\{ \omega _1 \; : \; (X_1(t_1), .. X_1(t_n)) (\omega _1) \in A \right\} = \mathbb {P}_2 \left\{ \omega _2 \; : \; (X_2(t_1), .. X_2(t_n)) (\omega _2) \in A \right\} . $$

In this case we write $\mathcal {L}_{\mathbb {P}_1} (X_1(\cdot )) = \mathcal {L}_{\mathbb {P}_2} (X_2(\cdot ))$ on D. Often we will just write $\mathcal {L}_{\mathbb {P}_1} (X_1(\cdot )) = \mathcal {L}_{\mathbb {P}_2} (X_2(\cdot ))$, which should be understood as meaning that the finite-dimensional distributions are the same on some set of full measure.

Theorem 1.134

Let H be a separable Hilbert space. Let $(\Omega _i, \mathscr {F}_i, \mathbb {P}_i)$ for $i= 1,2$ be two complete probability spaces, and $({\tilde{\Omega }}, {\tilde{\mathscr {F}}})$ be a measurable space. Let $\xi _i :\Omega _i \rightarrow {\tilde{\Omega }}, i=1,2$ be two random variables, and $f_i :[t,T] \times \Omega _i \rightarrow H, i=1,2$, be two processes satisfying

$$ \mathbb {P}_1 \left( \int _t^T |f_1(s)| {d} s<+\infty \right) =\mathbb {P}_2 \left( \int _t^T |f_2(s)| {d} s<+\infty \right) =1 $$

and, for some subset $D\subset [t, T]$ of full measure,

$$ \mathcal {L}_{\mathbb {P}_1} \left( f_1(\cdot ), \xi _1 \right) = \mathcal {L}_{\mathbb {P}_2} \left( f_2(\cdot ), \xi _2 \right) \; \text { on }D. $$

Then

$$\begin{aligned} \mathcal {L}_{\mathbb {P}_1} \left( \int _t^\cdot f_1(s) {d} s, \xi _1 \right) = \mathcal {L}_{\mathbb {P}_2} \left( \int _t^\cdot f_2(s) {d} s, \xi _2 \right) \; \text { on }[t, T]. \end{aligned}$$

(1.46)

Proof

See [471] Theorem 8.3, where the theorem was proved for a more general case of Banach space-valued processes. $\square$

Theorem 1.135

Let $\left( \Omega _1 ,\mathscr {F}_1,\mathscr {F}_{s}^{1,t},\mathbb {P}_1,W_{Q, 1} \right)$ and $\left( \Omega _2 ,\mathscr {F}_2,\mathscr {F}_{s}^{2,t},\mathbb {P}_2,W_{Q, 2}\right)$ be two generalized reference probability spaces. Let $\Phi _i:[t, T] \times \Omega _i \rightarrow \mathcal {L}_2(\Xi _0,H)$, $i=1,2$, be two $\mathscr {F}_{s}^{i, t}$-progressively measurable processes satisfying

$$ \mathbb {P}_1 \left( \int _t^T \Vert \Phi _1(s)\Vert _{\mathcal {L}_2(\Xi _0,H)}^2 {d} s<+\infty \right) =\mathbb {P}_2 \left( \int _t^T \Vert \Phi _2(s)\Vert _{\mathcal {L}_2(\Xi _0,H)}^2 {d} s<+\infty \right) =1. $$

Let $({\tilde{\Omega }},{\tilde{\mathscr {F}}})$ be a measurable space and $\xi _i :\Omega _i \rightarrow {\tilde{\Omega }}, i=1,2$, be two random variables. Assume that, for some subset $D\subset [t, T]$ of full measure,

$$ \mathcal {L}_{\mathbb {P}_1} \left( \Phi _1(\cdot ), W_{Q, 1} (\cdot ),\xi _1 \right) = \mathcal {L}_{\mathbb {P}_2} \left( \Phi _2(\cdot ), W_{Q, 2} (\cdot ),\xi _2 \right) \; \text {on }D. $$

Then

$$\begin{aligned} \mathcal {L}_{\mathbb {P}_1} \left( \int _t^\cdot \Phi _1(s) {d} W_{Q, 1} (s), \xi _1 \right) = \mathcal {L}_{\mathbb {P}_2} \left( \int _t^\cdot \Phi _2(s) {d} W_{Q, 2} (s), \xi _2 \right) \; \text { on }[t, T]. \end{aligned}$$

(1.47)

Proof

See [471] Theorem 8.6. $\square$

Consider now an operator A and mappings $b,\sigma$ satisfying Hypothesis 1.125, and $x\in H$. Let $\left( \Omega _1,\mathscr {F}_1,\mathscr {F}_{s}^{1,t},\mathbb {P}_1,W_{Q, 1} \right)$ and $\left( \Omega _2 ,\mathscr {F}_2,\mathscr {F}_{s}^{2,t},\mathbb {P}_2,W_{Q, 2}\right)$ be as in Theorem 1.135. For $i=1,2$ consider an $\mathscr {F}_{s}^{i, t}$-progressively measurable process $a_i:[t, T] \times \Omega _i \rightarrow \Lambda$.

Let $p> 2$ and let $\zeta _i\in L^p(\Omega _i,\mathscr {F}_t^{i, t}, \mathbb {P}_i)$, $i=1,2$. Denote by $\mathcal {H}_{p, i}$ the Banach space of all $\mathscr {F}_{s}^{i, t}$-progressively measurable processes $Z_i:[t, T]\times \Omega _i \rightarrow H$ such that

$$ \left( \sup _{s\in [t, T]} \mathbb {E}_i |Z_i(s)|^p \right) ^{1/p} <+\infty . $$

Let $\mathcal {K}_i:\mathcal {H}_{p,i} \rightarrow \mathcal {H}_{p, i}$ be the continuous map (see [180], p. 189) defined as

$$\begin{aligned}&\mathcal {K}_i(Z_i(\cdot ))(s) := e^{(s-t)A} \zeta _i + \int _t^s e^{(s-r) A} b(r,Z_i(r), a_i(r)) {d} r \nonumber \\&\qquad \qquad \quad \qquad \qquad \qquad + \int _t^s e^{(s-r) A} \sigma (r,Z_i(r),a_i(r)) {d} W_{Q, i}(r).\quad \end{aligned}$$

(1.48)

Lemma 1.136

Consider the setting described above, and let $\theta _i:[t, T] \times \Omega _i \rightarrow H, i=1,2$, be stochastic processes. If

$$ \mathcal {L}_{\mathbb {P}_1}(Z_1(\cdot ), a_1(\cdot ),W_{Q, 1}(\cdot ),\theta _1(\cdot ),\zeta _1)=\mathcal {L}_{\mathbb {P}_2}(Z_2(\cdot ), a_2(\cdot ),W_{Q, 2}(\cdot ),\theta _2(\cdot ),\zeta _2) $$

on some subset $D\subset [t, T]$ of full measure, then

$$\begin{aligned}&\mathcal {L}_{\mathbb {P}_1}(\mathcal {K}_1(Z_1(\cdot ))(\cdot ), a_1(\cdot ),W_{Q, 1}(\cdot ),\theta _1(\cdot ),\zeta _1) \\&\qquad \qquad \qquad \qquad \qquad =\mathcal {L}_{\mathbb {P}_2}(\mathcal {K}_2(Z_2(\cdot ))(\cdot ), a_2(\cdot ),W_{Q, 2}(\cdot ),\theta _2(\cdot ),\zeta _2) \,\,\,\, \text {on }D. \end{aligned}$$

Proof

Observe that, since we only have to check the finite-dimensional distributions, the claims of Theorems 1.134 and 1.135 hold even if $\xi _1$ and $\xi _2$ are stochastic processes, with (1.46) and (1.47) then being true on some set of full measure. Let us choose a partition $(t_1, .., t_n)$, with $t\le t_1< t_2< ... <t_n \le T, t_k\in D, k=1,..., n$. We need to show that

$$\begin{aligned}&\mathcal {L}_{\mathbb {P}_1}(\mathcal {K}_1(Z_1(\cdot ))(t_k), a_1(t_k),W_{Q, 1}(t_k), \theta _1(t_k),\zeta _1:k=1,..., n) \nonumber \\&\qquad \qquad =\mathcal {L}_{\mathbb {P}_2}(\mathcal {K}_2(Z_2(\cdot ))(t_k), a_2(t_k),W_{Q, 1}(t_k), \theta _2(t_k),\zeta _2:k=1,..., n). \end{aligned}$$

(1.49)

Define $f^i(r): = \mathbf{1}_{[t, t_1]}(r) e^{(t_1- r)A} b(r, Z_i(r), a_i(r))$ and $\Phi ^i(r): = \mathbf{1}_{[t, t_1]} (r) e^{(t_1 -r)A} \sigma (r, Z_i(r), a_i(r)), i=1,2$. We have

$$\begin{aligned}&\mathcal {L}_{\mathbb {P}_1}(f^1(\cdot ),\Phi _1(\cdot ), Z_1(\cdot ), a_1(\cdot ),W_{Q, 1}(\cdot ),\theta _1(\cdot ),\zeta _1) \\&\qquad \qquad \qquad \qquad \quad =\mathcal {L}_{\mathbb {P}_2}(f^2(\cdot ),\Phi _2(\cdot ), Z_2(\cdot ), a_2(\cdot ),W_{Q, 2}(\cdot ),\theta _2(\cdot ),\zeta _2) \;\; \text {on }D, \end{aligned}$$

and thus, by Theorem 1.134 applied with

$$ \xi _1(\cdot ) = (f^1(\cdot ),\Phi ^1(\cdot ), Z_1(\cdot ), a_1(\cdot ), W_{Q, 1} (\cdot ), \theta _1(\cdot ),\zeta _1), $$

$$ \xi _2(\cdot ) = (f^2(\cdot ),\Phi ^2(\cdot ), Z_2(\cdot ), a_2(\cdot ), W_{Q, 2} (\cdot ),\theta _2(\cdot ),\zeta _2), $$

$$\begin{aligned}&\mathcal {L}_{\mathbb {P}_1}\left( \int _t^{t_1} f^1(s) {d} s, f^1(\cdot ),\Phi ^1(\cdot ), Z_1(\cdot ), a_1(\cdot ),W_{Q, 1}(\cdot ), \theta _1(\cdot ),\zeta _1\right) \\&\qquad \quad =\mathcal {L}_{\mathbb {P}_2}\left( \int _t^{t_1} f^2(s) {d} s, f^2(\cdot ),\Phi ^2(\cdot ), Z_2(\cdot ), a_2(\cdot ),W_{Q, 2}(\cdot ), \theta _2(\cdot ),\zeta _2\right) \;\; \text {on }D. \end{aligned}$$

Now, applying Theorem 1.135 with

$$ \xi _1(\cdot ) = \left( \int _t^{t_1} f^1(s) {d} s, f^1(\cdot ),\Phi ^1(\cdot ), Z_1(\cdot ), a_1(\cdot ), W_{Q, 1} (\cdot ), \theta _1(\cdot ),\zeta _1\right) , $$

$$ \xi _2(\cdot ) = \left( \int _t^{t_1} f^2(s) {d} s, f^2(\cdot ),\Phi ^2(\cdot ), Z_2(\cdot ), a_2(\cdot ), W_{Q, 2} (\cdot ),\theta _2(\cdot ),\zeta _2\right) , $$

we obtain

$$\begin{aligned}&\quad \mathcal {L}_{\mathbb {P}_1}\left( \int _t^{t_1} f^1(s) {d} s,\int _t^{t_1} \Phi ^1(s){d} W_{Q, 1}(s), f^1(\cdot ),\Phi ^1(\cdot ), Z_1(\cdot ), a_1(\cdot ),W_{Q, 1}(\cdot ),\theta _1(\cdot ),\zeta _1\right) \\&=\mathcal {L}_{\mathbb {P}_2}\left( \int _t^{t_1} f^2(s) {d} s,\int _t^{t_1} \Phi ^2(s){d} W_{Q, 2}(s), f^2(\cdot ),\Phi ^2(\cdot ), Z_2(\cdot ), a_2(\cdot ),W_{Q, 2}(\cdot ), \theta _2(\cdot ),\zeta _2\right) \end{aligned}$$

on D (we recall that the stochastic convolution terms in (1.48) and the stochastic integrals above have continuous trajectories a.e.). In particular, this implies that

$$\begin{aligned}&\mathcal {L}_{\mathbb {P}_1}(\mathcal {K}_1(Z_1(\cdot ))(t_1), f^1(\cdot ),\Phi ^1(\cdot ), Z_1(\cdot ) , a_1(\cdot ), W_{Q, 1}(\cdot ), \theta _1(\cdot ),\zeta _1) \\&\qquad \qquad =\mathcal {L}_{\mathbb {P}_2}(\mathcal {K}_2(Z_2(\cdot ))(t_1), f^2(\cdot ),\Phi ^2(\cdot ), Z_2(\cdot ) , a_2(\cdot ), W_{Q, 2}(\cdot ), \theta _2(\cdot ),\zeta _2) \;\; \text {on }D. \end{aligned}$$

We now repeat the above procedure for $t_2,..., t_n$ which will yield (1.49) as its consequence. $\square$

Proposition 1.137

Let the operator A and the mappings $b,\sigma$ satisfy Hypothesis 1.125. Let $\left( \Omega _1 ,\mathscr {F}_1, \mathscr {F}_{s}^{1,t},\mathbb {P}_1,W_{Q, 1} \right)$ and $\left( \Omega _2 ,\mathscr {F}_2,\mathscr {F}_{s}^{2,t},\mathbb {P}_2,W_{Q, 2}\right)$ be two generalized reference probability spaces. Let $a_i:[t, T] \times \Omega _i \rightarrow \Lambda , i=1,2$ be an $\mathscr {F}_{s}^{i, t}$-progressively measurable process, and let $\zeta _i\in L^p(\Omega _i,\mathscr {F}_t^{i, t}, \mathbb {P}_i), i=1,2, p>2$. Let $\mathcal {L}_{\mathbb {P}_1} (a_1(\cdot ), W_{Q, 1}(\cdot ),\zeta _1)= \mathcal {L}_{\mathbb {P}_2} (a_2(\cdot ),W_{Q, 1}(\cdot ),\zeta _2)$ on some subset $D\subset [0,T]$ of full measure. Denote by $X_i(\cdot ), i=1,2,$ the unique mild solution of

$$\begin{aligned} \left\{ \begin{array}{l} {d} X_i(s) = \left( A X_i(s) + b(s, X_i(s), a_i(s)) \right) {d} s + \sigma (s, X_i(s), a_i(s)) {d} W_{Q, i}(s)\\ X_i(t)=\zeta _i \end{array} \right. \end{aligned}$$

(1.50)

on [t, T]. Then $\mathcal {L}_{\mathbb {P}_1}(X_1(\cdot ), a_1(\cdot )) = \mathcal {L}_{\mathbb {P}_2}(X_2(\cdot ), a_2(\cdot ))$ on D.

Proof

It is known (see [180], proof of Theorem 7.2, pp. 188–193) that the map $\mathcal {K}_i$ is a contraction in $\mathcal {H}_{p, i}$ if [t, T] is small enough. Thus if we divide [t, T] into such small intervals $[t, T_1],...[T_k, T]$, $X_i(\cdot )$ on $[t, T_1]$ is obtained as the limit in $\mathcal {H}_{p, i}$ (restricted to $[t, T_1]$) of the iterates $(\mathcal {K}_i^n (x))(\cdot )$. Therefore, using Lemma 1.136 and passing to the limit as $n\rightarrow +\infty$ we obtain

$$ \mathcal {L}_{\mathbb {P}_1} (\mathbf{1}_{[t, T_1]}(\cdot )X_1(\cdot ), a_1(\cdot ), W_{Q, 1}(\cdot ))= \mathcal {L}_{\mathbb {P}_2} ( \mathbf{1}_{[t, T_1]}(\cdot )X_2(\cdot ), a_2(\cdot ),W_{Q, 1}(\cdot ))\,\,\,\text {on } D. $$

Without loss of generality we may assume that $T_1\in D$. The solutions on $[T_1,T_2]$ are obtained as the limits in $\mathcal {H}_{p, i}$ (restricted to $[T_1,T_2]$) of the iterates $(\mathcal {K}_i^n (X_i(T_1)))(\cdot )$, where now

$$\begin{aligned}&\mathcal {K}_i(Z_i(\cdot ))(s) := e^{(s-T_1)A}X_i(T_1) + \int _{T_1}^s e^{(s-r) A} b(r, Z_i(r), a_i(r)) {d} r \\&\quad \qquad \qquad \qquad \qquad \qquad \qquad + \int _{T_1}^s e^{(s-r) A} \sigma (r,Z_i(r),a_i(r)) {d} W_{Q, i}(r). \end{aligned}$$

Thus, again using Lemma 1.136 and passing to the limit as $n\rightarrow +\infty$, it follows that

$$ \mathcal {L}_{\mathbb {P}_1} (\mathbf{1}_{[t, T_2]}(\cdot )X_1(\cdot ), a_1(\cdot ), W_{Q, 1}(\cdot ))= \mathcal {L}_{\mathbb {P}_2} ( \mathbf{1}_{[t, T_2]}(\cdot )X_2(\cdot ), a_2(\cdot ),W_{Q, 1}(\cdot ))\,\,\,\text {on } D. $$

We repeat the procedure to obtain the required claim. $\square$

1.5 Further Existence and Uniqueness Results in Special Cases

Throughout this section $T>0$ is a fixed constant, $H,\Xi , Q$, and the generalized reference probability space $\mu =(\Omega , \mathscr {F}, \{\mathscr {F}_s\}_{s\in [0,T]}, \mathbb {P}, W_Q)$ are as in Sect. 1.3 (with $t=0$), A is the infinitesimal generator of a $C_0$-semigroup on H, and $\Lambda$ is a Polish space. As in previous sections we will only consider equations on the interval [0, T], however all results would be the same if instead of [0, T] we took an interval [t, T], for $0\le t<T$.

1.5.1 SDEs Coming from Boundary Control Problems

In this section we study SDEs that include equations coming from optimal control problems with boundary control and noise. To see how they arise the reader can look at the examples in Sects. 2.6.2 and 2.6.3, and Appendix C. We consider the following SDE in H:

$$\begin{aligned} \left\{ \begin{array}{l} {d} X(s) = \left( A X(s) + b(s, X(s), a(s)) + (\lambda I- A)^{\beta } G a_b(s) \right) {d} s \\ \quad \qquad \qquad \qquad \qquad + \sigma (s, X(s), a(s)) {d} W_Q(s), \qquad s \in (0,T] \\ X(0)=\xi . \end{array} \right. \end{aligned}$$

(1.51)

Hypothesis 1.138

• (i) A generates an analytic semigroup $e^{tA}$ for $t \ge 0$ and $\lambda$ is a real constant such that $(\lambda I- A)^{-1} \in \mathcal {L}(H)$.

• (ii) $a:[0,T] \times \Omega \rightarrow \Lambda$ is progressively measurable, $b(\cdot ,\cdot ,\cdot )$ satisfies (1.31) and (1.33).

• (iii) $\Lambda _b$ is a Hilbert space and $a_b(\cdot ):[0,T]\times \Omega \rightarrow \Lambda _b$ is progressively measurable.

• (iv) $G\in \mathcal {L}(\Lambda _b, H)$.

• (v) $\beta \in \left[ 0, 1 \right)$.

• (vi) $\gamma$ is a constant belonging to the interval $\left[ 0, \frac{1}{2} \right)$, $\sigma$ is a mapping such that $(\lambda I- A)^{-{\gamma }} \sigma : [0,T] \times H \times \Lambda _b \rightarrow \mathcal {L}_{2} (\Xi _0, H)$ is continuous. There exists a constant $C >0$ such that

  $$ \Vert (\lambda I- A)^{-{\gamma }} \sigma (s,x, a )\Vert _{\mathcal {L}_2(\Xi _0, H)} \le C(1+|x|) $$

  for all $s\in [0,T], \; x \in H, \; a \in \Lambda$ and

  $$ \Vert (\lambda I- A)^{-{\gamma }} [\sigma (s, x_{1},a )- \sigma (s, x_{2}, a )]\Vert _{\mathcal {L}_2(\Xi _0,H)} \le C|x_{1}-x_{2}| $$

  for all $s\in [0,T], \; x_{1}, x_{2} \in H, \; a \in \Lambda$.

Remark 1.139

Part (i) of Hypothesis 1.138 implies, thanks to (B.18), that for every $\theta \ge 0$ there exists an $M_{\theta }>0$ such that

$$\begin{aligned} |(\lambda I- A)^{\theta }e^{tA}x|\le \frac{M_{\theta }}{t^{\theta }}|x|,\;\; \text{ for } \text{ every }\; t\in (0,T],\; x\in H. \end{aligned}$$

(1.52)

$\blacksquare$

Following Remark 1.120, the definition of a mild solution of (1.51) is given by Definition 1.119 in which the term

$$ \int _0^s e^{(s-r)A} (\lambda I- A)^{\beta }G a_b(r) {d} r $$

is interpreted as

$$ \int _0^s (\lambda I- A)^{\beta }e^{(s-r)A} G a_b(r) {d} r, $$

and the term

$$ \int _{0}^{s}e^{(s-r)A} \sigma (r, X(r), a(r)) dW_Q(r) $$

as

$$ \int _{0}^{s}(\lambda I- A)^{\gamma }e^{(s-r)A} (\lambda I- A)^{-\gamma } \sigma (r, X(r), a(r))dW_Q(r). $$

This is natural since $(\lambda I- A)^{\beta }e^{(s-r)A}$ is an extension of $e^{(s-r)A} (\lambda I- A)^{\beta }$ and $(\lambda I- A)^{\gamma }e^{(s-r)A} (\lambda I- A)^{-\gamma }=e^{(s-r)A}$.

Remark 1.140

SDEs of type (1.51) appear most frequently in optimal control problems of parabolic equations on a domain $\mathcal {O}\subset \mathbb {R}^n$ with boundary control/noise, see Sect. 2.6.2. More precisely, the cases $\beta \in \left( \frac{3}{4} , 1 \right)$ and $\beta \in \left( \frac{1}{4} , \frac{1}{2} \right)$ are related respectively to the Dirichlet and Neumann boundary control problems when one takes $\Lambda _b = L^2(\partial \mathcal {O})$ (or some subset of it) and $H = L^2(\mathcal {O})$. $\gamma \in \left( \frac{1}{4} , \frac{1}{2} \right)$ arises when one treats problems with boundary noise of Neumann type where again $\Lambda _b = L^2(\partial \mathcal {O})$ and $H = L^2(\mathcal {O})$. $\gamma , \beta \in \left( \frac{1}{2} - {\varepsilon } , \frac{1}{2} \right)$ arise in some specific Dirichlet boundary control/noise problems when one considers $\Lambda _b = L^2(\partial \mathcal {O})$ and a suitable weighted $L^2$ space as H. $\blacksquare$

Theorem 1.141

Assume that Hypothesis 1.138 holds, $p\ge 2$, and let $\alpha := \frac{1}{2} - \gamma$. Suppose that

$$\begin{aligned} p>\frac{1}{\alpha } \end{aligned}$$

(1.53)

and $a_b(\cdot ) \in M^{q}_\mu (0,T; \Lambda _b)$ for some $q\ge p$, $q>\frac{1}{1-\beta }$. Then, for every initial condition $\xi \in L^{2}(\Omega , \mathscr {F}_0, \mathbb {P})$, there exists a unique mild solution $X(\cdot )=X(\cdot ;0,\xi , a(\cdot ), a_b(\cdot ))$ of (1.51) in $\mathcal {H}_{2}(0,T;H)$ with continuous trajectories $\mathbb {P}$-a.s. If there exists a constant $C >0$ such that

$$\begin{aligned} \Vert (\lambda I- A)^{-{\gamma }} \sigma (s,x, a )\Vert _{\mathcal {L}_2(\Xi _0, H)} \le C \end{aligned}$$

(1.54)

for all $s\in [0,T]$, $x \in H$, $a \in \Lambda$, then the solution has continuous trajectories $\mathbb {P}$-a.s. without the restriction $p>\frac{1}{\alpha }$. If $\xi \in L^{p}(\Omega , \mathscr {F}_0, \mathbb {P})$ then $X(\cdot )\in \mathcal {H}_{p}(0,T;H)$ and there exists a constant $C_{T, p}$ independent of $\xi$ such that

$$\begin{aligned} \sup _{s \in [0,T]} \mathbb {E}|X(s)|^{p} \le C_{T, p} (1 + \mathbb {E} |\xi |^{p}). \end{aligned}$$

(1.55)

Proof

Assume first that $\xi \in L^{p}(\Omega , \mathscr {F}_0, \mathbb {P})$ where $p\ge 2$ without the restriction (1.53). Similarly to the proof of Theorem 1.127, we will show that for some $T_0\in (0,T]$ the map

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \mathcal {K} :\mathcal {H}_{p}(0,T_0) \rightarrow \mathcal {H}_{p}(0,T_0), \\ \displaystyle \mathcal {K}(Y)(s)= e^{sA}\xi + \int _0^s e^{(s-r)A} b(r,Y(r), a(r)) {d} r + \int _0^s (\lambda I- A)^{\beta }e^{(s-r)A} G a_b(r) {d} r \\ \displaystyle \qquad \qquad \quad \qquad +\int _{0}^{s}(\lambda I- A)^{\gamma }e^{(s-r)A} (\lambda I- A)^{-\gamma } \sigma (r, Y(r), a(r)){d} W_Q(r) \end{array} \right. \end{aligned}$$

(1.56)

is well defined and is a contraction. The only difference between our case here and that considered in Theorem 1.127 is the last two terms in (1.56).

First we prove that $\mathcal {K}$ maps $\mathcal {H}_{p}(0,T_0)$ into $\mathcal {H}_{p}(0,T_0)$. We only show how to deal with the non-standard terms. For the third term in (1.56) we can argue as follows. If $M_\beta$ is the constant from (1.52) for $\theta =\beta$, using (1.52), Hölder and Jensen’s inequalities, and $q\ge p$, $q>\frac{1}{1-\beta }$, we obtain

$$\begin{aligned}&\sup _{s\in [0,T_0]} \mathbb {E}\left| \int _{0}^{s}(\lambda I- A)^\beta e^{(s-r)A} Ga_{b} (r)dr \right| ^{p} \nonumber \\&\qquad \qquad \qquad \le \sup _{s\in [0,T_0]} M_\beta ^{p} \Vert G\Vert ^{p}\mathbb {E}\left( \int _{0}^{s} \frac{1}{(s-r)^\beta }|a_b(r)|{d} r\right) ^p \nonumber \\&\qquad \quad \le M_\beta ^{p} \Vert G\Vert ^{p} \left( \int _{0}^{T_0} \frac{1}{(T_0-r)^\frac{\beta q}{q-1}}{d} r \right) ^{\frac{p(q-1)}{q}} \mathbb {E} \left[ \int _0^{T_0} |a_b(r)|^{q} {d} r \right] ^{\frac{p}{q}} \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \le C_1\left( \mathbb {E} \left[ \int _0^{T_0} |a_b(r)|^{q} {d} r \right] \right) ^{\frac{p}{q}}<+\infty . \end{aligned}$$

(1.57)

As regards the stochastic integral term, using Theorem 1.111, (1.52), and Hypothesis 1.138-(vi), we estimate

$$\begin{aligned}&\sup _{s\in [0,T_0]} {\mathbb {E}} \left| \int _{0}^{s}(\lambda I- A)^\gamma e^{(s-r)A} (\lambda I- A)^{-\gamma } \sigma (r, Y(r), a(r))dW_Q(r) \right| ^{p} \nonumber \\&\quad \le \sup _{s\in [0,T_0]}C_1{\mathbb {E}}\left| \int _{0}^{s} \frac{1}{(s-r)^{{2}\gamma }} \Vert (\lambda I- A)^{-\gamma }\sigma (r, Y(r), a(r))\Vert ^{{2}}_{{\mathcal L}_{2}(\Xi _0,H)} {d} r\right| ^{\frac{p}{2}} \nonumber \\&\quad \le \sup _{s\in [0,T_0]} C_2\left( \int _{0}^{T_0} \frac{1}{(T_0-r)^{{2}\gamma }} {d} r \right) ^{\frac{p}{2}-1} \int _{0}^{s} \frac{1}{(s-r)^{{2}\gamma }} {\mathbb {E}} [(1+ |Y(r)|)^{{p}}] {d} r \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \le C_3 \left( 1 + |Y|^{p}_{\mathcal {H}_{p}(0,T_0)} \right) \end{aligned}$$

(1.58)

for some constant $C_3$. Progressive measurability of all the terms appearing in the definition of $\mathcal {K}(Y)(\cdot )$ can be proved by using estimates similar to (1.57) and (1.58) and arguing as in Remark 1.123.

Regarding the proof that, for $T_0$ small enough, $\mathcal {K}$ is a contraction, the only non-standard term to check is the stochastic convolution term, since the third term in (1.56) does not depend on X. Arguing as before we have that for $X, Y \in \mathcal {H}_{p}(0,T_0)$, thanks to Theorem 1.111, (1.52), Hypothesis 1.138-(vi), and Jensen’s inequality,

$$\begin{aligned}&\quad \sup _{s\in [0,T_0]} {\mathbb {E}} \left| \int _{0}^{s} (\lambda I- A)^\gamma e^{(s-r)A} (\lambda I- A)^{-\gamma } \left[ \sigma (r,X(r), a(r)) - \sigma (r, Y(r), a(r)) \right] dW_Q(r) \right| ^{p} \nonumber \\&\le \sup _{s\in [0,T_0]}C_1{\mathbb {E}} \left( \int _{0}^{s}\frac{1}{(s-r)^{2 \gamma }} \left\| (\lambda I- A)^{-\gamma } \left[ \sigma (r,X(r), a(r)) - \sigma (r, Y(r), a(r)) \right] \right\| ^{{2}}_{{\mathcal L}_{2}(\Xi _0,H)} {d} r \right) ^{\frac{p}{2}} \nonumber \\&\qquad \qquad \qquad \le \sup _{s\in [0,T_0]}C_2{\mathbb {E}}\left( \int _{0}^{s} \frac{1}{(s-r)^{2 \gamma }} | X(r) - Y(r) |^{2} {d} r \right) ^{\frac{p}{2}} \nonumber \\&\qquad \le \sup _{s\in [0,T_0]} C_2\left( \int _{0}^{T_0} \frac{1}{(T_0-r)^{{2}\gamma }} {d} r \right) ^{\frac{p}{2}-1} \int _{0}^{s} \frac{1}{(s-r)^{{2}\gamma }} {\mathbb {E}} [ | X(r) - Y(r) |^{{p}}] {d} r \nonumber \\&\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \le \omega (T_0) | X - Y |_{\mathcal {H}_{p}(0,T_0)}^p, \end{aligned}$$

(1.59)

where $\omega (r)\xrightarrow {r\rightarrow 0^+}0$. So for $T_0$ small enough (which is independent of the initial condition) we can apply the Banach fixed point theorem in $\mathcal {H}_{p}(0,T_0)$ as in the proof of Theorem 1.127 (see also the proof of [180], Theorem 7.2, p. 188). The process can now be reapplied on intervals $[T_0, 2T_0],...,[kT_0,T]$, where $k=[T/T_0]$, to obtain the existence of a unique mild solution in $\mathcal {H}_{p}(0,T)$ in the sense of the integral equality being satisfied for a.e. $s\in [0,T]$.

Estimate (1.55) follows from similar arguments using the growth assumptions on $b,\sigma$ in Hypothesis 1.138 and Gronwall’s lemma in the form given in Proposition D.30.

We will now prove the continuity of the trajectories if condition (1.53) is satisfied. We will only prove the continuity of the stochastic convolution term in (1.56) since the continuity of the other terms is easier to show. In particular, the continuity of the trajectories of the third term in (1.56) follows from Lemma 1.115-(ii).

Let now $p>\frac{1}{\alpha }$. Hence there is an $0<\alpha '<\alpha$ such that $p>\frac{1}{\alpha '}$. Then, for $r\in [t, T]$, using (1.52), (1.55), Hypothesis 1.138-(vi), and Jensen’s inequality

$$\begin{aligned}&\quad \mathbb {E} \left( \int _0^r (r-h)^{-2\alpha '} \left\| (\lambda I- A)^\gamma e^{(r-h)A} (\lambda I- A)^{-\gamma } \sigma (h, X(h), a(h)) \right\| ^2_{\mathcal {L}_2(\Xi _0, H)} {d} h \right) ^{\frac{p}{2}} \nonumber \\&\le \mathbb {E} \left( \int _0^r (r-h)^{-2\alpha '} \Vert (\lambda I- A)^\gamma e^{(r-h)A}\Vert ^2_{\mathcal {L}(H)} \left\| (\lambda I- A)^{-\gamma } \sigma (h, X(h), a(h)) \right\| ^2_{\mathcal {L}_2(\Xi _0, H)} {d} s \right) ^{\frac{p}{2}} \nonumber \\&\qquad \qquad \qquad \le C_1\mathbb {E}\left( \int _0^r (r-h)^{-2\alpha '} (r-h)^{-2\gamma }(1+|X(h)|)^2 {d} h \right) ^{\frac{p}{2}} \nonumber \\&\quad \le C_1\left( \int _0^T (T-h)^{-2\alpha '} (T-h)^{-2\gamma } {d} h \right) ^{\frac{p}{2}} \sup _{h\in [0,T]}\mathbb {E}[(1+|X(h)|)^p] =: C_2 < +\infty . \end{aligned}$$

(1.60)

Observe that $C_2$ does not depend on $r\in [0,T]$. This proves (1.25) and thus the claim follows from Proposition 1.116. When (1.54) holds, estimate (1.60) is easier and can be done for any exponent $p'>1/\alpha$ in place of p, and thus (1.25) is always satisfied.

Finally, we need to discuss the continuity of the trajectories if $\xi \in L^{2}(\Omega , \mathscr {F}_0, \mathbb {P})$. We argue as in the proof of Theorem 7.2 of [180]. For $n\ge 1$ we define the random variables

$$ \xi _n=\left\{ \begin{array}{l} \xi \quad \text {if}\,\,|\xi |\le n \\ 0\quad \,\text {if}\,\,|\xi |> n. \end{array} \right. $$

The solutions $X(\cdot ;0,\xi , a(\cdot ), a_b(\cdot ))$ and $X(\cdot ;0,\xi _n, a(\cdot ), a_b(\cdot ))$ on $[0,T_0]$ are obtained as fixed points in $\mathcal {H}_2(0,T_0)$ and $\mathcal {H}_p(0,T_0)$, with p large enough, of the same contraction map (1.56) with the second map having the term $e^{sA}\xi _n$ in place of $e^{sA}\xi$. Therefore both solutions can be obtained as limits of successive iterations starting, say, from processes $e^{sA}\xi$ and $e^{sA}\xi _n$, respectively. It is then easy to see that we have $X(\cdot ;0,\xi , a(\cdot ), a_b(\cdot ))=X(\cdot ;0,\xi _n, a(\cdot ), a_b(\cdot ))$, $\mathbb {P}$-a.s. on $\{\omega :|\xi (\omega )|\le n\}$. However, the solutions $X(\cdot ;0,\xi _n, a(\cdot ), a_b(\cdot ))$ have continuous trajectories. Thus $X(\cdot ;0,\xi , a(\cdot ), a_b(\cdot ))$ has continuous trajectories $\mathbb {P}$-a.s. on $[0,T_0]$ and we can then continue the argument on intervals $[T_0, 2T_0],...$. $\square$

Proposition 1.142

Let the assumptions of Theorem 1.141 be satisfied. Denote the unique mild solution of (1.51) in $\mathcal {H}_{p}(0,T;H)$ by $X(\cdot )=X(\cdot ;0,\xi , a(\cdot ), a_b(\cdot ))$.

• (i) If $\xi ^1=\xi ^2$ $\mathbb {P}$-a.s., $a^1(\cdot )=a^2(\cdot )$ ${d} t \otimes \mathbb {P}$-a.s. $a_b^1(\cdot )=a_b^2(\cdot )$ ${d} t \otimes \mathbb {P}$-a.s., then $\mathbb {P}$-a.s., $X(\cdot ;0,\xi ^1, a^1(\cdot ), a^1_b(\cdot )) = X(\cdot ;0,\xi ^2, a^2(\cdot ), a^2_b(\cdot ))$ on [0, T].

• (ii) Let $\left( \Omega _1 ,\mathscr {F}_1,\mathscr {F}_{s}^{1},\mathbb {P}_1,W_{Q, 1} \right)$ and $\left( \Omega _2 ,\mathscr {F}_2,\mathscr {F}_{s}^{2},\mathbb {P}_2,W_{Q, 2}\right)$ be two generalized reference probability spaces. Let $\zeta _i\in L^p(\Omega _i,\mathscr {F}_0^{i}, \mathbb {P}_i), i=1,2$. Let $(a^i, a^i_b): [0,T] \times \Omega _i \rightarrow \Lambda \times \Lambda _b, i=1,2$ be $\mathscr {F}_{s}^{i}$-progressively measurable processes satisfying the assumptions of Theorem 1.141. Suppose that $\mathcal {L}_{\mathbb {P}_1} (a^1(\cdot ), a^1_b(\cdot ), W_{Q, 1}(\cdot ),\zeta _1)= \mathcal {L}_{\mathbb {P}_2} (a^2(\cdot ), a^2_b(\cdot ),W_{Q, 1}(\cdot ),\zeta _2)$ on some subset $D\subset [t, T]$ of full measure. Then $\mathcal {L}_{\mathbb {P}_1}(X(\cdot ;0,\zeta _1, a^1(\cdot ), a^1_b(\cdot )), a^1(\cdot ), a_b^1(\cdot )) = \mathcal {L}_{\mathbb {P}_2}(X(\cdot ;0,\zeta _2, a^2(\cdot ), a^2_b(\cdot )), a^2(\cdot ), a_b^2(\cdot ))$ on D.

• (iii) The solution of (1.51) is unique in $M^p_\mu (0,T;H)$ as well.

Proof

(i) If $X_i (\cdot ) := X\left( \cdot ;0,\xi ^i,a^{i}(\cdot ), a^i_b(\cdot )\right)$, arguing as in (1.59) and using Hölder’s inequality, we obtain, for $s\in [0,T]$,

$$ \mathbb {E} |X_1(s) - X_2(s) |^p \le C_T \int _0^s {\mathbb {E}}|X_1(r) - X_2(r) |^p {d} r, $$

and the claim follows by using Gronwall’s lemma (Proposition D.29), and the continuity of the trajectories.

(ii) The argument is the same as the one used to prove Lemma 1.136 and Proposition 1.137, since in the current case the solution is also found by iterating the map $\mathcal {K}$.

(iii) The uniqueness in $M^p_\mu (0,T_0;H)$ follows from the estimate in Part (i) above and Proposition D.29. $\square$

1.5.2 Semilinear SDEs with Additive Noise

In this section we give more precise results for some semilinear SDEs with additive noise, i.e. for Eq. (1.28) when the coefficient $\sigma$ is constant and we have possible unboundedness in the drift.

Hypothesis 1.143

• (i) The linear operator A is the generator of a strongly continuous semigroup $\left\{ e^{t A} \, ,\ , t\ge 0\right\}$ in H and, for suitable $M\ge 1$ and $\omega \in {\mathbb {R}}$,

  $$\begin{aligned} |e^{tA}x|\le Me^{\omega t}|x|, \quad \forall t \ge 0,\;x \in H. \end{aligned}$$

  (1.61)

• (ii) $Q \in \mathcal {L}^+(\Xi )$, $\sigma \in {\mathcal L}(\Xi , H)$ and $e^{sA}\sigma {Q}\sigma ^{*}e^{sA^{*}}\in {\mathcal L}_1(H)$ for all $s > 0$. Moreover, for all $t\ge 0$,

  $$ \int _{0}^{t} \mathrm{Tr}\left[ e^{sA}\sigma {Q}\sigma ^{*}e^{sA^{*}}\right] ds <+\infty , $$

  so the symmetric positive operator

  $$\begin{aligned} Q_t:H \rightarrow H, \qquad Q_{t}:=\int _{0}^{t}e^{sA}\sigma {Q}\sigma ^{*}e^{sA^{*}}ds, \end{aligned}$$

  (1.62)

  is of trace class for every $t\ge 0$, i.e.

  $$\begin{aligned} \text{ Tr }\;[Q_{s}]<+ \infty . \end{aligned}$$

  (1.63)

Let $W_Q$ be a Q-Wiener process in $\Xi$ and consider the stochastic convolution process defined, for $s\ge 0$, as follows:

$$\begin{aligned} W^{A}(s) = \int _{0}^{s} e^{(s-r)A} \sigma {d} W_Q(r). \end{aligned}$$

(1.64)

Proposition 1.144

Suppose that Hypothesis 1.143 is satisfied. Then the process $W^A(\cdot )$ defined in (1.64) is a Gaussian process with mean 0 and covariance operator $Q_{s}$, is mean square continuous and $W^A(\cdot )\in \mathcal {H}_{p}^\mu (0,T;H)$ for every $p\ge 2$. Moreover, if there exists a $\gamma >0$ such that

$$\begin{aligned} \int _{0}^{T} s^{-\gamma } \mathrm{Tr} \left[ e^{sA}\sigma {Q} \sigma ^*e^{sA^{*}} \right] {d} s <\infty , \end{aligned}$$

(1.65)

then $W^{A} (\cdot )$ has continuous trajectories⁵ and, for $p>0$,

$$ \mathbb {E}\left[ \sup _{0\le s \le T} |W^{A}(s)|^{p} \right] < + \infty . $$

Proof

See [180] Chap. 5, Theorems 5.2 and 5.11. The fact that $W^A(\cdot )\in \mathcal {H}_{p}^\mu (0,T;H)$ for every $p\ge 2$ follows from Theorem 1.111. The last estimate can be found, for example, as a particular case of Proposition 3.2 in [284]. $\square$

A completely analogous result holds for the stochastic convolution starting at a point $t \ge 0$, i.e.

$$\begin{aligned} W^{A}(t, s) := \int _{t}^{s} e^{(s-r)A}\sigma {d} W_Q(r), \qquad s\ge t. \end{aligned}$$

(1.66)

Let$T>0$. We consider the SDE

$$\begin{aligned} \left\{ \begin{array}{l} {d} X(s) = \left( AX(s) + b(s, X(s)) \right) {d} s + \sigma {d} W_Q(s), \quad s >0 \\ X(0) = \xi . \end{array} \right. \end{aligned}$$

(1.67)

Hypothesis 1.145

$p\ge 1$ and $b(s, x)=b_0 (s,x, a_1(s))+ a_2 (s)$, where:

• (i) The process $a_1(\cdot ):[0,T]\times \Omega \rightarrow \Lambda$ (where $\Lambda$ is a given Polish space) is $\mathscr {F}_s$-progressively measurable. The map $b_{0}:[0,T] \times H \times \Lambda \rightarrow H$ is Borel measurable and there exists a non-negative function $f \in L^{1}(0,T;\mathbb {R})$ such that

  $$ |b_{0}(s,x, a_1)| \le f (s)(1+|x|) \qquad \forall s \in [0,T], \; x \in H\; \text { and } a_1 \in \Lambda . $$
  $$\begin{aligned} |b_{0}(s, x_{1}, a_1) - b_{0}(s, x_{2}, a_1)| \le f (s)&|x_{1}- x_{2}| \\&\forall s \in [0,T], \; x_{1}, x_{2} \in H\; \text { and } a_1 \in \Lambda . \end{aligned}$$

• (ii) The process $a_2(\cdot )$ is such that for all $t>0$, the process $(s,\omega ) {\rightarrow } e^{tA}a_2(s,\omega )$, when interpreted properly, is $\mathscr {F}_s$-progressively measurable on $[0,T]\times \Omega$ with values in H, and

  $$\begin{aligned} |e^{tA}a_2(s,\omega )| \le t^{-\beta }g(s,\omega ) \qquad \forall (t, s,\omega ) \in [0,T]\times [0,T]\times \Omega , \end{aligned}$$

  (1.68)

  for some $\beta \in [0,1)$ and $g\in M^q_\mu (0,T;{\mathbb {R}})$, where $q\ge p$ and $q>\frac{1}{1-\beta }$.

Hypothesis 1.145 covers some cases which are not standard and for which a separate proof of existence and uniqueness of mild solutions of (1.67) is required.

Remark 1.146

Hypothesis 1.145-(ii) is satisfied, for example, when A is the generator of an analytic $C_0$-semigroup and the process $a_2(\cdot )$ is of the form $a_2(s) = (\lambda I- A)^{\beta } a_3(s)$, where $\lambda \in {\mathbb {R}}$ is such that $(\lambda I- A)$ is invertible, $\beta \in (0,1)$, $a_3(\cdot )\in M^q_\mu (0,T;H)$, $q\ge p, q>\frac{1}{1-\beta }$. In such cases the definition of a mild solution of (1.67) is given by Definition 1.119 in which the formal term

$$ \int _0^s e^{(s-r)A} a_2 (r){d} r = \int _0^s e^{(s-r)A} (\lambda I- A)^\beta a_3 (r){d} r $$

appearing in the definition of a mild solution is interpreted as

$$ \int _{0}^{s} (\lambda I- A)^\beta e^{(s-r)A} a_3 (r){d} r. $$

This is natural since $(\lambda I- A)^{\beta }e^{(s-r)A}$ is an extension of $e^{(s-r)A} (\lambda I- A)^{\beta }$.

Another more general case where Hypothesis 1.145-(ii) is satisfied is when $a_2(\cdot ):[0,T]\times \Omega \rightarrow V^*$ is progressively measurable, where $V^*$ denotes the topological dual of $V=D(A^*)$. In such a case the semigroup $e^{tA}$ may be extended, by a standard construction (see e.g. [232]), to the space $V^*$. Denoting this extension still by $e^{tA}$, the process $e^{tA}a_2(\cdot ):[0,T]\times \Omega \rightarrow V^*$ is well defined. If we further assume that $e^{tA}a_2(\cdot )$ takes values in H and satisfies (1.68) for some $\beta \in (0,1)$, then Hypothesis 1.145-(ii) is satisfied. A similar and even slightly more general case has been studied in [232] in a deterministic context. $\blacksquare$

Proposition 1.147

Let $\xi \in L^p (\Omega , \mathscr {F}_0, \mathbb {P})$ and Hypotheses 1.143 and 1.145 be satisfied. Then Eq. (1.67) has a unique mild solution $X(\cdot ;0 ,\xi ) \in \mathcal {H}_{p}^\mu (0,T;H)$. The solution satisfies, for some $C_p(T)>0$ independent of $\xi$,

$$\begin{aligned} \sup _{s\in [0,T]} \mathbb {E} \left[ |X(s;0,\xi )|^{p} \right] \le C_p(T) (1+ \mathbb {E}[|\xi |^{p}]). \end{aligned}$$

(1.69)

Moreover, if $\xi _1,\xi _2\in L^p (\Omega , \mathscr {F}_0, \mathbb {P})$, we have, $\mathbb P$-a.s.,

$$\begin{aligned} | X(s;0,\xi _1) - X(s;0,\xi _2)| \le Me^{\omega T}|\xi _1-\xi _2| e^{Me^{\omega T}\int _{0}^{s}f (r ) {d} r }, \qquad s\in [0,T]. \end{aligned}$$

(1.70)

Finally, if (1.65) also holds for some $\gamma >0$, then the solution $X(\cdot ; 0,\xi )$ has $\mathbb {P}$-a.s. continuous trajectories, and if $\xi =x\in H$ is deterministic we then have

$$\begin{aligned} {\mathbb {E}}(\sup _{s\in [0,T]}|X(s)|^p)\le C_p(T)(1+|x|^p) \end{aligned}$$

(1.71)

for a suitable constant $C_p(T)>0$ independent of x. In particular, if g in Hypothesis 1.145-(ii) is in $M^q_\mu (0,T;{\mathbb {R}})$ for every $q\ge 1$, then estimate (1.69) holds for every $p> 0$ and the same is true for (1.71) if $\xi =x\in H$.

Proof

The proof of existence and uniqueness uses the same techniques employed in the Lipschitz case (Theorem 1.127) but contains a small additional difficulty due the presence of the term $a_2(\cdot )$ and possible singularities in s of the Lipschitz norm of $b_{0}(s,\cdot )$. We will write $\mathcal {H}_{p}(0,T)$ for $\mathcal {H}_{p}^\mu (0,T;H)$. For $Y \in \mathcal {H}_{p}(0,T)$ we set

$$\begin{aligned} \mathcal {K}(Y)(s) = e^{(s-t)A} \xi +\int _{0}^{s} e^{(s-r )A} b_{0}(r ,Y(r ), a_1(r)) {d} r + \int _{0}^{s} e^{(s-r)A} a_2 (r){d} r + W^{A}(s). \end{aligned}$$

(1.72)

$W^A$ belongs to $\mathcal {H}_{p}(0,T)$ thanks to Proposition 1.144. Hypotheses 1.145-(i) and 1.145-(ii) ensure, respectively, that the second and third term in the definition of the map $\mathcal {K}$ belong to $\mathcal {H}_{p}(0,T)$ as well (one can use the same arguments as these to obtain (1.57) when $\beta \in (0,1)$ and Hölder’s inequality if $\beta =0$). So $\mathcal {K}$ maps $\mathcal {H}_{p}(0,T)$ into itself. For $Y_{1}$, $Y_{2} \in \mathcal {H}_{p}(0,T)$, $s \in [0,T]$,

$$ |\mathcal {K} (Y_{1})(s) - \mathcal {K} (Y_{2})(s)| \le Me^{\omega T} \int _{0}^{s}f (r)|Y_{1}(r) - Y_{2}(r)|{d} r, $$

which yields, for $T_{0} \in (0,T]$,

$$\begin{aligned}&| \mathcal {K} (Y_{1}) - \mathcal {K} (Y_{2})|_{\mathcal {H}_p(0,T_{0})}^{p} \le Me^{\omega T} \sup _{s \in [0,T_{0}]} \mathbb {E} \left[ \int _{0}^{s}f (r)|Y_{1}(r) - Y_{2}(r)|{d} r \right] ^{p} \nonumber \\&\qquad \qquad \qquad \le Me^{\omega T} \left[ \int _{0}^{T_{0}}f (r){d} r \right] ^{p} \sup _{s \in [0,T_0]} \mathbb {E} |Y_{1}(s) - Y_{2}(s)|^{p} \nonumber \\&\qquad \qquad \qquad \qquad \quad \qquad =Me^{\omega T} \left[ \int _{0}^{T_{0}}f (r){d} r \right] ^{p} |Y_{1} - Y_{2}|_{\mathcal {H}_p(0,T_{0})}^{p}. \end{aligned}$$

(1.73)

Therefore, if $T_{0}$ is sufficiently small, we can apply the contraction mapping principle to find the unique mild solution of (1.67) in $\mathcal {H}_p(0,T_{0})$. The existence and uniqueness of a solution on the whole interval [0, T] follows, as usual, by repeating the procedure a finite number of times, since the estimate (1.73) does not depend on the initial data, and the number of steps does not blow up since f is integrable. Estimate (1.69) follows from (1.72) applied to the solution X if we perform estimates similar to those above and use Gronwall’s Lemma.

To show (1.70) we observe that if $Z(s) = X(s;0,\xi _1) - X(s;0,\xi _2)$, then for $s \in [0,T]$

$$ Z(s)=e^{sA}(\xi _1-\xi _2) +\int _{0}^{s}e^{(s -r )A} [b_{0}(r , X(r ;0,\xi _1), a_1(r)) -b_{0}(r , X(r ;0,\xi _2), a_1(r))] {d} r. $$

By Hypothesis 1.145 we thus have

$$ |Z(s)| \le Me^{\omega T}|\xi _1-\xi _2| +Me^{\omega T}\int _{0}^{s} f (r)|Z(r )| {d} r, \quad s \in [0,T] $$

so that, by Gronwall’s inequality (see Proposition D.29),

$$ |Z(s)| \le Me^{\omega T}|\xi _1-\xi _2| e^{ Me^{\omega T}\int _{0}^{s}f (r ) {d} r }, $$

which gives the claim. The continuity of trajectories follows from Proposition 1.144, Hypothesis 1.145 and Lemma 1.115 for the second and fourth terms in (1.72), and from Lemma 1.117 for the $\int _{0}^{s} e^{(s-r)A} a_2 (r){d} r$ term.

The last estimate (1.71) follows by standard arguments (see the proof of (1.37) in Theorem 1.130) if we use Proposition 1.144. This implies that if $g\in M^q_\mu (0,T;{\mathbb {R}})$ for any $q>0$, (1.71) holds for any $p\ge 2$. For $p \in (0,2)$, defining $Z_r(s):=\sup _{s\in [0,T]}|X(s) |^r$, we have

$$ \mathbb {E} (Z_p(s))\le [\mathbb {E}(Z_p(s)^{2/p})]^{p/2}\le (C(1+|x|^2))^{p/2} \le C_1(1+|x|^p). $$

$\square$

Proposition 1.148

Assume that Hypotheses 1.143, 1.145, together with (1.65), are satisfied, and let $a_2(\cdot )$ be as in Remark 1.146. Then:

• (i) Let $\xi _1,\xi _2\in L^2 (\Omega , \mathscr {F}_0, \mathbb {P})$, $\xi _1=\xi _2$ $\mathbb {P}$-a.s. Let $(a^1_1(\cdot ), a^1_3(\cdot )),(a^2_1(\cdot ), a^2_3(\cdot ))$ be two processes satisfying Hypothesis 1.145, together with Remark 1.146, such that $(a^1_1(\cdot ), a^1_3(\cdot ))=(a^2_1(\cdot ), a^2_3(\cdot ))$, ${d} t \otimes \mathbb {P}$-a.s. Then, denoting by $X^i(\cdot ;0,\xi _i)$ the solution of (1.67) for $b(s, x)=(\lambda -A)^\beta a_3^i(s) + b_0(s,x, a_1^i(s))$, we have $X^1(\cdot ;0,\xi _1)=X^2(\cdot ;0,\xi _2)$, $\mathbb {P}$-a.s. on [0, T].

• (ii) Let $\left( \Omega _1 ,\mathscr {F}_1,\mathscr {F}_{s}^{1},\mathbb {P}_1,W_{Q, 1} \right)$ and $\left( \Omega _2 ,\mathscr {F}_2,\mathscr {F}_{s}^{2},\mathbb {P}_2,W_{Q, 2}\right)$ be two generalized reference probability spaces. Let $\xi _i\in L^2(\Omega _i,\mathscr {F}_0^{i}, \mathbb {P}_i)$, $i=1,2$. Let $a^i_1(\cdot ), a^i_3(\cdot )$, $i=1,2$, be processes on $[0,T] \times \Omega _i$ satisfying Hypothesis 1.145, together with Remark 1.146. Suppose that $\mathcal {L}_{\mathbb {P}_1} (a^1_1(\cdot ), a^1_3(\cdot ), W_{Q, 1}(\cdot ),\xi _1)= \mathcal {L}_{\mathbb {P}_2} (a^2_1(\cdot ), a^2_3(\cdot ), W_{Q, 2}(\cdot ),\xi _2)$. Then $\mathcal {L}_{\mathbb {P}_1}(X^1(\cdot ;0,\xi _1), a^1_1(\cdot ), a^1_3(\cdot )) = \mathcal {L}_{\mathbb {P}_2}(X(\cdot ;0,\xi _2), a^2_1(\cdot ), a^2_3(\cdot ))$.

• (iii) If $f\in L^2 (0,T;{\mathbb {R}})$ then the solution of (1.67) ensured by Proposition 1.147 is unique in $M^2_\mu (0,T;H)$ as well.

Proof

Parts (i) and (ii) are proved similarly as Proposition 1.142 (i)–(ii). Part (iii) follows from (1.70), which is also true in this case. We also point out that if $p=2$, $f\in L^2 (0,T;{\mathbb {R}})$ then $\mathcal {K}$ maps $M^2_\mu (0,T;H)$ into itself and is a contraction in $M^2_\mu (0,T_0;H)$ for small $T_0$. $\square$

1.5.3 Semilinear SDEs with Multiplicative Noise

This section contains a result for a class of semilinear SDEs with multiplicative noise. Let $T>0$, and let H, $\Xi$, $\Lambda$ and a generalized reference probability space $\left( \Omega , \mathscr {F},\left\{ \mathscr {F}_s\right\} _{s \in [0,T]}, \mathbb {P}, W\right)$ be as in Sect. 1.3, where W(t), $t\in [0,T]$, is a cylindrical Wiener process (so here $\Xi _0=\Xi$). We consider the following SDE in H for $s\in [0,T]$:

$$\begin{aligned} \left\{ \begin{array}{l} dX(s) = AX(s)\; ds+b(s,X(s), a(s))\; ds +\sigma (s,X(s), a(s))\; dW(s),\\ X(0)=\xi . \end{array}\right. \end{aligned}$$

(1.74)

Hypothesis 1.149

• (i) The operator A generates a strongly continuous semigroup $e^{tA}$ for $t \ge 0$ in H.

• (ii) $a(\cdot )$ is a $\Lambda$-valued progressively measurable process.

• (iii) b is a function such that, for all $s \in (0,T]$, $e^{sA}b:[0,T]\times H \times \Lambda \rightarrow H$ is measurable and there exist $L\ge 0$ and $\gamma _1\in [0,1)$ such that, with $f_1(s)=Ls^{-\gamma _1}$,

  $$\begin{aligned} |e^{sA} b(t,x, a)| \le f_1(s) (1+|x|), \end{aligned}$$

  (1.75)
  $$\begin{aligned} |e^{sA} ( b(t,x,a)-b(t,y, a)) |\le f_1(s) |x-y|, \end{aligned}$$

  (1.76)

  for any $s\in (0,T],\; t\in [0,T],\;x, y\in H,\; a\in \Lambda$.

• (iv) The function $\sigma :[0,T]\times H\times \Lambda \rightarrow \mathcal {L}(\Xi , H)$ is such that, for every $v\in \Xi$, the map $\sigma (\cdot , \cdot , \cdot ) v:[0,T]\times H\times \Lambda \rightarrow H$ is measurable and, for every $s>0$, $t\in [0,T]$, $a\in \Lambda$ and $x\in H$, $e^{sA}\sigma (t,x, a)$ belongs to $\mathcal {L}_2(\Xi , H)$. Moreover, there exists a $\gamma _2\in [0,1/2)$ such that, with $f_2(s)=Ls^{-\gamma _2}$,

  $$\begin{aligned} |e^{sA}\sigma (t,x, a)|_{\mathcal {L}_2(\Xi , H)}\le & {} f_2(s) (1+|x|), \end{aligned}$$

  (1.77)
  $$\begin{aligned} |e^{sA}\sigma (t,x,a)-e^{sA}\sigma (t,y, a)|_{\mathcal {L}_2(\Xi , H)}\le & {} f_2(s)|x-y|, \end{aligned}$$

  (1.78)

  for every $s\in (0,T],\; t\in [0,T],\;x, y\in H, \; a\in \Lambda$.

Remark 1.150

Hypothesis 1.149-(iii) covers some cases where the term b is unbounded, which arise, for example, from a stochastic heat equation with a non-zero boundary condition which may also depend on the state variable x (see the last part of Example 4.222).

Moreover, Hypothesis 1.149-(iv) applies to cases, such as reaction-diffusion equations (see e.g. [177], Chap. 11 or, in our Chap. 2, Sect. 2.6.1 and, in particular, Eqs. (2.79) and (2.83), where the operator $\sigma$ is a nonlinear Nemytskii type operator. Indeed, in such cases it is known that, when the underlying space is $L^2(\mathcal {O})$ ($\mathcal {O} {\subset } {\mathbb {R}}^n$, open), the operator $\sigma (t,\cdot ):H\rightarrow {\mathcal L}(H)$ is never Lipschitz continuous while $e^{sA}\sigma (t,\cdot ):H\rightarrow {\mathcal L}_2(H)$ is so (see e.g. [177], proof of Theorem 11.2.4 and Sect. 11.2.1, or [283], Remark 2.2). $\blacksquare$

Remark 1.151

If in Hypothesis 1.125 we set $W_Q=Q^{1/2}\tilde{W}$ for a suitable cylindrical Wiener process $\tilde{W}$ in ${\tilde{\Xi }}=R(Q^{-1/2})$ and we substitute $\sigma$ with ${\tilde{\sigma }}= \sigma Q^{1/2}$, it is easy to see that Hypothesis 1.149 is more general. However, we need to replace $\Xi$ by ${\tilde{\Xi }}$. A cylindrical Wiener process W in $\Xi$ may not be adapted to the original filtration. Similarly, Hypothesis 1.149 is more general than Hypotheses 1.143 and 1.145, together with (1.65), if we take f bounded and $a_2(\cdot )\equiv 0$ there. $\blacksquare$

The solution of Eq. (1.74) is defined in the mild sense of Definition 1.119, where the convolution term

$$ \int _0^s e^{(s-r)A} \sigma (r,X(r), a(r))\; dW(r),\qquad s\in [0,T], $$

makes sense thanks to (1.77) and Remark 1.123. Moreover, since $s{\rightarrow } e^{sA}b(t,x, a)$ is continuous on (0, T] for every $t\in [0,T], x\in H, a\in \Lambda$, we have from Lemma 1.18 that $e^{\cdot A}b$ is $\mathcal {B}([0,T])\otimes \mathcal {B}([0,T])\otimes \mathcal {B}(H)\otimes \mathcal {B}(\Lambda )/\mathcal {B}(H)$-measurable.

Theorem 1.152

Let Hypothesis 1.149 hold and let $a(\cdot )$ be a $\Lambda$-valued, progressively measurable process. Let $p\in [2,\infty )$. Then, for every initial condition $\xi \in L^{p}(\Omega , \mathscr {F}_0, \mathbb {P})$, the SDE (1.74) has a unique mild solution $X(\cdot )$ in $\mathcal {H}_p(0,T;H)$. The solution satisfies

$$\begin{aligned} \sup _{s\in [0,T]} \mathbb {E} \left[ |X(s)|^{p} \right] \le C_0 (1+\mathbb {E}[|\xi |^{p}]) \end{aligned}$$

(1.79)

for some constant $C_0>0$ independent of $\xi$ and $a(\cdot )$. The mild solution $X(\cdot )$ has continuous trajectories and, when $\xi \equiv x \in H$, we have

$$\begin{aligned} {\mathbb {E}}\left[ \sup _{s\in [0,T]}|X(s) |^p\right] \le C(1+|x|^p),\quad \text {for all}\,\,\, p>0, \end{aligned}$$

(1.80)

for some constant C depending only on $p,\gamma _1, \gamma _2,T, L$ and $M_T:=\sup _{s\in [0,T]}|e^{s A}|$.

Finally, when b and $\sigma$ do not depend on a, mild solutions of (1.74) defined on different generalized reference probability spaces have the same laws.

Proof

Let $p\ge 2$. The existence of a unique solution is proved using the Banach contraction mapping theorem in $\mathcal {H}_{p}(0,T_0)$ for some $T_0 \in (0, T)$ small enough. We define $\mathcal {K} :\mathcal {H}_{p}(0,T) \rightarrow \mathcal {H}_{p}(0,T)$ by

$$\begin{aligned} \displaystyle \mathcal {K}(Y)(s):= e^{sA}\xi + \int _0^s e^{(s-r)A} b(r,Y(r), a(r)) {d} r + \int _0^s e^{(s-r)A} \sigma (r,Y(r), a(r)) {d} W(r). \end{aligned}$$

(1.81)

We observe first that this expression belongs to $\mathcal {H}_{p}(0,T)$. Thanks to (1.75), (1.77) and Theorem 1.111, we have

$$\begin{aligned}&\mathbb {E} \left| \int _0^s e^{(s-r)A} b(r,Y(r), a(r)) {d} r + \int _0^s e^{(s-r)A} \sigma (r,Y(r), a(r)) {d} W(r) \right| ^p \nonumber \\&\qquad \qquad \quad \qquad \le C_{p} \bigg ( \mathbb {E} \left| \int _0^s\left[ f_1(s-r) (1+ |Y(r)|) \right] {d} r \right| ^p \nonumber \\&\qquad \qquad \qquad \quad + \mathbb {E}\left| \int _0^s e^{(s-r)A} \sigma (r,Y(r), a(r)) {d} W(r) \right| ^p \bigg ) \nonumber \\&\qquad \qquad \qquad \quad \le C_{p} \left[ \int _0^T f_1(r) {d} r \right] ^p \sup _{r\in [0,T]} \mathbb {E} (1+ |Y(r)|)^p \nonumber \\&\qquad \qquad \qquad \qquad \qquad \quad + C_{p} \left[ \int _0^T f_2^2(r){d} r \right] ^{\frac{p}{2}}\sup _{r\in [0,T]} \mathbb {E}(1 + |Y(r)|)^p, \end{aligned}$$

(1.82)

where the constant $C_{p}$ depends only on p. Therefore, for any $Y\in \mathcal {H}_{p}(0,T)$, $\mathcal {K}(Y)\in \mathcal {H}_{p}(0,T)$. The estimates showing that $\mathcal {K}$ is a contraction on $\mathcal {H}_{p}(0,T_0)$ for $T_0 \in (0, T]$ small enough are essentially the same. Using (1.76) and (1.78) instead of (1.75) and (1.77) we obtain, for all $Y_1,Y_2\in \mathcal {H}_{p}(0,T_0)$,

$$\begin{aligned}&|\mathcal {K} (Y_{1}) - \mathcal {K} (Y_{2})|_{\mathcal {H}_{p}(0,T_0)}^p \le C_{p} \left( \left[ \int _0^{T_0}f_1(r) {d} r \right] ^p \right. \\&\qquad \qquad \qquad \qquad \qquad \left. + \left[ \int _0^{T_0} f_2^2(r){d} r \right] ^{\frac{p}{2}}\right) \sup _{r\in [0,T_0]} \mathbb {E} (|Y_1(r) - Y_2(r)|^p), \end{aligned}$$

and thus $\mathcal {K}$ is a contraction in $\mathcal {H}_{p}(0,T_0)$ if $T_0 \in (0, T]$ is small enough. The existence and uniqueness of solution in $\mathcal {H}_{p}(0,T)$ follows, as usual, by repeating the procedure a finite number of times, since the estimate does not depend on the initial data, and the number of steps does not blow up since $f_1$ and $f_2^2$ are integrable. Estimate (1.79) follows in a standard way by applying estimates like those in (1.82) to the fixed point of the map $\mathcal {K}$ and using Gronwall’s lemma (see also the proof of Theorem 7.5 in [180]).

The continuity of the trajectories and (1.80) are proved using the factorization method similarly to the way it is done in the proof of Proposition 6.9 for $p>2$. We extend (1.80) to $0<p\le 2$ in the same way as in the proof of Proposition 1.147. Uniqueness in law is proved similarly as in Proposition 1.137. $\square$

Proposition 1.153

Assume that Hypothesis 1.149 holds. Let $(t_1,x_1)$, $(t_2,x_2)\in [0,T]\times H$ with $t_1\le t_2$. Denote by $X(\cdot ; t_1,x_1,a(\cdot )), X(\cdot ; t_2,x_2,a(\cdot ))$ the corresponding mild solutions of (1.74) with the same progressively measurable process $a(\cdot )$ and initial conditions $X(t_i)=x_i\in H$, $i=1,2$. Then, for all $s \in [t_2,T]$ we have, setting $\gamma _3:=[2(1-\gamma _1)]\wedge [1-2\gamma _2]$,

$$\begin{aligned} \begin{array}{l} {\mathbb {E}}[|X(s;t_1,x_1,a(\cdot ))-X(s;t_2,x_2,a(\cdot ))|^2] \le \\ \\ \qquad \quad \le C_2 \left[ |x_1-x_2|^2 + (1+|x_1|^2)|t_2 - t_1|^{\gamma _3} + |e^{(t_2-t_1)A}x_1 -x_1|^2 \right] \end{array} \end{aligned}$$

(1.83)

for some constant $C_2$ depending only on $\gamma _1, \gamma _2,T, L$ and $M:=\sup _{s\in [0,T]}|e^{s A}|$. Moreover, the term $|e^{(t_2-t_1)A}x_1 -x_1|^2$ can be replaced by $|e^{(t_2-t_1)A}x_2 -x_2|^2$.

Proof

To simplify the notation we define $X_i(s):=X(s;t_i,x_i,a(\cdot )), b(r,X_i(r)):=b(r,X_i(r),a(r)),\sigma (r,X_i(r)):=\sigma (r,X_i(r), a(r))$, $i=1,2$. By the definition of a mild solution we have, for $s\in [t_i, T]$,

$$ X_i(s)=e^{(s-t_i)A}x_i+ \int _{t_i}^s e^{(s-r)A}b(r,X_i(r))dr + \int _{t_i}^s e^{(s-r)A}\sigma (r, X_i(r))dW(r), $$

hence

$$ |X_1(s)-X_2(s)| \le |e^{(s-t_1)A}x_1 -e^{(s-t_2)A}x_2| $$

$$ +\left| \int _{t_1}^{t_2} e^{(s-r)A}b(r, X_1(r))dr \right| +\left| \int _{t_2}^{s} e^{(s-r)A}\left( b(r, X_1(r))-b(r, X_2(r))\right) dr \right| $$

$$ + \left| \int _{t_1}^{t_2} e^{(s-r)A}\sigma (r, X_1(r))dW(r)\right| +\left| \int _{t_2}^{s} e^{(s-r)A}\left( \sigma (r, X_1(r))-\sigma (r, X_2(r))\right) dW(r) \right| . $$

Therefore

$$\begin{aligned}&\,\,\, {\mathbb {E}}|X_1(s)-X_2(s)|^2 \le 5|e^{(s-t_1)A}x_1 -e^{(s-t_2)A}x_2|^2 \nonumber \\&+5{\mathbb {E}}\left| \int _{t_1}^{t_2} e^{(s-r)A}b(r, X_1(r))dr \right| ^2 +5{\mathbb {E}}\left| \int _{t_2}^{s} e^{(s-r)A}\left( b(r, X_1(r))-b(r, X_2(r))\right) dr \right| ^2 \nonumber \\&\quad \qquad \qquad \qquad +5 {\mathbb {E}}\left| \int _{t_1}^{t_2} e^{(s-r)A}\sigma (r, X_1(r))dW(r)\right| ^2 \nonumber \\&\qquad \qquad \qquad +5{\mathbb {E}}\left| \int _{t_2}^{s} e^{(s-r)A}\left( \sigma (r, X_1(r))-\sigma (r, X_2(r))\right) dW(r) \right| ^2. \end{aligned}$$

(1.84)

To estimate the second and the third terms we use Jensen’s inequality applied to the inner integral. Using Hypothesis 1.149-(ii) and (1.80) we then obtain

$$\begin{aligned}&{\mathbb {E}}\left| \int _{t_1}^{t_2} e^{(s-r)A}b(r, X_1(r))dr \right| ^2 \le L^2{\mathbb {E}}\left| \int _{t_1}^{t_2} (s-r)^{-\gamma _1} (1+|X_1(r)|) dr \right| ^2 \\&\qquad \qquad \le L^2 \left( \int _{t_1}^{t_2} (s-r)^{-\gamma _1} dr\right) \int _{t_1}^{t_2} (s-r)^{-\gamma _1} {\mathbb {E}}(1+|X_1(r)|)^2 dr \\&\qquad \qquad \qquad \le 2L^2 [1+C(1+ |x_1|^2)]\left( \int _{t_1}^{t_2} (s-r)^{-\gamma _1} dr\right) ^2\\&\qquad \qquad \qquad \qquad \qquad \qquad \le 2L^2 [1+C(1+ |x_1|)^2)] \frac{1}{1-\gamma _1} (t_1-t_2)^{2(1-\gamma _1)}. \end{aligned}$$

In the same way we estimate the third term obtaining, by Hypothesis 1.149-(ii),

$$\begin{aligned}&{\mathbb {E}}\left| \int _{t_2}^{s} e^{(s-r)A}\left( b(r, X_1(r))-b(r, X_2(r))\right) dr \right| ^2\\&\qquad \qquad \le L^2 \left( \int _{t_2}^{s} (s-r)^{-\gamma _1} dr\right) \int _{t_2}^{s} (s-r)^{-\gamma _1} {\mathbb {E}}|X_1(r)-X_2(r)|^2 dr \\&\qquad \qquad \qquad \qquad \qquad \le \frac{L^2(s-t_2)^{1-\gamma _1}}{1-\gamma _1} \int _{t_2}^{s} (s-r)^{-\gamma _1} {\mathbb {E}}|X_1(r)-X_2(r)|^2 dr. \end{aligned}$$

The fourth and the fifth term of (1.84) are estimated using the isometry formula. We have

$$ {\mathbb {E}}\left| \int _{t_1}^{t_2} e^{(s-r)A}\sigma (r, X_1(r))dW(r)\right| ^2 = \int _{t_1}^{t_2} {\mathbb {E}}|e^{(s-r)A}\sigma (r, X_1(r))|^2_{\mathcal {L}_2(\Xi , H)}dr $$

$$ \le L^2\int _{t_1}^{t_2} (s-r)^{-2\gamma _2}{\mathbb {E}}(1+|X_1(r)|)^2dr \le 2L^2[1+C(1+|x_1|^2)] \int _{t_1}^{t_2} (s-r)^{-2\gamma _2}dr $$

$$ \le 2L^2 [1+C(1+ |x_1|^2)] \frac{1}{1-2\gamma _2} (t_1-t_2)^{1-2\gamma _2} $$

and

$$\begin{aligned}&{\mathbb {E}}\left| \int _{t_2}^{s} e^{(s-r)A}\left( \sigma (r, X_1(r))-\sigma (r, X_2(r))\right) dW(r) \right| ^2\\&\qquad \qquad \,\, =\int _{t_2}^{s} {\mathbb {E}}|e^{(s-r)A}\left( \sigma (r, X_1(r))-\sigma (r, X_2(r))\right| ^2_{\mathcal {L}_2(\Xi , H)}dr \\&\qquad \qquad \qquad \qquad \qquad \qquad \quad \le L^2\int _{t_2}^{s} (s-r)^{-2\gamma _2}{\mathbb {E}}|X_1(r)-X_2(r)|^2 dr. \end{aligned}$$

Using all these estimates in (1.84) we obtain, for a suitable constant $C_1>0$, for $\gamma _3:=[2(1-\gamma _1)]\wedge [1-2\gamma _2]$ and $\gamma _4:=\gamma _1 \vee [2\gamma _2]$,

$$ {\mathbb {E}}|X_1(s)-X_2(s)|^2 \le 5|e^{(s-t_1)A}x_1 -e^{(s-t_2)A}x_2|^2 + C_1(1+|x_1|^2)|t_2 - t_1|^{\gamma _3}+ $$

$$+ C_1 \int _{t_2}^{s} (s-r)^{-\gamma _4} {\mathbb {E}}|X_1(r)-X_2(r)|^2 dr. $$

Observing that

$$ |e^{(s-t_1)A}x_1 -e^{(s-t_2)A}x_2|\le M|x_1-x_2| + |e^{(s-t_2)A}(e^{(t_2-t_1)A}x_1 -x_1)|, $$

we can thus apply Gronwall’s lemma in the form of Proposition D.30. It gives us

$$ {\mathbb {E}}|X_1(s)-X_2(s)|^2\le C_2 \left[ |x_1-x_2|^2 + (1+|x_1|^2)|t_2 - t_1|^{\gamma _3} + |e^{(t_2-t_1)A}x_1 -x_1|^2 \right] $$

for some $C_2>0$ with the required properties. $\square$

Lemma 1.154

Assume that Hypothesis 1.149 holds. Fix a $\Lambda$-valued progressively measurable process $a(\cdot )$. Let X be the unique mild solution of (1.74) described in Theorem 1.152 with initial condition $X(0)=x\in H$. Define, for $s\in [0,T]$, $\psi (s)=b(s,X(s), a(s)), \Phi (s)=\sigma (s, X(s), a(s))$. Let $\{ e_i \}_{i\in \mathbb {N}}$ be an orthonormal basis of $\Xi$ and, for any $k\in \mathbb {N}$, let $P^k:\Xi \rightarrow \Xi$ be the orthogonal projection onto $\mathrm{span}\{e_1,..., e_k\}$. Let $X^k$ be the unique mild solution of

$$\begin{aligned} \left\{ \begin{array}{l} {d} X^k(s) =(AX^k(s)+ e^{\frac{1}{k}A} \psi (s)) {d} s + e^{\frac{1}{k}A} \Phi (s) P^k {d} W(s), \\ X^k(0)=x. \end{array} \right. \end{aligned}$$

(1.85)

Then, for any $p>0$, there exists an $M_p>0$ such that

$$\begin{aligned} \sup _{k\in \mathbb {N}} \, \mathbb {E}\left[ \sup _{s\in [0,T]} |X^k(s)|^p \right] \le M_p. \end{aligned}$$

(1.86)

Moreover, for every $s\in [0,T]$,

$$\begin{aligned} \lim _{k\rightarrow \infty }\mathbb {E} \left[ |X^k(s)-X(s)|^2\right] =0 \end{aligned}$$

(1.87)

and, for every $\varphi \in C_{m}(H)$ ($m\ge 0)$,

$$\begin{aligned} \lim _{k\rightarrow \infty } \mathbb {E}\left[ \varphi (X^k(s))\right] = \mathbb {E}\left[ \varphi (X(s))\right] , \qquad s\in [0,T]. \end{aligned}$$

(1.88)

Proof

It is easy to see, by using (1.80), that (1.86) is satisfied.

We now prove (1.87). We have, for $s\in [0,T]$,

$$\begin{aligned}&\mathbb {E} \left| X(s)-X^k(s)\right| ^2 \le 2 \mathbb {E}\left| \int _{0}^s e^{(s-r)A}\left( \psi (r)- e^{\frac{1}{k}A}\psi (r)\right) dr \right| ^2 \nonumber \\&\quad \qquad \qquad \qquad + 4 \mathbb {E} \left| \int _{0}^s e^{(s-r)A} \Phi (r) (I-P^k ) dW(r) \right| ^2\nonumber \\&\quad \qquad \qquad \qquad + 4 \mathbb {E} \left| \int _{0}^s (I-e^{\frac{1}{k}A})e^{(s-r)A} \Phi (r) P^k dW(r) \right| ^2=I_1+I_2+I_3.\nonumber \end{aligned}$$

We have for any k,

$$ \left| e^{(s-r)A}\left( \psi (r)- e^{\frac{1}{k}A}\psi (r)\right) \right| \le 2L(s-r)^{-\gamma _1}(1+|X(r)|) $$

which is integrable on [0, s] for a.e. $\omega$. Moreover,

$$ \left| e^{(s-r)A}\left( \psi (r)- e^{\frac{1}{k}A}\psi (r)\right) \right| \rightarrow 0\quad \text {as}\,\, k\rightarrow +\infty $$

$dr\otimes \mathbb {P}$-a.s. Therefore it follows from the dominated convergence theorem that

$$ \int _{0}^s e^{(s-r)A}\left( \psi (r)- e^{\frac{1}{k}A}\psi (r)\right) dr \rightarrow 0 \quad \text {as}\,\, k\rightarrow +\infty $$

$\mathbb {P}$-a.s. Now by Hölder’s inequality

$$\begin{aligned}&\left| \int _{0}^s e^{(s-r)A}\left( \psi (r)- e^{\frac{1}{k}A}\psi (r)\right) dr \right| ^2 \\&\quad \qquad \qquad \qquad \le 4L^2\left( \int _{0}^s (s-r)^{-\gamma _1}dr\right) \left( \int _{0}^s (s-r)^{-\gamma _1}(1+|X(r)|)^2dr\right) \end{aligned}$$

which is integrable on $\Omega$. Thus, using the dominated convergence theorem again we conclude that $\lim _{k\rightarrow \infty } I_1=0$.

Recall that $\Xi _0=\Xi$. To estimate $I_2$, we set $Q^k:=I-P^k$. We have

$$\begin{aligned}&I_2 = 4 \mathbb {E} \left| \int _{0}^s e^{(s-r)A} \Phi (r) (I-P^k ) dW(r) \right| ^2 \nonumber \\&\quad \qquad \qquad \qquad = 4 \int _{0}^s \mathbb {E}\left\| e^{(s-r)A} \Phi (r) Q^k \right\| ^2_{{\mathcal L}_{2}(\Xi , H)} dr \nonumber \\&\qquad \qquad \qquad =4 \int _{0}^s \mathbb {E} \sum _{i\in \mathbb {N}} \left\langle e^{(s-r)A} \Phi (r) Q^k e_i, e^{(s-r)A} \Phi (r) Q^k e_i \right\rangle {d} r =:\eta (k). \nonumber \end{aligned}$$

Observe that

$$\begin{aligned}&\sum _{i\in \mathbb {N}} \left\langle e^{(s-r)A} \Phi (r) Q^k e_i, e^{(s-r)A} \Phi (r) Q^k e_i \right\rangle \nonumber \\&\qquad \qquad \qquad \quad = \sum _{i = k+1}^{+\infty } \left\langle e^{(s-r)A} \Phi (r) e_i, e^{(s-r)A} \Phi (r) e_i \right\rangle \nonumber \\&\qquad \qquad \quad \le \sum _{i \in \mathbb {N}} \left\langle e^{(s-r)A} \Phi (r) e_i, e^{(s-r)A} \Phi (r) e_i \right\rangle = \left\| e^{(s-r)A} \Phi (r) \right\| ^2_{{\mathcal L}_{2}(\Xi , H)}. \nonumber \end{aligned}$$

Since the series above has nonnegative terms, we obtain

$$ \lim _{k\rightarrow \infty } \left\| e^{(s-r)A} \Phi (r) Q^k \right\| ^2_{{\mathcal L}_{2}(\Xi , H)}=0\quad \;\; {d} r \otimes \mathbb {P}\text {-a.s.} $$

Therefore, thanks to (1.80), Hypothesis 1.149 and the dominated convergence theorem, we obtain

$$ \lim _{k\rightarrow \infty } I_2 \le \lim _{k\rightarrow \infty } \eta (k)=0. $$

The term $I_3$ is estimated similarly.

Thanks to (1.87), for any subsequence of $X^k(s)$ we can extract a sub-subsequence converging to X(s) almost everywhere and then, thanks to (1.86), (1.80) and the dominated convergence theorem, we obtain (1.88) along the sub-subsequence. This implies (1.88) for the whole sequence $X^k(s)$. $\square$

Remark 1.155

Observe that if b and $\sigma$ satisfy Hypothesis 1.149, the functions $e^{\frac{1}{k}A} b(s,x, a)$ and $e^{\frac{1}{k}A} \sigma (s,x, a)P_k$ satisfy Hypothesis 1.125. $\blacksquare$

The last lemma concerns the additive noise case of Sect. 1.5.2, however we included it here since its proof is similar to the proof of Lemma 1.154.

Let $W_Q$ be from Sect. 1.5.2. We know (see (1.12)) that $W_Q(s)=\sum _{n=1}^{+\infty } g_n\beta _n(s)$, $s\ge 0$, where $\{g_n\}$ is an orthonormal basis of $\Xi _0$. Define $e_n=Q^{-1/2}g_n, n\in \mathbb {N}$. Then $\{e_n\}$ is an orthonormal basis of $\Xi$. Let $\tilde{P}^k$ be the orthogonal projection in $\Xi _0$ onto $\mathrm{span}\{g_1,..., g_k\}$ and $P^k$ be the orthogonal projection in $\Xi$ onto $\mathrm{span}\{e_1,..., e_k\}$, $k\in \mathbb {N}$. It is easy to see that $\tilde{P}^k Q^{1/2}=Q^{1/2}P^k$ as operators on $\Xi$.

Lemma 1.156

Let Hypotheses 1.143 and 1.145 be satisfied and let $q\ge 2$. Let X be the unique mild solution of (1.67) described in Proposition 1.147 with initial condition $X(0)=x\in H$. Define for $k, m\in \mathbb {N}$, $B_k=\{(s,\omega ):|b_0(s,X(s), a_1(s))|\le k\}, D_m=\{(s,\omega ):|g(s,\omega )|\le m\}$. There exists a sequence $m_k$ such that the sequence $X^k$ of the solutions of the SDE

$$\begin{aligned} \left\{ \begin{array}{l} {d} X^k(s) = \left( AX^k(s) + \psi _k(s) \right) {d} s + \sigma \tilde{P}^k{d} W_Q(s), \quad s >0, \\ X^k(0) = x, \end{array} \right. \end{aligned}$$

(1.89)

where $\psi _k(s)= b_0(s,X(s), a_1(s))\mathbf{1}_{B_k}(s,\omega )+ e^{\frac{1}{k}A}a_2(s)\mathbf{1}_{D_{m_k}}(s,\omega )$, satisfies the following.

• (i) For any $p\in [2,q]$ there exists an $M_p>0$ such that

  $$\begin{aligned} \sup _{k} \sup _{s\in [0,T]} \mathbb {E} \left[ |X^k(s)|^p \right] , \, \sup _{s\in [0,T]} \mathbb {E} \left[ |X(s)|^p \right] \le M_p. \end{aligned}$$

  (1.90)

• (ii) For every $s\in [0,T]$

  $$ \lim _{k\rightarrow \infty } \mathbb {E} \left[ |X^k(s) - X(s)|^2 \right] = 0. $$

Proof

Part (i). The moment estimates are uniform in k (regardless of the choice of $m_k$) thanks to the following facts:

• (a) Define $W^{A, k}(s) := \int _{0}^{s} e^{(s-r)A} \sigma \tilde{P}^k {d} W_Q(r)$, $s\in [0,T]$. Given an orthonormal basis $\{w_n\}$ of H, for any $k\in \mathbb {N}$ and $s\in [0,T]$, we have

  $$\begin{aligned}&0 \le \mathrm{Tr} \left( \left( e^{sA} \sigma \tilde{P}^k Q^{1/2}\right) \left( e^{sA} \sigma \tilde{P}^k Q^{1/2}\right) ^*\right) \nonumber \\&\!\quad = \mathrm{Tr} \left( \left( e^{sA} \sigma Q^{1/2} P^k \right) \left( e^{sA} \sigma Q^{1/2} P^k \right) ^*\right) \nonumber \\&\!\quad = \sum _{n\in \mathbb {N}} | P_k Q^{1/2} \sigma ^* e^{sA^*} w_n|^2 \le \sum _{n\in \mathbb {N}} | Q^{1/2} \sigma ^* e^{sA^*} w_n|^2 = \sum _{n\in \mathbb {N}} \mathrm{Tr} \left( e^{sA} \sigma Q \sigma ^* e^{sA^*} \right) . \end{aligned}$$

  (1.91)

  Thus, by Theorem 1.111, it follows that for any $k\in \mathbb {N}$ and $p\ge 1$,

  $$ \sup _{k} \sup _{s\in [0,T]} \mathbb {E} \left[ |W^{A, k}(s)|^p \right] < +\infty . $$

  Using (1.91) we also have, by the Lebesgue dominated convergence theorem,

  $$\begin{aligned} \int _0^T \Vert e^{s A } \sigma \tilde{P}^k - e^{s A } \sigma \Vert ^2_{\mathcal {L}_2(\Xi _0, H)} {d} s = \int _0^T\sum _{n\in \mathbb {N}} | (P_k - I)Q^{1/2} \sigma ^* e^{sA^*} w_n|^2 {d} s\rightarrow 0. \end{aligned}$$

  (1.92)

• (b) By the definition

  $$ |e^{tA}\psi _k(s)| \le f(s)(1+|X(s)|)+t^{-\beta }g(s,\omega )\quad \text {for}\,\,t, s\in [0,T],\omega \in \Omega . $$

Part (ii). The scheme of the proof is similar to that of (1.87). We choose $m_k$ such that

$$\begin{aligned} \mathbb {E}\left| \int _{0}^T k^\beta g(r,\omega )|1-\mathbf{1}_{D_{m_k}}(r,\omega )|dr\right| ^2\le \frac{1}{k}. \end{aligned}$$

(1.93)

We have for every $s\in [0,T]$,

$$\begin{aligned}&\mathbb {E} \left| X(s)-X^k(s)\right| ^2 \le 4 \mathbb {E} \left| \int _{0}^s e^{(s-r)A} b_0(r,X(r), a_1(r))(1-\mathbf{1}_{B_k}(r,\omega ))dr\right| ^2\nonumber \\&\qquad \qquad \qquad +4 \mathbb {E} \left| \int _{0}^s e^{(s-r)A} (a_2(r)- e^{\frac{1}{k}A}a_2(r))dr\right| ^2\nonumber \\&\qquad \qquad \quad +4 \mathbb {E} \left| \int _{0}^s e^{(\frac{1}{k}+s-r)A}a_2(r)(1-\mathbf{1}_{D_{m_k}}(r,\omega ))dr\right| ^2\nonumber \\&\qquad \qquad \qquad \qquad \qquad + 4\mathbb {E} \left| W^{A, k}(s)-W^A(s) \right| ^2=J_1+J_2+J_3+J_4.\nonumber \end{aligned}$$

The term $J_1$ converges to 0 as $k\rightarrow +\infty$ by Hypothesis 1.145, Hölder’s inequality, (1.69) for $p=2$ and the dominated convergence theorem. The term $J_2$ converges to 0 by the same arguments as for the term $I_1$ in the proof of Lemma 1.154. The term $J_3$ converges to 0 by (1.93) and finally $J_4\rightarrow 0$ by (1.92). $\square$

1.6 Transition Semigroups

Let $T\in (0,+\infty ]$ and recall that, as before, when $T=+\infty$ the notation [0, T] and [t, T] means $[0,+\infty )$ and $[t,+\infty )$. Let $H,\Xi , Q$, and the generalized reference probability space $\mu =(\Omega , \mathscr {F}, \{\mathscr {F}_s\}_{s\in [0,T]}, \mathbb {P}, W_Q)$ be the same as in Sect. 1.3. Consider for $t \in [0,T]$ the following SDE with non-random coefficients

$$\begin{aligned} \left\{ \begin{array}{l} {d} X(s) = \left( AX(s)+ b(s,X(s)) \right) {d} s + \sigma (s, X(s)) {d} W_Q(s) \\ X(t)=x \in H, \end{array} \right. \end{aligned}$$

(1.94)

where $b:[0,T] \times H \rightarrow H$ and $\sigma :[0,T]\times H \rightarrow \mathcal {L}_2(\Xi _0,H)$. If Hypothesis 1.125, where we drop the dependence on a in all conditions, (respectively, Hypotheses 1.143 and 1.145 with $a_2(\cdot )\equiv 0$ and with no dependence on $a_1$, respectively, Hypothesis 1.149 with no dependence on a) is satisfied, then Theorem 1.127 (respectively, Proposition 1.147, respectively, Theorem 1.152) ensures that (1.94) has a unique mild solution $X(\cdot ;t, x)$. Moreover, we also have uniqueness in law of the solutions.

We will be using the spaces $B_b(H)$ of bounded Borel measurable functions on H and $B_m(H), m>0$, of Borel measurable functions on H with at most polynomial growth of order m, defined in Appendix A.2.

For any $\phi \in B_b(H)$ and $t \ge 0$, $s\in [t, T]$, we define

$$\begin{aligned} \left\{ \begin{array}{l} P_{t,s}[\phi ] :H \rightarrow \mathbb {R} \\ P_{t,s}[\phi ] :x {\rightarrow } \mathbb {E} [\phi (X(s;t, x))]. \end{array} \right. \end{aligned}$$

(1.95)

It is not obvious that $P_{t, s}[\phi ]\in B_b(H)$ and it has to be checked in each case. The general argument is the following and we illustrate it in the case when Hypothesis 1.149 is satisfied. First, using (1.83) it is easy to see that $P_{t, s}[\phi ]\in C_b(H)$ if $\phi \in UC_b(H)$. Then, using the functions constructed in the proof of Theorem 1.34 and the dominated convergence theorem, we get that $P_{t, s}[\phi ]\in B_b(H)$ for every $\phi =\mathbf{1}_A, A=\overline{A}\subset H$. This, together with Corollary 1.3 and the dominated convergence theorem, allows us to extend $P_{t, s}[\phi ]\in B_b(H)$ to every $\phi =\mathbf{1}_A, A\in B(H)$. We can then use Lemma 1.15 to conclude that $P_{t, s}[\phi ]\in B_b(H)$ for every $\phi \in B_b(H)$. Similar arguments can be applied in the cases when Hypotheses 1.143 and 1.145 hold or if Hypothesis 1.125 is satisfied. Moreover, thanks to estimates (1.36), (1.69) and (1.80), $P_{t, s}[\phi ]$ is then also well defined for any $\phi \in B_m(H)$, $m>0$.

Theorem 1.157

(Markov property) Let $T\in (0,+\infty ]$. Let Hypothesis 1.149 be satisfied with b and $\sigma$ independent of a. Then for every $\phi \in B_m(H)$ ($m\ge 0$) and $0\le t \le s \le r \le T$ (with the last inequality strict when $T=+\infty$),

$$ \mathbb {E} \phi (X(r;t,x)|\mathscr {F}_s) = P_{s,r}[\phi ](X(s;t, x)) \;\;\; \mathbb {P}-\text {almost surely}, $$

and

$$\begin{aligned} P_{t,r}[\phi ](x) = P_{t,s} \left[ P_{s, r} [\phi ] \right] (x) \qquad \text {for all }x\in H. \end{aligned}$$

(1.96)

The same result is true if Hypotheses 1.143 and 1.145 hold without dependence on $a_1$ and with $a_2(\cdot )=0$ or if Hypothesis 1.125 holds without the dependence on a in all conditions.

Proof

See [180], Theorem 9.14, p. 248, and Corollary 9.15, p. 249. The hypotheses are a little different from these in [180], however the same arguments can be easily adapted using the proof of Proposition 1.153. The proof in [180] is given for $\phi \in B_b(H)$ but the argument is exactly the same when $\phi \in B_m(H)$ ($m>0$) simply recalling that the operator $P_{t, s}$ is well defined on such functions thanks to estimate (1.80). $\square$

It follows from the uniqueness in law of the solutions of (1.94) that the operators $P_{t, s}$ do not depend on the choice of a generalized reference probability space $\mu$. As a consequence of the uniqueness in law we also have the following corollary.

Corollary 1.158

Let Hypothesis 1.149 be satisfied with b and $\sigma$ independent of a and of the time variable s. Equation (1.94) then reduces to

$$\begin{aligned} \left\{ \begin{array}{l} {d} X(s) = \left( AX(s)+ b(X(s)) \right) {d} s + \sigma (X(s)) {d} W_Q(s), \\ X(t)=x \in H. \end{array} \right. \end{aligned}$$

(1.97)

Denote by $X(\cdot ;t, x)$ the unique mild solution of this equation (defined on $[t,+\infty )$). In this case, for any $\phi \in B_m(H)$ ($m\ge 0$) and $0\le t\le s$, we have

$$\begin{aligned} P_{t, s}[\phi ](x) = P_{0,s-t}[\phi ]. \end{aligned}$$

(1.98)

Hence, defining $P_{s}[\phi ]$ as follows,

$$\begin{aligned} \left\{ \begin{array}{l} P_{s}[\phi ] :H \rightarrow \mathbb {R} \\ P_{s}[\phi ] :x {\rightarrow } \mathbb {E} \phi (X(s;0,x)), \end{array} \right. \end{aligned}$$

(1.99)

we have

$$\begin{aligned} P_{s+r}[\phi ](x) = P_{s} \left[ P_{r} [\phi ] \right] (x) \qquad \text {for all }x\in H, s, r \ge 0. \end{aligned}$$

(1.100)

The same result is true if Hypotheses 1.143 and 1.145 hold without dependence on $a_1$ and with $a_2(\cdot )=0$ or if Hypothesis 1.125 holds without the dependence on a in all conditions.

Proof

We only need to prove (1.98), which is an immediate consequence of the uniqueness in law of the mild solutions of (1.97). Indeed, by the uniqueness in law, for all $s \ge t \ge 0$ and $x \in H$, the random variables X(s; t, x) and $X(s-t;0,x)$ have the same distributions, hence

$$ P_{t,s}[\phi ] (x)= \mathbb {E} [\phi (X(s;t, x))] =\mathbb {E} [\phi (X(s-t;0,x))]= P_{0,s-t}[\phi ] (x). $$

$\square$

Definition 1.159

(Transition semigroup, (strong) Feller property) If (1.96) (respectively, (1.100)) is satisfied we call $P_{t, s}$ (respectively, $P_t$) the two-parameter transition semigroup (respectively, one-parameter transition semigroup) associated to Eq. (1.94).

We say that $P_{t, s}$ (respectively, $P_t$) possesses the Feller property if

$$ P_{t, s}(C_b(H)){\subset } C_b(H) \; \text {(respectively, } P_{t}(C_b(H)){\subset } C_b(H)\text {)} $$

and that $P_{t, s}$ (respectively, $P_t$) possesses the strong Feller property if

$$ P_{t, s}(B_b(H)){\subset } C_b(H) \; \text {(respectively, } P_{t}(B_b(H)){\subset } C_b(H)\text {)} $$

for all $0\le t<s\le T$ (respectively $t \in (0,T]$).

Lemma 1.160

Assume that (1.94) has unique mild solutions $X(\cdot ;t, x)$ which satisfy, for every $m\ge 0$, the estimate

$$\begin{aligned} {\mathbb {E}}[|X(s;t, x)|^{m}] \le C(m)(1 + |x|^{m}), \qquad t \ge 0, \; s \in [t, T], \; x \in H, \end{aligned}$$

(1.101)

for some constant C(m). If the Feller property holds for the associated two-parameter transition semigroup $P_{t, s}$ ($t\ge 0$, $s \in [t, T]$), then we also have

$$ P_{t, s}(C_m(H)){\subset } C_m(H) \qquad \forall m\ge 0 $$

while, if the strong Feller property holds, we also have

$$ P_{t, s}(B_m(H)){\subset } C_m(H) \qquad \forall m\ge 0. $$

Proof

Let $\phi \in B_m (H)$ and define, for $k\in \mathbb N$,

$$ \phi _k(x)=\phi (x)\mathbf{1}_{|x|\le k}+ \phi \left( k\frac{x}{|x|}\right) \mathbf{1}_{|x|> k}. $$

It is clear that $\phi _k \in B_b(H)$, it coincides with $\phi$ on $\{|x|\le k\}$ and if $\phi$ is continuous so is $\phi _k$. Moreover, when $k\rightarrow + \infty$, $\phi _k$ converges to $\phi$ uniformly on bounded sets. Assume now that the strong Feller property holds (the argument for the Feller property is exactly the same). In this case $P_{t, s}[\phi _k]$ is continuous, hence, to get the claim, it is enough to show that $P_{t, s}[\phi _k]$ converges to $P_{t, s}[\phi _k]$ uniformly on bounded sets. Indeed,

$$\begin{aligned}&P_{t,s}[\phi _k-\phi ] (x)=\mathbb {E} \left[ (\phi _k- \phi )(X(s;t,x))\right] \\&\qquad \qquad = \mathbb {E} \left[ \left( \phi \left( k\frac{X(s;t,x)}{|X(s;t,x)|}\right) - \phi (X(s;t, x))\right) \mathbf{1}_{|X(s;t, x)|> k}\right] \\&\qquad \qquad \qquad \qquad \qquad \quad \le 2\mathbb {E} \left[ \Vert \phi \Vert _{B_m}(1+|X(s;t, x)|^m) \mathbf{1}_{|X(s;t, x)|\ge k}\right] . \end{aligned}$$

Hence, for any $p>1$ we have by (1.101)

$$\begin{aligned}&P_{t, s}[\phi _k-\phi ] (x) \le 2 \Vert \phi \Vert _{B_m} \left[ \mathbb {E}(1+|X(s;t, x)|^m)^{p}\right] ^{1/p} \left[ \mathbb {E}{} \mathbf{1}_{|X(s;t, x)|\ge k}\right] ^{1-1/p} \\&\qquad \qquad \quad \le C(1+|x|^m) \left[ \frac{ \mathbb {E}|X(s;t, x)|}{k}\right] ^{1-1/p}\le C(1+|x|^{m})\left[ \frac{ 1+|x|}{k}\right] ^{1-1/p} \end{aligned}$$

which converges to 0 uniformly on bounded sets. $\square$

Remark 1.161

Estimate (1.101) is satisfied in two important cases:

• when Hypothesis 1.149 is satisfied with b and $\sigma$ independent of a;

• when Hypotheses 1.143 and 1.145 hold without dependence on $a_1$ and with $a_2(\cdot )=0$.

This follows from the growth estimates of Theorem 1.152 and Proposition 1.147.$\blacksquare$

Theorem 1.162

Assume that Hypothesis 1.149 is satisfied. Then for every $\phi \in C_m(H)$ ($m\ge 0$), the function $P_{t, s}[\phi ] :H \rightarrow \mathbb {R}$ belongs to $C_m(H)$. The same holds if we assume that Hypotheses 1.143 and 1.145 hold without dependence on $a_1$ and with $a_2(\cdot )=0$.

Proof

The result is a consequence of the continuous dependence and growth estimates of Theorem 1.152 and Propositions 1.153 and 1.147. $\square$

1.7 Itô’s and Dynkin’s Formulae

In this section we assume that $T>0$, $H,\Xi , Q$, and the generalized reference probability space $\mu =(\Omega , \mathscr {F}, \{\mathscr {F}_s\}_{s\in [0,T]}, \mathbb {P}, W_Q)$ are the same as in Sect. 1.3. The operator A is the generator of a $C_0$-semigroup on H, and $\Lambda$ is a Polish space. The various Itô’s and Dynkin’s formulae presented in this section are used in proving existence of viscosity solutions (Chap. 3) and verification theorems (Chaps. 4 and 5).

Given a function $F:[0,T] \times H \rightarrow \mathbb {R}$, we denote by $F_t$ the derivative of F(t, x) with respect to t and by DF and $D^2F$ the first and second-order Fréchet derivatives with respect to x.

Theorem 1.163

(Itô’s Formula) Assume that $\Phi$ is a process in $\mathcal {N}_Q^{2}(0,T;H)$, f is an H-valued progressively measurable ($\mathbb {P}$-a.s.) Bochner integrable process on [0, T], and define, for $s\in [0,T]$,

$$ X(s): = X(0) + \int _0^s f(r) {d} r + \int _0^s \Phi (r) {d} W_Q(r), $$

where X(0) is an $\mathscr {F}_{0}$-measurable H-valued random variable. Consider $F:[0,T] \times H \rightarrow \mathbb {R}$ and assume that F and its derivatives $F_t,DF, D^2 F$ are continuous and bounded on bounded subsets of $[0,T]\times H$. Let $\tau$ be an $\mathscr {F}_s$-stopping time. Then, for $\mathbb {P}$-a.e. $\omega$,

$$\begin{aligned}&F(s\wedge \tau , X(s\wedge \tau )) = F(0,X(0)) + \int _0^{s\wedge \tau } F_t(r, X(r)) {d} r \nonumber \\&\qquad + \int _0^{s \wedge \tau } \left\langle D F(r, X(r)), f(r) \right\rangle {d} r + \int _0^{s\wedge \tau } \left\langle D F(r, X(r)), \Phi (r) {d} W_Q(r) \right\rangle \nonumber \\&\,\, + \frac{1}{2} \int _0^{s\wedge \tau } \mathrm{Tr} \left[ \left( \Phi (r) {Q}^{1/2} \right) \left( \Phi (r) Q^{1/2} \right) ^* D^2 F(r, X(r)) \right] {d} r \qquad \text {on}\,\,[0,T]. \end{aligned}$$

(1.102)

Proof

See [294], Theorems 2.9 and 2.10. See also, under the assumption of uniform continuity on bounded sets of F and its derivatives, [180] Theorem 4.32, p. 106. $\square$

Proposition 1.164

Let$F:[0,T] \times H \rightarrow \mathbb {R}$ and $x\in H$. Assume that F and its derivatives $F_t,DF, D^2 F$ are continuous and bounded on bounded subsets of $[0,T]\times H$. Suppose that $D F:[0,T]\times H \rightarrow D(A^*)$ and that $A^*DF$ is continuous and bounded on bounded subsets of $[0,T]\times H$. Let $f\in M^{p}_\mu (0,T;H)$, $\Phi \in \mathcal {N}_Q^{p}(0,T;H)$ for some $p>2$. Let $X(\cdot )$ be the unique mild solution of (1.42) such that $X(0)=x$ and $\tau$ be an $\mathscr {F}_s$-stopping time. Then, for $\mathbb {P}$-a.e. $\omega$,

$$\begin{aligned}&F({s \wedge \tau }, X({s \wedge \tau })) = F(0,x) + \int _0^{s \wedge \tau } F_t(r, X(r)) {d} r \nonumber \\&\qquad \quad + \int _0^{s \wedge \tau } \left\langle A^* D F(r, X(r)), X(r) \right\rangle {d} r + \int _0^{s \wedge \tau } \left\langle D F(r, X(r)), f(r) \right\rangle {d} r \nonumber \\&\qquad \qquad + \frac{1}{2} \int _0^{s \wedge \tau } \mathrm{Tr} \left[ \left( \Phi (r) {Q}^{1/2} \right) \left( \Phi (r) Q^{1/2} \right) ^* D^2 F(r, X(r)) \right] {d} r \nonumber \\&\qquad \qquad \qquad \qquad + \int _0^{s \wedge \tau } \left\langle D F(r, X(r)), \Phi (r) {d} W_Q(r) \right\rangle \qquad \text {on}\,\,[0,T]. \end{aligned}$$

(1.103)

Proof

Since both sides of (1.103) are continuous processes, it is enough to prove the formula for a single s. We approximate $X(\cdot )$ by the sequence $X^n(\cdot )$ introduced in Proposition 1.132. By definition $X^n(\cdot )$ solves the integral equation

$$ X^n(s) = \int _0^s \left( A_n X^n(r) + f(r) \right) {d} r + \int _0^s \Phi (r) {d} W_Q(r). $$

For any $R>0$ such that $|x|<R$ define the stopping times

$$ {\hat{\tau }}^R := \inf \left\{ s \in [0,T] \; : \; |X(s)|> R \right\} ,\quad {\hat{\tau }}^R_n := \inf \left\{ s \in [0,T] \; : \; |X_n(s)| > R +1 \right\} $$

and denote by $\tau ^R$ and $\tau ^R_n$, respectively,

$$ \tau ^R := \min (\tau , {\hat{\tau }}^R),\quad \tau ^R_n := \min (\tau , {\hat{\tau }}^R, {\hat{\tau }}^R_n). $$

Observe that, thanks to (1.44), up to extracting a subsequence of $X_n$ (still denoted by $X_n$), $\sup _{s\in [0,T]} |X^n(s) - X(s)|^{p}$ converges to 0 on some set ${\tilde{\Omega }}$ with $\mathbb {P}({\tilde{\Omega }})=1$. It is then easy to see that on ${\tilde{\Omega }}$ we have

$$ \lim _{n\rightarrow \infty } \tau ^R_n = \tau ^R. $$

We deduce that, for $\omega \in {\tilde{\Omega }}$,

$$\begin{aligned} \lim _{n\rightarrow \infty } \mathbf{1}_{[0,s \wedge \tau ^R_n]} = \mathbf{1}_{[0,s \wedge \tau ^R]}, \quad \text {pointwise on }[0,T]. \end{aligned}$$

(1.104)

We can apply Itô’s formula (1.102) to the approximating problem ($A^*_n$ is the adjoint of $A_n$) obtaining, once we rewrite it using Lemma 1.110,

$$\begin{aligned}&F({s \wedge \tau ^R_n}, X^n({s \wedge \tau ^R_n})) = F(0,x) + \int _0^s \mathbf{1}_{[0,s \wedge \tau ^R_n]}(r) F_t(r, X^n(r)) {d} r \nonumber \\&\qquad \qquad \qquad + \int _0^s \mathbf{1}_{[0,s \wedge \tau ^R_n]}(r) \left\langle A_n^* D F(r, X^n(r)), X^n(r) \right\rangle {d} r \nonumber \\&\qquad \qquad \qquad \quad + \int _0^s \mathbf{1}_{[0,s \wedge \tau ^R_n]}(r) \left\langle D F(r, X^n(r)), f(r) \right\rangle {d} r \nonumber \\&\qquad + \frac{1}{2} \int _0^s \mathbf{1}_{[0,s \wedge \tau ^R_n]} (r) \text {Tr} \left[ \left( \Phi (r) {Q}^{1/2} \right) \left( \Phi (r) Q^{1/2} \right) ^* D^2 F(r, X^n(r)) \right] {d} r \nonumber \\&\qquad \qquad \qquad \qquad \quad + \int _0^s \mathbf{1}_{[0,s \wedge \tau ^R_n]} (r) \left\langle D F(r, X^n(r)), \Phi (r) {d} W_Q(r) \right\rangle . \end{aligned}$$

(1.105)

By the local boundedness of F and its derivatives, it follows that for $\mathbb {P}$-a.e. $\omega$ all the integrands of the deterministic integrals in (1.105) are dominated for $n\in \mathbb {N}$ by integrable functions. Regarding the term containing $A_n^* D F(r, X^n(r))$, recall from (B.11) that $A_n = J_n A$ are uniformly bounded as linear operators from D(A) (endowed with the graph norm) to H. Moreover, thanks to (1.104), (1.44) and the continuity of F and its derivatives, we know that these integrands converge to the corresponding ones in (1.103) (with $\tau _R$ instead of $\tau$) on [0, s], $\mathbb {P}$-a.s. We can thus conclude, by using the Lebesgue dominated convergence theorem, that the deterministic integrals in (1.105) converge to their counterparts in (1.103).

To justify the convergence of the stochastic integral we observe that, with

$$ I_n:= \int _0^s \mathbf{1}_{[0,s \wedge \tau ^R_n]} (r) \left\langle D F(r, X^n(r)), \Phi (r) {d} W_Q(r) \right\rangle , $$

$$ I:= \int _0^s \mathbf{1}_{[0,s \wedge \tau ^R]} (r) \left\langle D F(r, X(r)), \Phi (r) {d} W_Q(r) \right\rangle , $$

we have

$$\begin{aligned}&\quad \mathbb {E}\left| I_n-I\right| ^2 \\&\le \int _0^s\mathbb {E}\Vert \Phi (r)\Vert ^2_{\mathcal {L}_2(\Xi _0,H)}\left| \mathbf{1}_{[0,s \wedge \tau ^R_n]} (r) D F(r, X^n(r))-\mathbf{1}_{[0,s \wedge \tau ^R]} (r) D F(r, X(r))\right| ^2 {d} r\rightarrow 0 \end{aligned}$$

as $n\rightarrow +\infty$ by the dominated convergence theorem. Therefore, up to a subsequence, we have $\lim _{n\rightarrow +\infty }I_n=I$, $\mathbb {P}$-a.s.

It now remains to let $R\rightarrow +\infty$ to obtain the claim. $\square$

Proposition 1.165

Letb and $\sigma$ satisfy Hypothesis 1.125 and let $a:[t, T] \rightarrow \Lambda$ be a progressively measurable process. Let $X(\cdot )$ be the unique mild solution of (1.30) such that $X(0)=x\in H$. Consider $F:[0,T] \times H \rightarrow \mathbb {R}$. Assume that F and its derivatives $F_t,DF, D^2 F$ are continuous on $[0,T]\times H$. Suppose that $D F:[0,T]\times H \rightarrow D(A^*)$ and that $A^*DF$ is continuous on $[0,T]\times H$. Moreover, suppose that there exist $C\ge 0,N\ge 0$ such that

$$\begin{aligned} |F(s,x)|&+ |D F(s,x)| + |F_t(s, x)| + \Vert D^2 F(s,x)\Vert \nonumber \\&+|A^*DF(s, x)| \le C (1 + |x|)^N \end{aligned}$$

(1.106)

for all $x\in H$, $s\in [0,T]$. Let $\tau$ be an $\mathscr {F}_s$-stopping time. Then:

• (i) For $\mathbb {P}$-a.e. $\omega$,

  $$\begin{aligned}&F(s\wedge \tau , X(s\wedge \tau )) = F(0,x) + \int _0^{s\wedge \tau } F_t(r, X(r)) {d} r \nonumber \\&\quad + \int _0^{s\wedge \tau } \left\langle A^* D F(r, X(r)), X(r) \right\rangle {d} r + \int _0^{s\wedge \tau } \left\langle D F(r, X(r)), b(r, X(r), a(r)) \right\rangle {d} r \nonumber \\&+ \frac{1}{2} \int _0^{s\wedge \tau } \mathrm{Tr} \left[ \left( \sigma (r, X(r), a(r)) {Q}^{1/2} \right) \left( \sigma (r, X(r), a(r)) Q^{1/2} \right) ^* D^2 F(r, X(r)) \right] {d} r \nonumber \\&\qquad \qquad + \int _0^{s\wedge \tau } \left\langle D F(r, X(r)), \sigma (r, X(r), a(r)) {d} W_Q(r) \right\rangle \qquad \text {on}\,\,[0,T]. \end{aligned}$$

  (1.107)

• (ii) Let $\eta$ be a real process solving

  $$ \left\{ \begin{array}{l} {d} \eta (s) = \tilde{b} (s) {d} s\\ \eta (0) = \eta _0 \in \mathbb {R}, \end{array} \right. $$

  where $\tilde{b}:[0,T] \rightarrow \mathbb {R}$ is bounded and progressively measurable. Then, for $\mathbb {P}$-a.e. $\omega$,

  $$\begin{aligned}&\quad F(s\wedge \tau , X(s\wedge \tau ))\eta (s\wedge \tau ) = F(0,x)\eta _0 +\int _0^{s\wedge \tau } (F_t(r,X(r))\eta (r) + F(r, X(r)) \tilde{b}(r)) {d} r \nonumber \\&+ \int _0^{s\wedge \tau } \left\langle A^* D F(r, X(r)), X(r) \right\rangle \eta (r){d} r +\int _0^{s\wedge \tau } \left\langle D F(r, X(r)), b(r, X(r), a(r)) \right\rangle \eta (r){d} r \nonumber \\&\quad + \frac{1}{2} \int _0^{s\wedge \tau } \mathrm{Tr} \left[ \left( \sigma (r, X(r), a(r))Q^{\frac{1}{2}}\right) \left( \sigma (r, X(r), a(r)) Q^{\frac{1}{2}} \right) ^* D^2 F(r, X(r)) \right] \eta (r) {d} r \nonumber \\&\qquad \quad + \int _0^{s\wedge \tau } \left\langle D F(r, X(r))\eta (r), \sigma (r, X(r), a(r)) {d} W_Q(r) \right\rangle \qquad \text {on}\,\,[0,T]. \end{aligned}$$

  (1.108)

  In particular, for $s\in [0,T]$,

  $$\begin{aligned}&\quad \mathbb {E} \left[ F(s\wedge \tau , X(s\wedge \tau ))\eta (s\wedge \tau ) \right] = F(0,x)\eta _0 +\mathbb {E} \int _0^{s\wedge \tau } (F_t(r,X(r))\eta (r) + F(r, X(r)) \tilde{b}(r)) {d} r \nonumber \\&+\mathbb {E} \int _0^{s\wedge \tau } \left\langle A^* D F(r, X(r)), X(r) \right\rangle \eta (r){d} r + \mathbb {E} \int _0^{s\wedge \tau } \left\langle D F(r, X(r)), b(r, X(r), a(r)) \right\rangle \eta (r){d} r \nonumber \\&+ \frac{1}{2} \mathbb {E} \int _0^{s\wedge \tau } \mathrm{Tr} \left[ \left( \sigma (r, X(r), a(r))Q^{\frac{1}{2}}\right) \left( \sigma (r, X(r), a(r)) Q^{\frac{1}{2}} \right) ^* D^2 F(r, X(r)) \right] \eta (r) {d} r. \end{aligned}$$

  (1.109)

Proof

Part (i) follows directly from Proposition 1.164 applied with $f(s):= b(s, a(s), X(s))$ and $\Phi (s):= \sigma (s, a(s), X(s))$, $s\in [0,T]$, by noticing that, thanks to (1.33), (1.34) and (1.37), we have $f\in M^p_\mu (0,T;H)$ and $\Phi \in \mathcal {N}^p_Q(0,T;H)$ for every $p\ge 1$.

Part (ii) is a corollary of (i). We introduce the Hilbert space $\hat{H} := H \times \mathbb {R}$ (with the usual inner product), and set

$$ \hat{A}= \left( \begin{matrix} A\\ 0 \end{matrix} \right) , \,\,\, \hat{b}= \left( \begin{matrix} b\\ \tilde{b} \end{matrix} \right) , \,\,\,{\hat{\sigma }}= \left( \begin{matrix} \sigma &{}\ &{}0\\ 0&{}&{}0 \end{matrix} \right) . $$

Then the process

$$ \hat{X}(s) = \left( \begin{matrix} X(s)\\ \eta (s) \end{matrix} \right) $$

is the mild solution of the SDE

$$ \left\{ \begin{array}{l} {d} \hat{X}(s) = \left( \hat{A} \hat{X}(s) + \hat{b}(s, \hat{X}(s),a(s)) \right) {d} s + {\hat{\sigma }} (s, \hat{X}(s), a(s)) {d} W_Q(s) \\ \hat{X}(0) = \left( \begin{matrix} x\\ \eta _0 \end{matrix} \right) . \end{array} \right. $$

Therefore, (1.108) follows from (1.107) applied to the function $\hat{F}(s,\hat{x})= F(s, x) \eta _0$, where $\hat{x}=(x,\eta _0)$. Taking expectation in (1.108) we obtain (1.109). $\square$

Proposition 1.166

Let Hypothesis 1.125 be satisfied and A be maximal dissipative. Let $a:[t, T] \rightarrow \Lambda$ be a progressively measurable process. Let $X(\cdot )$ be the unique mild solution of (1.30) such that $X(0)=x\in H$. Let $F\in C^{1,2}([0,T] \times H)$ be of the form $F(t, x) = \varphi (t,|x|)$ for some $\varphi (t, r)\in C^{1,2} ([0,T] \times \mathbb {R})$, where $\varphi (t,\cdot )$ is even and non-decreasing on $[0,+\infty )$. Moreover, suppose that there exist $C\ge 0,N\ge 0$ such that

$$\begin{aligned} |F(s,x)| + |D F(s,x)| + |F_t(s, x)| + \Vert D^2 F(s, x)\Vert \le C (1 + |x|)^N \end{aligned}$$

(1.110)

for all $x\in H$, $s\in [0,T]$. Let $\tau$ be an $\mathscr {F}_s$-stopping time. Then:

• (i) For $\mathbb {P}$-a.e. $\omega$,

  $$\begin{aligned}&F(s\wedge \tau , X(s\wedge \tau )) \le F(0,x) + \int _0^{s\wedge \tau } \bigg [F_t(r, X(r)) + \left\langle b(r, X(r), a(r)), DF (r, X(r)) \right\rangle \nonumber \\&\qquad + \frac{1}{2} \mathrm{Tr}\left[ \left( \sigma (r, X(r), a(r)) Q^{\frac{1}{2}}\right) \left( \sigma (r, X(r), a(r))Q^{\frac{1}{2}}\right) ^* D^2F(r, X(r)) \right] \bigg ]{d} r \nonumber \\&\qquad \qquad + \int _0^{s\wedge \tau } \left\langle D F(r, X(r)), b(r, X(r), a(r)) {d} W_Q(r) \right\rangle \qquad \text {on}\,\,[0,T]. \end{aligned}$$

  (1.111)

• (ii) If $\eta$ is as in part (ii) of Proposition 1.165 and $\eta$ is positive then, for $\mathbb {P}$-a.e. $\omega$,

  $$\begin{aligned}&\,\, F(s\wedge \tau , X(s\wedge \tau ))\eta (s\wedge \tau ) \le F(0,x)\eta _0 +\int _0^{s\wedge \tau } (F_t(r,X(r))\eta (r) + F(r, X(r)) \tilde{b}(r)) {d} r \nonumber \\&\qquad \qquad \qquad +\int _0^{s\wedge \tau } \left\langle D F(r, X(r)), b(r, X(r), a(r)) \right\rangle \eta (r){d} r \nonumber \\&+ \frac{1}{2} \int _0^{s\wedge \tau } \mathrm{Tr} \left[ \left( \sigma (r, X(r), a(r))Q^{\frac{1}{2}}\right) \left( \sigma (r, X(r), a(r)) Q^{\frac{1}{2}} \right) ^* D^2 F(r, X(r)) \right] \eta (r) {d} r \nonumber \\&\quad \qquad + \int _0^{s\wedge \tau } \left\langle D F(r, X(r))\eta (r), \sigma (r, X(r), a(r)) {d} W_Q(r) \right\rangle \qquad \text {on}\,\,[0,T]. \end{aligned}$$

  (1.112)

  In particular, for $s\in [0,T]$,

  $$\begin{aligned}&\quad \mathbb {E} \left[ F(s\wedge \tau , X(s\wedge \tau ))\eta (s\wedge \tau ) \right] \le F(0,x)\eta _0 \nonumber \\&\quad \qquad \qquad \qquad +\mathbb {E} \int _0^{s\wedge \tau } (F_t(r,X(r))\eta (r) + F(r, X(r)) \tilde{b}(r)) {d} r \nonumber \\&\quad \qquad \qquad \qquad + \mathbb {E} \int _0^{s\wedge \tau } \left\langle D F(r, X(r)), b(r, X(r), a(r)) \right\rangle \eta (r){d} r \nonumber \\&+ \frac{1}{2} \mathbb {E} \int _0^{s\wedge \tau } \mathrm{Tr} \left[ \left( \sigma (r, X(r), a(r))Q^{\frac{1}{2}}\right) \left( \sigma (r, X(r), a(r)) Q^{\frac{1}{2}} \right) ^* D^2 F(r, X(r)) \right] \eta (r) {d} r. \end{aligned}$$

  (1.113)

Proof

(i) We set, for $s\in [0,T]$, $f(s):= b(s, a(s), X(s))$ and $\Phi (s):= \sigma (s, a(s), X(s))$ and consider the approximation $X^n(\cdot )$ of $X(\cdot )$ as in Proposition 1.132. Observe that, thanks to (1.33), (1.34) and (1.37) we have $f\in M^p_\mu (0,T;H)$ and $\Phi \in \mathcal {N}^p_Q(0,T;H)$ for every $p\ge 1$ so the assumptions of Proposition 1.132 are satisfied.

We observe that $DF(s, x) = \frac{\partial \varphi }{\partial r} (s, |x|) \frac{x}{|x|}$ and, since $\varphi (s,\cdot )$ is non-decreasing on $[0,+\infty )$, $\frac{\partial \varphi }{\partial r} (s, r) \ge 0$. Therefore, since A, and thus $A_n$, is dissipative,

$$\begin{aligned} \left\langle A_n X^n(s), DF (r, X^n(s)) \right\rangle = \frac{\partial \varphi }{\partial r} (s, |X^n(s)|) \frac{1}{|X^n(s)|} \left\langle A_n X^n(s), X^n(s) \right\rangle \le 0 \end{aligned}$$

(1.114)

for every $s\ge 0$.

Hence, defining for any $R>|x|$ the stopping times $\tau ^R_n$ as in Proposition 1.164, applying Itô’s formula for $X^n(\cdot )$ and using (1.114), we obtain

$$\begin{aligned}&F(s\wedge \tau ^R_n, X^n(s\wedge \tau ^R_n)) = F(0,x) + \int _0^{s\wedge \tau ^R_n} \bigg [F_t(r, X^n(r)) + \left\langle A_n X^n(r), DF (r, X^n(r)) \right\rangle \nonumber \\&\quad + \left\langle f(r), DF (r, X^n(r)) \right\rangle + \frac{1}{2} \mathrm{Tr}\left[ \left( \Phi (r) Q^{\frac{1}{2}}\right) \left( \Phi (r)Q^{\frac{1}{2}}\right) ^* D^2F(r, X^n(r)) \right] \bigg ] {d} r \nonumber \\&\quad \qquad \qquad + \int _0^{s\wedge \tau ^R_n} \left\langle D F(r, X^n(r)), b(r, X^n(r), a(r)) {d} W_Q(r) \right\rangle \nonumber \\&\qquad \qquad \le F(0,x) + \int _0^{s\wedge \tau ^R_n} \bigg [F_t(r, X^n(r)) + \left\langle f(r), DF (r, X^n(r)) \right\rangle \nonumber \\&\qquad \qquad \quad + \frac{1}{2} \mathrm{Tr}\left[ \left( \Phi (r) Q^{\frac{1}{2}}\right) \left( \Phi (r)Q^{\frac{1}{2}}\right) ^* D^2F(r, X^n(r)) \right] \bigg ]{d} r \nonumber \\&\qquad \qquad \qquad \quad + \int _0^{s\wedge \tau ^R_n} \left\langle D F(r, X^n(r)), b(r, X^n(r), a(r)) {d} W_Q(r) \right\rangle . \end{aligned}$$

(1.115)

It remains to pass to the limit as $n\rightarrow +\infty$ and $R\rightarrow +\infty$ in (1.115). This is done following the same arguments as in the proof of Proposition 1.164.

(ii) The proof combines the proof of (i) with the arguments used in the proof of Proposition 1.165-(ii). $\square$

Remark 1.167

Propositions 1.165 and 1.166 are used to work with viscosity solution test functions in Chap. 3. In particular, parts (ii) of them are useful when discount factors are present (see e.g. Lemma 3.65).$\blacksquare$

The next two non-standard versions of Dynkin’s formula will be used to prove verification theorems in Chaps. 4 and 5.

Proposition 1.168

Let $Q=I$. Assume that Hypothesis 1.149 is satisfied. Assume that there exists $\lambda \in \mathbb {R}$, $\lambda \in \varrho (A)$ such that $(\lambda I- A)^{-1}b:[0,T]\times H\times \Lambda \rightarrow H$ is measurable. Suppose moreover that there exists a $C>0$ such that, for all $(t,x, a)\in [0,T]\times H\times \Lambda$,

$$\begin{aligned} \left\{ \begin{array}{l} |(\lambda I- A)^{-1} b(t,x, a)| \le C ( 1 + |x|)\\ \Vert \sigma (t,x,a) \Vert _{\mathcal {L}(\Xi , H)} \le C ( 1 + |x|). \end{array} \right. \end{aligned}$$

(1.116)

Fix a $\Lambda$-valued progressively measurable process $a(\cdot )$. Let X be the unique mild solution of (1.74) described in Theorem 1.152 such that $X(0)=x\in H$. Let $F:[0,T] \times H \rightarrow \mathbb {R}$ be such that F and its derivatives $F_t,D F, D^2 F$ are continuous in $[0,T]\times H$. Suppose that $D F:[0,T]\times H \rightarrow D(A^*)$, that $A^*DF$ is continuous in $[0,T]\times H$, that $D^2 F:[0,T]\times H\rightarrow \mathcal {L}_1(H)$ is continuous, and that there exist $C>0$ and $N\ge 1$ such that

$$\begin{aligned} |F(s,x)| + |D F(s,x)| + |F_t(s, x)|&+ \Vert D^2 F(s, x)\Vert _{\mathcal {L}_1(H)} \nonumber \\&+ |A^*D F(s, x)| \le C (1 + |x|)^N. \end{aligned}$$

(1.117)

Then, for any $s\in [0,T]$,

$$\begin{aligned}&\mathbb {E} \left[ F(s, X(s)) \right] = F(0,x) +\mathbb {E} \int _0^s F_t(r, X(r)) {d} r +\mathbb {E} \int _0^s \left\langle A^* D F(r, X(r)), X(r) \right\rangle {d} r \nonumber \\&\qquad \quad + \mathbb {E} \int _0^s \left\langle (\lambda I- A^*)D F(r, X(r)), (\lambda I- A)^{-1}b(r, X(r), a(r)) \right\rangle {d} r \nonumber \\&\qquad \quad + \frac{1}{2} \mathbb {E} \int _0^s \mathrm{Tr} \left[ \sigma (r, X(r), a(r)) \sigma (r, X(r), a(r))^* D^2 F(r, X(r)) \right] {d} r. \end{aligned}$$

(1.118)

Proof

We approximate the process $X(\cdot )$ by the processes $X^k(\cdot )$ from Lemma 1.154.

Observe that, thanks to Hypothesis 1.149 and to (1.80), the processes $r{\rightarrow } e^{\frac{1}{k}A} b(r, X(r), a(r))$ and $r{\rightarrow } e^{\frac{1}{k}A} \sigma (r, X(r), a(r))$ belong respectively to $M^p_\mu (0,T;H)$ and $\mathcal {N}_I^p(0,T;H)$ for all $p\ge 1$. Thus we can apply Proposition 1.164 obtaining, for $s\in [0,T]$,

$$\begin{aligned}&\mathbb {E} \left[ F(s, X^k(s)) \right] = F(0,x) + \int _0^s \mathbb {E}\, F_t(r, X^k(r)) {d} r \nonumber \\&\quad + \int _0^s \mathbb {E} \left\langle A^* D F(r, X^k(r)), X^k(r) \right\rangle {d} r + \int _0^s \mathbb {E} \left\langle D F(r, X^k(r)), e^{\frac{1}{k}A} b(r) \right\rangle {d} r \nonumber \\&\qquad \quad + \frac{1}{2} \int _0^s \mathbb {E}\, \mathrm{Tr} \left[ \left( e^{\frac{1}{k}A} \sigma (r) P^k \right) \left( e^{\frac{1}{k}A} \sigma (r) P^k \right) ^* D^2 F(r, X^k(r)) \right] {d} r, \end{aligned}$$

(1.119)

where we use the notation $b(r):=b(r, X(r), a(r))$, $\sigma (r):=\sigma (r, X(r), a(r))$. The claim will follow if we can pass to the limit as $k\rightarrow +\infty$ in each term of this expression. We will only show how to prove the convergence of the last two terms since the arguments for the other terms are similar and simpler.

Using (1.80), (1.86), (1.87) and the dominated convergence theorem it is easy to see that

$$ \lim _{k\rightarrow \infty }|X(\cdot )-X^k(\cdot )|_{M^2_\mu (0,T;H)}=0. $$

Therefore we can find a subsequence, still denoted by $X^k(\cdot )$, that converges to $X(\cdot )$ $dt\otimes \mathbb {P}$-a.e.

Using the assumptions it is obvious that

$$\begin{aligned}&\left\langle D F(r, X^k(r)), e^{\frac{1}{k}A}b(r) \right\rangle =\left\langle (\lambda I- A^*)D F(r, X^k(r)), e^{\frac{1}{k}A} (\lambda I- A)^{-1}b(r) \right\rangle \nonumber \\&\qquad \qquad \qquad \rightarrow \left\langle (\lambda I- A^*)D F(r, X(r)),(\lambda I- A)^{-1}b(r) \right\rangle \quad dt\otimes \mathbb {P}-a.e. \nonumber \end{aligned}$$

as $k\rightarrow +\infty$. Moreover, thanks to (1.80), (1.86), (1.116) and (1.117),

$$\begin{aligned}&\int _0^s \mathbb {E} \left| \left\langle (\lambda I- A^*)D F(r, X^k(r)), e^{\frac{1}{k}A} (\lambda I- A)^{-1}b(r) \right\rangle \right| ^2 {d} r\nonumber \\&\qquad \quad \qquad \qquad \qquad \le C_1 \int _0^s \mathbb {E}\left[ \left( 1 + |X^k(r)|^{2N} \right) \left( 1 +|X(r)|^{2} \right) \right] {d} r \le C_2\nonumber \end{aligned}$$

for some $C_1$ and $C_2$ independent of k. Similarly we obtain

$$ \int _0^s \mathbb {E} \left| \left\langle (\lambda I- A^*)D F(r, X(r)), (\lambda I- A)^{-1}b(r) \right\rangle \right| ^2 {d} r\le C_3 $$

for some $C_3$. Therefore it follows from Lemma 1.51 that

$$\begin{aligned}&\lim _{k\rightarrow +\infty } \int _0^s \mathbb {E} \left\langle D F(r, X^k(r)), e^{\frac{1}{k}A} b(r) \right\rangle {d} r\\ \nonumber&\qquad \qquad \quad \qquad \qquad = \int _0^s \mathbb {E} \left\langle (\lambda I- A^*)D F(r, X(r)),(\lambda I- A)^{-1}b(r) \right\rangle {d} r. \nonumber \end{aligned}$$

Regarding the last term in (1.119),

$$\begin{aligned}&\mathrm{Tr} \left[ e^{\frac{1}{k}A}\sigma (r)P^k (e^{\frac{1}{k}A}\sigma (r)P^k)^* D^2 F(r, X^k(r)) \right] -\mathrm{Tr} \left[ \sigma (r) \sigma (r)^* D^2 F(r, X(r)) \right] \\&= I_1+I_2:= \mathrm{Tr} \left[ e^{\frac{1}{k}A}\sigma (r)P^k (e^{\frac{1}{k}A}\sigma (r)P^k)^* \left( D^2 F(r, X^k(r)) -D^2 F(r, X(r)) \right) \right] \\&\qquad \qquad \quad +\mathrm{Tr} \left[ \left( e^{\frac{1}{k}A}\sigma (r)P^k (e^{\frac{1}{k}A}\sigma (r)P^k)^*-\sigma (r) \sigma (r)^*\right) D^2 F(r, X(r))\right] . \end{aligned}$$

By Proposition B.26, (1.116) and the assumptions for $D^2F$ we have

$$ |I_1|\le C_4(1+|X(r)|)^2\Vert D^2 F(r, X^k(r))-D^2 F(r, X(r))\Vert _{\mathcal {L}_1(H)}\rightarrow 0\quad \text {as}\;\;k\rightarrow +\infty $$

$dt\otimes \mathbb {P}$-a.e. Let $\{e_1,e_2,...\}$ be an orthonormal basis of eigenvectors of $D^2F(r, X(r))$ and $\lambda _1,\lambda _2,...$ be the corresponding eigenvalues. Then

$$\begin{aligned}&\mathrm{Tr} \left[ e^{\frac{1}{k}A}\sigma (r)P^k (e^{\frac{1}{k}A}\sigma (r)P^k)^*D^2 F(r, X(r))\right] \\&\qquad \quad \qquad =\sum _{n=1}^\infty \lambda _n\left| P^k\sigma (r)^*e^{\frac{1}{k}A^*}e_n\right| ^2_{\Xi } \rightarrow \sum _{n=1}^\infty \lambda _n\left| \sigma (r)^*e_n\right| ^2_{\Xi } \\&\qquad \quad \qquad \qquad \qquad \qquad =\mathrm{Tr} \left[ \sigma (r)\sigma (r)^*D^2 F(r, X(r))\right] \quad \text {as}\;\;k\rightarrow +\infty \end{aligned}$$

$dt\otimes \mathbb {P}$-a.e. Therefore $\lim _{k\rightarrow +\infty } (I_1+I_2)=0$ $dt\otimes \mathbb {P}$-a.e. Since, by (1.80), (1.86), (1.116) and (1.117), we also have

$$ \int _0^s \mathbb {E}\, |I_1+I_2|^2 {d} r\le C_5 $$

for some constant $C_5$ independent of k, the convergence of the last term in (1.119) now follows from Lemma 1.51. $\square$

Proposition 1.169

Let Hypotheses 1.143 and 1.145 be satisfied and let $q\ge 2$. Consider $\lambda \in \mathbb {R}$ such that $(\lambda I - A)$ is invertible and $(\lambda I - A)^{-1}\in \mathcal {L}(H)$. Assume that $(\lambda I - A)^{-1}a_2(\cdot )\in M^1_\mu (0,T;H)$. Let X be the unique mild solution of (1.67) described in Proposition 1.147 such that $X(0)=x\in H$. Let $F:[0,T] \times H \rightarrow \mathbb {R}$ be such that F and its derivatives $F_t,D F, D^2 F$ are continuous in $[0,T]\times H$. Suppose that $D F:[0,T]\times H \rightarrow D(A^*)$, $A^*DF$ is continuous in $[0,T]\times H$, $D^2 F:[0,T]\times H\rightarrow \mathcal {L}_1(H)$ is continuous and there exists a $C>0$ such that (1.117) holds with $N=0$. Then, for any $s\in [0,T]$,

$$\begin{aligned}&\mathbb {E} \left[ F(s, X(s)) \right] = F(0,x) +\mathbb {E} \int _0^s F_t(r, X(r)) {d} r \nonumber \\&\,\, +\mathbb {E} \int _0^s \left\langle A^* D F(r, X(r)), X(r) \right\rangle {d} r + \mathbb {E} \int _0^s \left\langle D F(r, X(r)), b_0(r, X(r), a_1(r)) \right\rangle {d} r \nonumber \\&\qquad \qquad \quad + \mathbb {E} \int _0^s \left\langle (\lambda I- A^*)D F(r, X(r)), (\lambda I- A)^{-1}a_2(r) \right\rangle {d} r \nonumber \\&\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + \frac{1}{2} \mathbb {E} \int _0^s \mathrm{Tr} \left[ \sigma Q \sigma ^* D^2 F(r, X(r)) \right] {d} r. \nonumber \end{aligned}$$

Proof

We approximate X using the processes $X^k$ defined in Lemma 1.156. It is immediate to see that $\psi _k \in M^p_\mu (0,T;H)$ and $\sigma \tilde{P}^k \in \mathcal {N}_Q^p(0,T;H)$ for all $p\ge 1$ so we can apply Proposition 1.164 obtaining for every $s\in [0,T]$,

$$\begin{aligned}&\mathbb {E} \left[ F(s, X^k(s)) \right] = F(0,x) + \mathbb {E} \int _0^s F_t(r, X^k(r)) {d} r\ + \mathbb {E} \int _0^s \left\langle A^* D F(r, X^k(r)), X^k(r) \right\rangle {d} r \nonumber \\&\qquad \qquad + \mathbb {E} \int _0^s \mathbf{1}_{B_k}(r,\omega ) \left\langle D F(r, X^k(r)), b_0(r, X(r), a_1(r)) \right\rangle {d} r \nonumber \\&\qquad + \mathbb {E} \int _0^s \mathbf{1}_{D_{m_k}}(r,\omega ) \left\langle (\lambda I- A^*)D F(r,{X^k}(r)), e^{\frac{1}{k}A}(\lambda I- A)^{-1}a_2(r) \right\rangle {d} r \nonumber \\&\qquad \qquad \qquad + \frac{1}{2} \mathbb {E} \int _0^s \mathrm{Tr} \left[ (\sigma Q^{1/2}P^k)(\sigma Q^{1/2}P^k)^* D^2 F(r,{X^k}(r)) \right] {d} r, \end{aligned}$$

(1.120)

where $B_k$, $D_{m_k}$ and $P^k$ are introduced in Lemma 1.156 and in the paragraph before it.

We need to check the convergence of each term of this expression. Using parts (i) and (ii) of Lemma 1.156 we have

$$ \lim _{k\rightarrow \infty }|X(\cdot )-X^k(\cdot )|_{M^1_\mu (0,T;H)}=0. $$

Therefore we can find a subsequence of $X^k$, still denoted by $X^k$, that converges $dt\otimes \mathbb {P}$-a.e. to X. The proof proceeds using the same arguments (and even simpler) as those in the proof of Proposition 1.169. We only look at the two middle terms of the right-hand side of (1.120) that are a little different. We observe that

$$ \left| \mathbb {E} \int _0^s \left\langle (\lambda I- A^*)D F(r,{X^k}(r)), \left( 1 - \mathbf{1}_{D_{m_k}}(r,\omega ) e^{\frac{1}{k}A}\right) (\lambda I- A)^{-1} a_2(r) \right\rangle {d} r \right| $$

converges to zero thanks to the dominated convergence (recall that, by assumption, (1.117) holds with $N=0$). Regarding the fourth term observe that

$$ \mathbf{1}_{B_k}(r,\omega ) \left\langle D F(r, X^k(r)), b_0(r, X(r), a_1(r)) \right\rangle $$

converges to

$$ \left\langle D F(r, X(r)), b_0(r, X(r), a_1(r)) \right\rangle $$

$dt\otimes \mathbb {P}$-a.e. as $k\rightarrow +\infty$. Moreover, since DF is bounded, Hypothesis 1.145-(i) implies

$$ \left| \mathbf{1}_{B_k}(r,\omega ) \left\langle D F(r, X^k(r)), b_0(r, X(r), a_1(r)) \right\rangle \right| \le C f(r)(1+|X(r)|) $$

for all $k\in \mathbb {N}$. Thus the result follows by the dominated convergence theorem. $\square$

1.8 Bibliographical Notes

Section 1.1 contains elements of basic probability and measure theory. Classical references include, for example, [18, 58, 61, 267, 370, 478, 520]. We refer in particular to [58, 61, 267, 370] for the general theory of measure and probability (Sect. 1.1.1) and to [58, 61, 267, 520] for results on measurability (Sects. 1.1.2 and 1.1.3). For the Bochner integral and the integration of Banach-valued functions (Sect. 1.1.3), the reader can consult [190, 191, 194, 397]; some results, useful from the stochastic calculus perspective are contained in [180]. For Sects. 1.1.4 and 1.1.5 conditional expectation for Banach-valued random variables the reader can refer to [180, 356, 370, 478, 572]. Gaussian measure in Hilbert spaces (Sect. 1.1.6) and Fourier transform are nicely introduced in [153, 180] and a more extended study of the subject is contained in [391].

Generalities about stochastic processes, martingales and stopping times in Sect. 1.2 can be found in many different books, e.g. [356, 372, 384, 447–449, 503, 508, 572], while for Hilbert-valued martingales (Sect. 1.2.2) the reader may consult [180, 294, 487]. For standard Wiener and Q-Wiener processes and related results we refer to [124, 180, 294, 372, 447, 448, 452]. The material of Sect. 1.2.4 is based on [180]. Definition 1.92, which not contained in the standard literature, is introduced here because it is useful to study stochastic control problems. The presentation of Lemma 1.94 is based on [372, 513]. The material of Sect. 1.2.5 is loosely based on [180, 294, 372].

The material of Sect. 1.3 is based on [177, 180, 294] (see also [124, 491]). These books present the theory in Hilbert spaces while [447, 448] (see also [192]) present the Banach space case.

The presentation of Sect. 1.4 on solutions of stochastic differential equations in Hilbert spaces is also based on [180, 294]. In particular, [180] is a standard reference in the theory. Other references on strong and mild solutions are, for example, in [124, 177, 413] while a good introduction to variational solutions is in [124, 387, 413, 491, 519]. The reader is also referred to [180] for more on weak mild solutions. Section 1.4.4, containing some results about uniqueness in law, uses the approach of [471]. For a different approach to weak uniqueness based on the theorem of Yamada–Watanabe, we refer the reader to [491], Appendix E.

Section 1.5 contains existence and uniqueness results for stochastic differential equations with special unbounded terms and cylindrical additive noise. They are more or less common knowledge, however we presented proofs since no complete references seem to be available in the literature.

Classical results on transition semigroups (Sect. 1.6) can be found in [180]. The statements here are a little modified and extended so that they may be used in our applications to optimal control, mainly in Chap. 4.

Section 1.7 contains various versions of Itô’s and Dynkin’s formulae (Propositions 1.164–1.166) in connection with mild solutions for functions that have properties of test functions used in the definition of a viscosity solution (Definition 3.32). Such results have been known and used in the viscosity solution literature, however complete proofs are available only in [374]. The statements here are slightly more general than those in [374] and we presented proofs for the reader’s convenience. The last two results of Sect. 1.7 (Propositions 1.168 and 1.169) are used to prove the verification theorems of Sects. 4.8 and 5.5. They have been used in the literature (e.g. in [306]) but without complete proofs, hence we provide them for completeness. We finally recall that Itô’s formula related to variational solutions of linear stochastic parabolic equations is proved in [467].