Abstract
In this chapter we discuss so-called “backward stochastic differential equations”, BSDEs for short. Linear BSDEs first appeared a long time ago, both as the equations for the adjoint process in stochastic control, as well as the model behind the Black and Scholes formula for the pricing and hedging of options in mathematical finance. These linear BSDEs can be solved more or less explicitly (see Proposition 5.31 below). However, the first published paper on nonlinear BSDEs, appeared only in 1990, see Pardoux and Peng [51]. Since then, the interest in BSDEs has increased regularly, due to the connections of this subject with mathematical finance, stochastic control, and partial differential equations.
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5.1 Introduction
In this chapter we discuss so-called “backward stochastic differential equations”, BSDEs for short. Linear BSDEs first appeared a long time ago, both as the equations for the adjoint process in stochastic control, as well as the model behind the Black and Scholes formula for the pricing and hedging of options in mathematical finance. These linear BSDEs can be solved more or less explicitly (see Proposition 5.31 below). However, the first published paper on nonlinear BSDEs, appeared only in 1990, see Pardoux and Peng [51]. Since then, the interest in BSDEs has increased regularly, due to the connections of this subject with mathematical finance, stochastic control, and partial differential equations. We refer the interested reader to El Karoui et al. [29] and [30], Pham [60] and the references therein for developments on the use of BSDEs as models in mathematical finance, as well as the connection of BSDEs with stochastic control (see also [28] and [37]). BSDEs are also an efficient tool for constructing \(\Gamma \)-martingales on manifolds with prescribed limit, see Darling [19]. The connection of BSDEs with semi linear PDEs was initiated in Pardoux, Peng [54], see also among the now vast literature on the subject [6, 48, 52] and [53].
We shall present both the abstract theory of BSDEs, and the connection of BSDEs with semilinear PDEs (both parabolic and elliptic). Let us motivate the notion of a BSDE via an associated semilinear parabolic PDE.
To each \((t,x) \in \mathbb{R}_{+} \times \mathbb{R}^{d}\), we associate the Markov diffusion process {X s t, x, s ≥ t} which is a solution of the SDE
where the Brownian motion B has dimension k. The associated infinitesimal generator reads
Let T > 0 be an arbitrary final time, \(\kappa \in C(\mathbb{R}^{d})\) and \(F \in C([0,T] \times \mathbb{R}^{d} \times \mathbb{R} \times \mathbb{R}^{k})\). We consider the following backward semilinear second order PDE
Suppose that this equation has a classical solution \(u \in C^{1,2}([0,T] \times \mathbb{R}^{d})\). It then follows from Itô’s formula that for any 0 ≤ t < s ≤ T,
Considering the pair of adapted processes
we have that for each \((t,x) \in [0,T] \times \mathbb{R}^{d}\),
and Y t t, x is a deterministic quantity which equals u(t, x). The solution u of the above semilinear parabolic PDE is expressed in terms of the solution of this last backward stochastic differential equation (BSDE). We will see below that this is indeed an extension of the Feynman–Kac formula (in the sense that if F is affine, then the Feynman–Kac formula is a consequence of the above representation). Note that the above computation can be applied to a system of PDEs, rather than a single PDE. We shall consider only the case where the same second order PDE operator \(\mathcal{A}\) is applied to each coordinate u i of u. A probabilistic representation for more general systems of semilinear PDEs, with a different \(\mathcal{A}\) for each coordinate of u, can be found in [55], see also [52] and [58].
Let us now write an abstract version of the above BSDE. Let t = 0, and forget about the superscript x. Suppose now that we are given a probability space with filtration \((\Omega,\mathcal{F},\mathcal{F}_{t}, \mathbb{P})\) and for each \((y,z) \in \mathbb{R} \times \mathbb{R}^{k}\), a measurable process {F(t, y, z), 0 ≤ t ≤ T} (F being jointly measurable), together with an \(\mathcal{F}_{T}\) random variable η.
We formulate the problem of solving a BSDE as follows: find a pair of adapted processes {(Y t , Z t ), 0 ≤ t ≤ T} such that
Note that, since the boundary condition for \(\{Y _{t}: t \in \left [0,T\right ]\}\) is given at the terminal time T, it is not really natural for the solution {Y t } to be adapted at each time t to the past of the Brownian motion {B s } before time t. The price we have to pay for such a severe constraint to be satisfied is to have the coefficient of the Brownian motion – the process {Z t } – to be chosen independently of {Y t }, hence the solution of the BSDE is a pair of processes. Note that in the case \(F \equiv 0\), \(Y _{t} = \mathbb{E}(\eta \vert \mathcal{F}_{t})\) and Z is given by the martingale representation theorem from Sect. 2.4.
One may also think of a “backward SDE” as an inverse problem for an SDE, namely we are looking for a point \(y \in \mathbb{R}\), and an adapted process {Z t }, such that the solution {Y t } of
satisfies Y T = η.
Finally, note that while the above presentation treats T as a deterministic quantity, an important alternative is to replace it by a stopping time (or else by + ∞). This is essential when giving probabilistic representations of semilinear elliptic PDEs.
In this chapter, we suppose given a stochastic basis \((\Omega,\mathcal{F}, \mathbb{P},\left \{\mathcal{F}_{t}\right \}_{t\geq 0})\) with {B t ; t ≥ 0} a k-dimensional Brownian motion and the filtration \(\left \{\mathcal{F}_{t}\right \}_{t\geq 0}\) being the natural filtration of {B t : t ≥ 0}, i.e. for all t ≥ 0:
5.2 Basic Inequalities
For convenience we rewrite in this context the Itô formula (2.14) and we give a basic inequality. First we introduce a notation used in this chapter.
Notation 5.1.
For p ≥ 1 we define
Let \(\left (Y,Z\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) satisfy for all \(t \in \left [0,T\right ],\quad \mathbb{P}\mbox{ -}a.s\).:
where
-
\(\diamond \) K ∈ S m 0,
-
\(\diamond \) \(K_{\cdot }\left (\omega \right ) \in BV _{loc}\left (\left [0,\infty \right [; \mathbb{R}^{m}\right ),\; \mathbb{P}\text{-a.s.}\;\omega \in \Omega \).
5.2.1 Backward Itô’s Formula
If \(\varphi \in C^{1,2}\left (\left [0,T\right ] \times \mathbb{R}^{m}\right )\) , then \(\mathbb{P}\mbox{ -}a.s\)., for all \(t \in \left [0,T\right ]\):
From Corollary 2.29 we get for all \(p \in \mathbb{R}\),
where
We have \(\left \vert U_{s}^{\left (p,\varepsilon \right )}\right \vert \leq \left (\left \vert Y _{s}\right \vert ^{2}+\varepsilon \right )^{\left (p-1\right )/2}\) and
where \(n_{p}\mathop{ =}\limits^{ \mathit{def }}1 \wedge \left (p - 1\right )\) and \(m_{p}\mathop{ =}\limits^{ \mathit{def }}1 \vee \left (p - 1\right )\).
Moreover
In particular for p ≥ 1 and \(\varepsilon \searrow 0\) we obtain
where
and \(\left \{L_{t}: t \geq 0\right \}\) is an increasing continuous progressively measurable stochastic process such that for all t ≥ 0 (in the sense of convergence in probability)
The stochastic process \(\left \{L_{t}: t \geq 0\right \}\) has the following property:
for every interval \(\left [s,t\right ] \subset \left \{r \geq 0: Y _{r}\left (\omega \right ) = 0\right \}\), or \(\left [s,t\right ] \subset \mathrm{ int}\left \{r \geq 0: Y _{r}\left (\omega \right )\neq 0\right \}\).
Moreover, we have
Since
it follows that for every p ≥ 1 and 0 ≤ t ≤ T:
In fact we deduce from Lemma 2.37 a more general inequality:
\(\diamond \) if \(\psi \,: \left [0,T\right ] \times \mathbb{R}^{m} \rightarrow \mathbb{R}\) is a function of class C 1, convex in its second argument,then \(a.s.,\ \forall \ t \in \left [0,T\right ]\):
5.2.2 A Fundamental Inequality
Let \(\left (Y,Z\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) satisfy an identity of the form
where
-
\(\diamond \) \(K \in S_{m}^{0}\left (\left [0,T\right ]\right )\) and \(K_{\cdot }\left (\omega \right ) \in BV \left (\left [0,T\right ]; \mathbb{R}^{m}\right ),\; \mathbb{P}\text{-a.s.}\;\omega \in \Omega \).
-
\(\diamond \) Assume that there exist
-
(a)
\(D,R,N\quad \mathcal{P}\)-m.i.c.s.p., D 0 = R 0 = N 0 = 0;
-
(b)
\(V \quad \mathcal{P}-\) m.b-v.c.s.p. V 0 = 0;
-
(c)
λ < 1 ≤ p,
such that as measures on \(\left [0,T\right ],\;a.s.\)
-
(a)
where
By Proposition 6.80, Corollary 6.81 and Corollary 6.82 from Annex C we have:
Proposition 5.2.
Let (5.6) and (5.7) be satisfied and moreover
-
(A)
If p > 1, then there exists a positive constant C p,λ , depending only upon \(\left (p,\lambda \right )\) , such that, \(\mathbb{P}\mbox{ -}a.s\) ., for all \(t \in \left [0,T\right ]\) :
$$\displaystyle\begin{array}{rcl} & & \mathbb{E}^{\mathcal{F}_{t} }\sup \limits _{r\in \left [t,T\right ]}\left \vert e^{V _{r} }Y _{r}\right \vert ^{p} + \mathbb{E}^{\mathcal{F}_{t} }\left (\int _{t}^{T}e^{2V _{r} }dD_{r}\right )^{p/2} + \mathbb{E}^{\mathcal{F}_{t} }\left (\int _{t}^{T}e^{2V _{r} }\left \vert Z_{r}\right \vert ^{2}\mathit{dr}\right )^{p/2} \\ & & \quad \leq C_{p,\lambda }\ \mathbb{E}^{\mathcal{F}_{t} }\ \left [\left \vert e^{V _{T} }Y _{T}\right \vert ^{p} + \left (\int _{ t}^{T}e^{2V _{r} }\mathbf{1}_{p\geq 2}\mathit{dR}_{r}\right )^{p/2} + \left (\int _{ t}^{T}e^{V _{r} }\mathit{dN}_{r}\right )^{p}\right ]. {}\end{array}$$(5.8) -
(B)
If p = 1 (and n p = 0), then \(\mathbb{P}\mbox{ -}a.s\) ., for all 0 ≤ t ≤ T
$$\displaystyle{e^{V _{t} }\left \vert Y _{t}\right \vert \leq \mathbb{E}^{\mathcal{F}_{t} }e^{V _{T} }\left \vert Y _{T}\right \vert + \mathbb{E}^{\mathcal{F}_{t} }\int _{t}^{T}e^{V _{r} }\mathit{dN}_{r}}$$and for all 0 < α < 1 there exists a positive constant C α , depending only upon α such that
$$\displaystyle{\begin{array}{r} \sup \limits _{r\in \left [t,T\right ]}\left [\mathbb{E}\left (e^{V _{r}}\left \vert Y _{r}\right \vert \right )\right ]^{\alpha } + \mathbb{E}\left (\sup \limits _{r\in \left [t,T\right ]}\ \left \vert e^{V _{r}}Y _{r}\right \vert ^{\alpha }\right ) + \mathbb{E}\left (\int _{t}^{T}e^{2V _{r}}\left \vert Z_{r}\right \vert ^{2}\mathit{dr}\right )^{\alpha /2} \\ + \mathbb{E}\left (\int _{t}^{T}e^{2V _{r} }\left \vert D_{r}\right \vert ^{2}\mathit{dr}\right )^{\alpha /2} \\ \leq C_{\alpha }\left [\Big(\mathbb{E}\left (e^{V _{T}}\left \vert Y _{T}\right \vert \right )\Big)^{\alpha } + \left (\mathbb{E}\int _{t}^{T}e^{V _{r}}\mathit{dN}_{r}\right )^{\alpha }\right ].\end{array} }$$ -
(C)
If p ≥ 1 and R = N = 0, then \(\mathbb{P}\mbox{ -}a.s\) ., for all \(t \in \left [0,T\right ]\) :
$$\displaystyle{ e^{pV _{t} }\left \vert Y _{t}\right \vert ^{p} \leq \mathbb{E}^{\mathcal{F}_{t} }e^{pV _{T} }\left \vert Y _{T}\right \vert ^{p}. }$$(5.9)
Corollary 5.3.
Under the assumptions of Proposition 5.2 , if there exists a c ≥ 0 such that \(\sup _{s\in \left [0,T\right ]}\left \vert V _{s}\right \vert \leq c\) , then \(\mathbb{P}\mbox{ -}a.s\) ., for all \(t \in \left [0,T\right ]\) :
5.3 BSDEs with Deterministic Final Time
Our main goal in this section is to study backward stochastic differential equations (abbreviated BSDEs) of the form
or equivalently, a.s. for all \(t \in \left [0,T\right ]\):
whose solution \(\left (Y _{t},Z_{t}\right )_{t\in \left [0,T\right ]}\) takes values in \(\mathbb{R}^{m} \times \mathbb{R}^{m\times k}\), and where we assume in this section that:
-
T > 0 is a fixed final deterministic time;
-
\(\eta: \Omega \rightarrow \mathbb{R}^{m}\), the final condition, is an \(\mathcal{F}_{T}\)-measurable random vector;
-
\(F: \Omega \times \left [0,T\right ] \times \mathbb{R}^{m} \times \mathbb{R}^{m\times k} \rightarrow \mathbb{R}^{m}\) is a \(\left (\mathcal{P}, \mathbb{R}^{m} \times \mathbb{R}^{m\times k}\right )\)-Carathéodory function, that is
$$\displaystyle{\begin{array}{l} F\left (\cdot,\cdot,y,z\right )\text{ is }\mathcal{P}\mbox{ -}\text{m.s.p., }\forall \ \left (y,z\right ) \in \mathbb{R}^{m}\times \mathbb{R}^{m\times k}; \\ F\left (\omega,t,\cdot,\cdot \right )\text{ is a continuous function,}\ d\mathbb{P} \otimes dt\mbox{ -}a.e.;\end{array} }$$ -
\(G: \Omega \times \left [0,T\right ] \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}\) is a \(\left (\mathcal{P}, \mathbb{R}^{m}\right )\)-Carathéodory function, i.e.
$$\displaystyle{\begin{array}{l} G\left (\cdot,\cdot,y\right )\text{ is }\mathcal{P}\mbox{ -}\text{m.s.p., }\forall \ y \in \mathbb{R}^{m}; \\ G\left (\omega,t,\cdot,\cdot \right )\text{ is a continuous function,}\ d\mathbb{P} \otimes dt\mbox{ -}a.e.;\end{array} }$$ -
A is a \(\mathcal{P}\)-m.i.c.s.p., A 0 = 0.
Note that, by Exercise 1.1, F is \(\left (\mathcal{P}\otimes \mathcal{B}_{m} \otimes \mathcal{B}_{m\times k},\mathcal{B}_{m}\right )\)-measurable and G is \(\left (\mathcal{P}\otimes \mathcal{B}_{m},\mathcal{B}_{m}\right )\)-measurable.
We state the following definition:
Definition 5.4.
A pair \(\left (Y,Z\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) is a solution of (5.10) if
and, a.s. for all \(t \in \left [0,T\right ]\):
5.3.1 A Priori Estimates and Uniqueness
We now consider the BSDE
where
-
\(\eta: \Omega \rightarrow \mathbb{R}^{m}\), the final condition, is an \(\mathcal{F}_{T}\)-measurable random vector;
-
\(\left (\omega,t,y,z\right )\longmapsto \Phi \left (\omega,t,y,z\right ): \Omega \times \left [0,T\right ] \times \mathbb{R}^{m} \times \mathbb{R}^{m\times k} \rightarrow \mathbb{R}^{m};\)
-
\(\left (\omega,t\right )\longmapsto Q_{t}\left (\omega \right ): \Omega \times \left [0,T\right ] \rightarrow \mathbb{R}\) is a \(\mathcal{P}\)-m.i.c.s.p. such that Q 0 = 0.
Note that the BSDE (5.10) can be written in this form with
where \(\left \{\alpha _{t}: t \geq 0\right \}\) and \(\left \{\beta _{t}: t \geq 0\right \}\) are two real positive \(\mathcal{P}\)-m.s.p. (given by the Radon–Nikodym representation theorem), α t +β t = 1, such that
We define for any ρ ≥ 0
The basic assumptions on \(\Phi \) are the following
-
\(\blacklozenge\) \(\forall \ y \in \mathbb{R}^{m}\), \(z \in \mathbb{R}^{m\times k}\), the function \(\Phi \left (\cdot,\cdot,y,z\right ): \Omega \times \left [0,T\right ] \rightarrow \mathbb{R}^{m}\) is \(\mathcal{P}\)-measurable;
-
\(\blacklozenge\) there exist three \(\mathcal{P}\)-m.s.p. \(\mu: \Omega \times \left [0,T\right ] \rightarrow \mathbb{R}\) and \(\ell,\alpha: \Omega \times \left [0,T\right ] \rightarrow \mathbb{R}_{+}\) such that, \(\mathbb{P}\mbox{ -}a.s\).
and for all \(y,y^{{\prime}}\in \mathbb{R}^{m}\) and \(\;z,z^{{\prime}}\in \mathbb{R}^{m\times k},\;d\mathbb{P} \otimes \mathit{dQ}_{t}\mbox{ -}a.e\).:
The assumptions on \(\Phi \) yield a continuity behaviour result which we leave as an exercise for the reader.
Lemma 5.5.
Under the assumption (5.15)
and the mapping
is continuous from \(S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) into \(S_{m}^{0}\left [0,T\right ]\) .
We shall show that the monotonicity of \(\Phi \) yields an inequality of the form (5.7).
Let (with a > 1 arbitrary)
We have:
Lemma 5.6.
Let a,p > 1, r 0 ≥ 0 and the assumptions ( 5.13 - BSDE-H \(_{\Phi }\) ) be satisfied. Let \(\left (Y,Z\right ),\;\left (\tilde{Y },\tilde{Z}\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) . Then, in the sense of signed measures on \(\left [0,T\right ]\) :
and
where
Proof.
The monotonicity property of \(\Phi \) implies that for any \(\mathbb{R}^{m}\)-valued stochastic process \(\left \{U_{s}: s \geq 0\right \}\), \(\left \vert U_{s}\right \vert \leq 1\):
Since
it follows that
Hence
(5.16) follows if we choose
The inequality (5.17) is obtained as follows:
■
Taking into account Proposition 5.2 with \(\mathit{dK}_{s} = \Phi \left (s,Y _{s},Z_{s}\right )\mathit{dQ}_{s}\), we deduce from (5.16), first with r 0 = 0 and then with r 0 > 0:
Proposition 5.7.
Let the assumptions ( 5.13 - BSDE-H \(_{\Phi }\) ) be satisfied. Then for every a,p > 1 there exists a constant C a,p such that for all solutions \(\left (Y,Z\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) of the BSDE (5.12) satisfying
where again
the following inequality holds, \(\mathbb{P}\text{-a.s.},\;\) for all \(t \in \left [0,T\right ]\) :
Moreover, if p ≥ 2, then for all r 0 > 0:
Corollary 5.8.
Let p = 1. Let the assumptions ( 5.13 - BSDE-H \(_{\Phi }\) ) be satisfied and \(\Phi \) be independent of \(z \in \mathbb{R}^{m\times k}\) ( \(\ell_{t} \equiv 0\) and \(V _{t} =\bar{\mu } _{t} =\int _{ 0}^{t}\mu _{s}\mathit{dQ}_{s}\) ). If \(\left (Y,Z\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) is a solution of the BSDE (5.12) satisfying
then the following inequality holds \(\mathbb{P}\mbox{ -}a.s\) ., for all \(t \in \left [0,T\right ]\) :
Moreover for all 0 < q < 1
Proof.
Since
the conclusions follow by Corollary 6.81. ■
From (5.19) we immediately have:
Corollary 5.9.
Let a,p > 1. If
and there exists a constant A ≥ 0 such that for all \(t \in \left [0,T\right ]\) :
then for all \(t \in \left [0,T\right ]\) :
Let \(\left (Y,Z\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) be a solution of the BSDE
where \(\Phi \) satisfies (5.13 \(-\mathbf{BSDE}\mbox{ -}\mathbf{H}_{\Phi })\) and \((\hat{Y },\hat{Z}) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) is a solution of the BSDE
where \(\hat{\Phi }\left (\cdot,\cdot,\cdot,\cdot \right ): \Omega \times \left [0,T\right ] \times \mathbb{R}^{m} \times \mathbb{R}^{m\times k} \rightarrow \mathbb{R}^{m}\) is \(\mathcal{P}\)-measurable with respect to \(\left (\omega,t\right ) \in \Omega \times \left [0,T\right ]\) and continuous with respect to \(\left (y,z\right ) \in \mathbb{R}^{m} \times \mathbb{R}^{m\times k}\). We clearly need to assume that
Note that
where
and by the assumptions (5.13 \(-\mathbf{BSDE}\mbox{ -}\mathbf{H}_{\Phi })\)
with, as above,
Hence by Proposition 5.2 we have:
Theorem 5.10 (Continuity and Uniqueness).
Let a,p > 1 and the assumptions ( 5.13 \(-\mathbf{BSDE}\mbox{ -}\mathbf{H}_{\Phi })\) be satisfied. Let
be solutions of the BSDEs (5.21) and (5.22) respectively. If
then there exists a positive C a,p such that:
If \(\Phi =\hat{ \Phi }\) , then for all 0 ≤ t ≤ s ≤ T,
In particular uniqueness follows in the space \(S_{m}^{p}\left (\left [0,T\right ];e^{V }\right ) \times \Lambda _{m\times k}^{0}\left (0,T\right )\) , where
Recall the notation
Theorem 5.11 (Continuity and Uniqueness).
Let p = 1. Assume that \(\Phi,\hat{\Phi }: \Omega \times \left [0,T\right ] \times \mathbb{R}^{m} \times \mathbb{R}^{m\times k} \rightarrow \mathbb{R}^{m}\) satisfy assumptions ( 5.13 ) and both are independent of \(z \in \mathbb{R}^{m\times k}\) ( \(\ell_{t} =\hat{\ell} _{t} \equiv 0\) ). If \(\left (Y,Z\right ),(\hat{Y },\hat{Z}) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) are two solutions of the BSDE (5.107) corresponding respectively to \(\left (\eta,\Phi \right )\) and \((\hat{\eta },\hat{\Phi })\) such that
and \(\Delta _{s}\mathop{ =}\limits^{ \mathit{def }}\Phi (s,\hat{Y }_{s},\hat{Z}_{s}) -\hat{ \Phi }(s,\hat{Y }_{s},\hat{Z}_{s})\) , then \(\mathbb{P}\mbox{ -}a.s\) ., for all \(t \in \left [0,T\right ]\) :
and for every \(q \in \left (0,1\right )\) there exists a constant C q such that
Proof.
Since
the conclusions follow by Corollary 6.81. ■
5.3.2 Complementary Results
In this subsection we generalize the uniqueness result and we shall give a scheme to obtain the solution as a limit of uniformly bounded solutions of approximate BSDEs.
Let a, p > 1 and
Define
Note that if 1 < a 1 < a 2 then \(V _{t}^{a_{1},p} \leq V _{t}^{a_{2},p}\) and consequently
Let
Remark 5.12.
If Q, μ and ℓ are deterministic functions, then for all a, p > 1:
and
Corollary 5.13.
Let the assumptions \((\mathbf{BSDE}\mbox{ -}\mathbf{H}_{\Phi })\) be satisfied. Then for each p > 1, the BSDE (5.12) has at most one solution
If, moreover,
then the BSDE (5.12) has at most one solution
Proof.
Let \(\left (Y,Z\right ),(\hat{Y },\hat{Z}) \in S_{m}^{0}\left (\left [0,T\right ]\right ) \times \Lambda _{m\times k}^{0}\left (0,T\right )\) be two solutions of the BSDE (5.12) corresponding to η.
-
(A)
Let p > 1 be such that \(\left (Y,Z\right ),(\hat{Y },\hat{Z}) \in S_{m}^{1+,p}\left (\left [0,T\right ];e^{V }\right ) \times \Lambda _{m\times k}^{0}\left (0,T\right )\). Then from (5.26) and the definition of \(S_{m}^{1+,p}\left (\left [0,T\right ];e^{V }\right )\) there exists an a > 1 such that
$$\displaystyle{\mathbb{E}\ \sup \limits _{t\in \left [0,T\right ]}\left \vert e^{V _{t}^{a,p} }Y _{t}\right \vert ^{p} < \infty \;\;\text{and}\;\;\mathbb{E}\ \sup \limits _{ t\in \left [0,T\right ]}\left \vert e^{V _{t}^{a,p} }\hat{Y }_{t}\right \vert ^{p} < \infty,}$$i.e. the condition (5.23) is satisfied; consequently the estimate (5.24) follows and uniqueness too.
-
(B)
If \(\left (Y,Z\right ),(\hat{Y },\hat{Z}) \in S_{m}^{1+,1+}\left (\left [0,T\right ];e^{V }\right ) \times \Lambda _{m\times k}^{0}\left (0,T\right )\) then there exist a 1, a 2, p 1, p 2 > 1 such that
$$\displaystyle{\mathbb{E}\ \sup \limits _{t\in \left [0,T\right ]}\left \vert e^{V _{t}^{a_{1},p_{1}} }Y _{t}\right \vert ^{p_{1} } < \infty \;\;\text{and}\;\;\mathbb{E}\ \sup \limits _{t\in \left [0,T\right ]}\left \vert e^{V _{t}^{a_{2},p_{2}} }\hat{Y }_{t}\right \vert ^{p_{2} } < \infty.}$$Let a > 1 and 1 < p < p 1 ∧ p 2. Put
$$\displaystyle{b_{i} = \dfrac{a} {2n_{p}} - \dfrac{a_{i}} {2n_{p_{i}}}.}$$Since
$$\displaystyle\begin{array}{rcl} V _{t}^{a,p}& =& \int _{ 0}^{t}\mu _{ s}\mathit{dQ}_{s} + \dfrac{a} {2n_{p}}\int _{0}^{t}\left (\ell_{ s}\right )^{2}\mathit{ds} {}\\ & =& V _{t}^{a_{i},p_{i} } + b_{i}\int _{0}^{t}\left (\ell_{ s}\right )^{2}\mathit{ds}, {}\\ \end{array}$$we get
$$\displaystyle{\begin{array}{l} \mathbb{E}\ \sup \limits _{t\in \left [0,T\right ]}\left \vert e^{V _{t}^{a,p} }Y _{t}\right \vert ^{p} = \mathbb{E}\ \left \{\sup \limits _{t\in \left [0,T\right ]}\left \vert e^{V _{t}^{a_{i},p_{i}} }Y _{t}\right \vert ^{p}\exp \left [pb_{i}\int _{0}^{T}\left (\ell_{ s}\right )^{2}\mathit{ds}\right ]\right \} \\ \quad \quad \quad \quad \leq \left (\mathbb{E}\ \sup \limits _{t\in \left [0,T\right ]}\left \vert e^{V _{t}^{a_{i},p_{i}} }Y _{t}\right \vert ^{p_{i}}\right )^{ \frac{p} {p_{i}} }\left \{\mathbb{E}\exp \left [\frac{p_{i}pb_{i}} {p_{i}-p}\int _{0}^{T}\left (\ell_{ s}\right )^{2}\mathit{ds}\right ]\right \}^{\frac{p_{i}-p} {p_{i}} } \\ \quad \quad \quad \quad < \infty.\end{array} }$$Similar we have
$$\displaystyle{\mathbb{E}\ \sup \limits _{t\in \left [0,T\right ]}\left \vert e^{V _{t}^{a,p} }\hat{Y }_{t}\right \vert ^{p} < \infty.}$$Hence the estimate (5.24) holds and the uniqueness follows. ■
The next Proposition will allow us to extend existence results from situations where the data satisfy the following strong boundedness condition: there exists a positive constant C such that for all \(t \in \left [0,T\right ]\):
where
Let
We have \(V _{s} - V _{t} \leq \hat{ V }_{s} -\hat{ V }_{t}\) for all 0 ≤ t ≤ s ≤ T.
Define, for \(n \in \mathbb{N}^{{\ast}}\),
and the stochastic processes
It is easy to verify that there exists a positive constant M n, p, a such that
Proposition 5.14.
Let a,p > 1 and the assumptions ( 5.13 - BSDE-H \(_{\Phi }\) ) be satisfied. Also assume that
If for each \(n \in \mathbb{N}^{{\ast}}\), \(\left (Y ^{n},Z^{n}\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) is a solution of the BSDE
such that e V Y n \(\in S_{m}^{p}\left [0,T\right ]\) , then
and there exists (a unique!) \(\left (Y,Z\right ) \in S_{m}^{p}\left (\left [0,T\right ];e^{V }\right ) \times \Lambda _{m\times k}^{p}\left (0,T;e^{V }\right )\) such that
and, \(\mathbb{P}\mbox{ -}a.s\) . for all \(t \in \left [0,T\right ]\) :
Proof.
In view of (5.19) we have for all \(t \in \left [0,T\right ]\):
and (5.28) follows.
For all \(n,i \in \mathbb{N}^{{\ast}}\):
where
and similarly for K t n+i.
Since
we deduce from Proposition 5.2 that
Hence there exists \(\left (Y,Z\right ) \in S_{m}^{p}\left (\left [0,T\right ];e^{V }\right ) \times \Lambda _{m\times k}^{p}\left (0,T;e^{V }\right )\) such that (5.29) holds. The last assertion follows from Lemma 5.16 below, whose proof is left as an exercise for the reader. ■
Clearly from the construction in Proposition 5.14 we have:
Corollary 5.15.
Suppose that the assumptions from ( 5.13 - BSDE-H \(_{\Phi }\) ) are satisfied. Then the existence of a solution under the conditions (5.27) with p = 2 and some a > 1 implies existence under the same conditions for any p > 1.
We end this subsection with a continuity result, the easy proof of which is left as an exercise for the reader.
Lemma 5.16.
Let the assumptions ( 5.13 - BSDE-H \(_{\Phi }\) ) be satisfied. If \(\left (Y,Z\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) , then
and the mapping
is continuous.
5.3.3 BSDEs with Lipschitz Coefficients
5.3.3.1 BSDEs with Deterministic Lipschitz Conditions
Consider the backward stochastic differential equation: \(\mathbb{P}\mbox{ -}a.s\)., for all \(t \in \left [0,T\right ]\)
under the assumptions
-
\(\diamond \) p > 1,
$$\displaystyle{ \eta \in L^{p}\left (\Omega,\mathcal{F}_{ T}, \mathbb{P}; \mathbb{R}^{m}\right ), }$$(5.32) -
\(\diamond \) the function \(F\left (\cdot,\cdot,y,z\right ): \Omega \times \left [0,T\right ] \rightarrow \mathbb{R}^{m}\) is \(\mathcal{P}\)-measurable for every \(\left (y,z\right ) \in \mathbb{R}^{m} \times \mathbb{R}^{m\times k}\),
-
\(\diamond \) there exist \(L \in L^{1}\left (0,T\right )\), \(\ell\in L^{2}\left (0,T\right )\) such that
We recall the notation
Theorem 5.17.
Let p > 1 and the assumptions (5.32) and (5.33) be satisfied. Then the BSDE (5.31) has a unique solution \(\left (Y,Z\right ) \in S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\) . Moreover uniqueness holds in \(S_{m}^{1+}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) .
Proof.
We first remark that if \(\left (Y,Z\right ) \in S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\) then
Indeed, since
then
Uniqueness follows from Corollary 5.13.
We prove existence.
Note that a solution of the Eq. (5.31) is a fixed point of the mapping \(\Gamma: S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right ) \rightarrow S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\) defined by
where
By Corollary 2.45 the mapping \(\Gamma \) is well defined.
Let \(M \in \mathbb{N}^{{\ast}}\) and 0 = T 0 < T 1 < ⋯ < T M = T, with \(T_{i} = \dfrac{iT} {M}\). Then
We show that \(\Gamma \) is a strict contraction on the Banach space \(S_{m}^{p}\left [T_{M-1},T\right ] \times \Lambda _{m\times k}^{p}\left (T_{M-1},T\right )\) with the norm
for M large enough.
Let \(\left (X,U\right ),\left (X^{{\prime}},U^{{\prime}}\right ) \in S_{m}^{p}\left [T_{M-1},T\right ] \times \Lambda _{m\times k}^{p}\left (T_{M-1},T\right )\). Then
where
Since
and
we have by (5.8), with D = 0, R = V = 0, λ = 0,
Let \(M_{0} \in \mathbb{N}^{{\ast}}\) be such that
Then \(\Gamma \) is a strict contraction on \(S_{m}^{p}\left [T_{M_{0}-1},T\right ] \times \Lambda _{m\times k}^{p}\left (T_{M_{0}-1},T\right )\) and consequently the Eq. (5.31) has a unique solution \(\left (Y,Z\right ) \in S_{m}^{p}\left [T_{M_{0}-1},T\right ] \times \Lambda _{m\times k}^{p}\left (T_{M_{0}-1},T\right )\). The next step is to solve the equation on the interval \(\left [T_{M_{0}-2},T_{M_{0}-1}\right ]\) with the final value \(Y \left (T_{M_{0}-1}\right )\). Repeating the same arguments, the proof is completed in M 0 steps. ■
Corollary 5.18.
Consider the BSDE: \(\;\forall \ t \in \left [0,T\right ],\ \; \mathbb{P}\text{-a.s.}\;\)
If p > 1, \(S \in S_{m}^{p}\left [0,T\right ]\), \(\eta \in L^{p}\left (\Omega,\mathcal{F}_{T}, \mathbb{P}; \mathbb{R}^{m}\right )\) and F satisfies the assumptions (5.33) , then the Eq. (5.34) has a unique solution \(\left (Y,Z\right ) \in S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\) .
Proof.
By the substitutions \(\hat{Y }_{t} = Y _{t} + S_{t}\), \(\hat{\eta }=\eta +S_{T}\) and \(\hat{F}\left (t,y,z\right ) = F\left (t,y - S_{t},z\right )\) the Eq. (5.34) is transformed into
which satisfies the assumptions of Theorem 5.17. ■
We now study the case p = 1, where we restrict ourselves to the case where F does not depend on z.
Corollary 5.19.
If \(S \in S_{d}^{1}\left [0,T\right ]\), \(\eta \in L^{1}\left (\Omega,\mathcal{F}_{T}, \mathbb{P}; \mathbb{R}^{m}\right )\) and \(F\left (t,y,z\right ) \equiv F\left (t,y\right )\) satisfies the assumptions (5.33) with p = 1, then the BSDE
has a unique solution \(\left (Y,Z\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) such that
and
Proof.
As in the proof of Corollary 5.18 we can reduce the problem to the case S = 0.
Let \(n,i \in \mathbb{N}^{{\ast}}\). By Theorem 5.17 there exists a unique pair \(\left (Y ^{n},Z^{n}\right )\) such that for all p ≥ 1
and \(\left (Y ^{n},Z^{n}\right )\) is solution of the equation
Note that
By (5.4) for Y s n − Y s n+i and p = 1 we infer
Denote by C, C ′ generic constants independent of n and i.
From (5.36)
is a martingale. Then \(\mathbb{E}M_{t}^{n,n+i} = 0\) and, taking the expectation in (5.38) we deduce from the backward Gronwall inequality (Corollary 6.62):
Using the Burkholder–Davis–Gundy inequality (1.18) and Doob’s inequality (1.11-A 3), we deduce for 0 < q < 1:
Recalling that M t n, n+i is a martingale then, once again by the Burkholder–Davis–Gundy inequality
then from (5.38) and (5.39) we obtain for every 0 < q < 1
Hence, there exist Y and Z such that
Passing to the limit in (5.37) we deduce that the pair \(\left (Y,Z\right )\) solves the problem.
Now, by Corollary 2.47, \(M_{t} = \int _{0}^{t}Z_{ s}\mathit{dB}_{s}\) is a martingale, because
Finally, the uniqueness is obtained in the same manner as the estimates for Y n − Y n+i and Z n − Z n+i. ■
5.3.3.2 BSDEs with Random Lipschitz Conditions
We now generalize Theorem 5.17 to a class of BSDEs with random Lipschitz constants.
We consider the BSDE (5.31) in a more general form:
We assume that
- (BSDE - A0):
-
:
-
(i)
\(\eta: \Omega \rightarrow \mathbb{R}^{m}\) is an \(\mathcal{F}_{T}\) -measurable random vector,
-
(ii)
Q is a \(\mathcal{P}\)-m.i.c.s.p. such that Q 0 = 0;
-
(i)
and the function \(\Phi: \Omega \times \left [0,T\right ] \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}\) satisfies
- (BSDE - \(\mathbf{LH}_{\Phi }\) ):
-
-
\(\blacktriangle \) for all \(y \in \mathbb{R}^{m}\), \(z \in \mathbb{R}^{m\times k}\), the function \(\Phi \left (\cdot,\cdot,y,z\right ): \Omega \times \left [0,T\right ] \rightarrow \mathbb{R}^{m}\) is \(\mathcal{P}\)-measurable;
-
\(\blacktriangle \) there exist \(\mathcal{P}\)-m.s.p. \(L,\ell,\alpha: \Omega \times \left [0,T\right ] \rightarrow \mathbb{R}_{+}\), such that
$$\displaystyle{\alpha _{t}\mathit{dQ}_{t} = dt\quad \quad \mathit{and}\quad \quad \int _{0}^{T}\left (L_{ t}\mathit{dQ}_{t} + \left (\ell_{t}\right )^{2}dt\right ) < \infty,\;\; \mathbb{P}\text{-a.s.};}$$for all \(t \in \left [0,T\right ]\), \(y,y^{{\prime}}\in \mathbb{R}^{m}\) and \(\;z,z^{{\prime}}\in \mathbb{R}^{m\times k},\; \mathbb{P}\mbox{ -}a.s\).:
$$\displaystyle{ \begin{array}{r@{\quad }l} \quad &\mathit{Lipschitz\ conditions} \\ \left (i\right )\quad \quad &\left \vert \Phi (t,y^{{\prime}},z) - \Phi (t,y,z)\right \vert \leq L_{t}\vert y^{{\prime}}- y\vert, \\ \left (\mathit{ii}\right )\quad \quad &\vert \Phi (t,y,z^{{\prime}}) - \Phi (t,y,z)\vert \leq \alpha _{t}\ell_{t}\vert z^{{\prime}}- z\vert, \\ \quad &\mathit{Boundedness\ condition:} \\ \left (\mathit{iii}\right )\quad \quad &\int _{0}^{T}\Phi _{\rho }^{\#}\left (t\right )\mathit{dQ}_{ t} < \infty,\;\;\forall \ \rho \geq 0,\end{array} }$$(5.41)where
$$\displaystyle{\Phi _{\rho }^{\#}\left (t\right )\mathop{ =}\limits^{ \mathit{def }}\sup _{\left \vert y\right \vert \leq \rho }\left \vert \Phi (t,y,0)\right \vert.}$$
-
Note that the condition α t dQ t = dt implies that \(\Phi \left (t,Y _{t},Z_{t}\right )\mathit{dQ}_{t} = F\left (t,Y _{t},Z_{t}\right )dt + G\left (t,Y _{t}\right )\mathit{dA}_{t}\), where G does not depend upon z.
We recall the following notations. For each fixed p > 1 let \(n_{p} = 1 \wedge \left (p - 1\right )\) and
The stochastic process V is that from Lemma 5.6 with a = 2. Therefore for all \(\left (Y,Z\right ),\;\left (\tilde{Y },\tilde{Z}\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) we have
and
Lemma 5.20.
Let p ≥ 2 and the assumptions ( BSDE - A0 ), ( BSDE - \(\mathbf{LH}_{\Phi })\) be satisfied. If moreover there exists a constant b > 0 such that
then the BSDE (5.40) has a unique solution \(\left (Y,Z\right ) \in S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\) .
Proof.
We have
Since
it follows that for every δ > 0 we can define on \(S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\) an equivalent norm by
Let \(\Gamma: S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right ) \rightarrow S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\) be defined by
We remark that for all \(X,U \in S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\),
and consequently \(S_{\cdot } = \int _{0}^{\cdot }\Phi \left (s,X,U\right )\mathit{dQ}_{ s} \in S_{m}^{p}\left [0,T\right ]\). By the martingale representation result from Corollary 2.45 it follows that \(\Gamma \) is well defined.
The fact that BSDE (5.40) has a unique solution \(\left (Y,Z\right ) \in S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\) will be a consequence of the fact that \(\Gamma \) is a strict contraction on the Banach space \(\left (S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right ),\left \Vert \cdot \right \Vert _{\delta }\right )\), for some δ > 0.
Let \(\left (Y,Z\right ) = \Gamma \left (X,U\right )\) and \((Y ^{{\prime}},Z^{{\prime}}) = \Gamma \left (X^{{\prime}},U^{{\prime}}\right )\). We have
where \(K_{t} = \int _{0}^{t}\left [\Phi \left (s,X_{ s},U_{s}\right ) - \Phi \left (s,X_{s}^{{\prime}},U_{ s}^{{\prime}}\right )\right ]\mathit{dQ}_{ s}\) and for all δ > 1
Then by Proposition 5.2-A,
for δ ≥ 1 + 4C p 2∕p. Hence
and the result follows. ■
Theorem 5.21.
Let p > 1, \(n_{p} = 1 \wedge \left (p - 1\right )\) and the assumptions ( BSDE - A0 ), \((\mathbf{BSDE} \boldsymbol{\text{-}}\mathbf{LH}_{\Phi })\) be satisfied. Let
Assume also that there exists \(\delta > \frac{p} {p-1}\) such that for \(q = \frac{p\delta } {p+\delta }\) and \(n_{q} = 1 \wedge \left (q - 1\right )\),
Then the BSDE (5.40) has a unique solution \(\left (Y,Z\right ) \in S_{m}^{0} \times \Lambda _{m\times k}^{0}\) such that
Moreover there exists a positive constant C p depending only on p such that for all \(t \in \left [0,T\right ]\)
Remark 5.22.
We remark that \(q\,=\, \frac{p\delta } {p+\delta }\) defined in Theorem 5.21 satisfies 1 < q < p. If q ≥ 2 then n q = n p = 1 and the condition (5.46-iii) is clearly satisfied.
Proof of Theorem 5.21.
Uniqueness follows from Theorem 5.10.
Existence. Let \(t \in \left [0,T\right ]\) and
Define, for \(n \in \mathbb{N}^{{\ast}}\),
By Lemma 5.20 we infer that the approximating BSDE
has a unique solution \(\left (Y ^{n},Z^{n}\right ) \in S_{m}^{q}\left [0,T\right ] \times \Lambda _{m\times k}^{q}\left (0,T\right )\), for all q ≥ 2.
Let
We have for all \(n,i \in \mathbb{N}\)
Therefore
and since
we obtain, by Proposition 5.2-A, that
By Beppo Levi’s monotone convergence Theorem 1.9 it follows by letting i → ∞ that for all \(t \in \left [0,T\right ]\),
Consequently by (5.46 − i) for all \(n \in \mathbb{N}^{{\ast}}\),
Let \(\delta > \frac{p} {p-1}\), \(q = \frac{p\delta } {p+\delta } \in \left (1,p\right )\), \(n_{q} = 1 \wedge \left (q - 1\right )\) and \(n_{p} = 1 \wedge \left (p - 1\right )\) satisfy (5.46 −ii, iii). If we define
we have for all \(n \in \mathbb{N}^{{\ast}}\)
Hence for all \(n,i \in \mathbb{N}^{{\ast}}\)
Since
by Proposition 5.2-A, we infer that
Note that
-
(a)
$$\displaystyle\begin{array}{rcl} & & \mathbb{E}\ \left [e^{qV _{T}^{\left (q\right )} }\left \vert \eta \right \vert ^{q}\mathbf{1}_{ (n,\infty )}\left (\beta _{T} + \left \vert \eta \right \vert \right )\right ]+\mathbb{E}\left (\int _{0}^{T}e^{V _{s}^{\left (q\right )} }\left \vert \Phi \left (t,0,0\right )\right \vert \ \mathbf{1}_{(n,\infty )}\left (\gamma _{s}\right )\mathit{dQ}_{s}\right )^{q} {}\\ & & \leq \Big [\mathbb{E}\exp \left (\delta \Delta _{T}\right )\Big]^{\left (p-q\right )/p}\Big(\mathbb{E}\ \left [e^{pV _{T} }\left \vert \eta \right \vert ^{p}\mathbf{1}_{ (n,\infty )}\left (\beta _{T} + \left \vert \eta \right \vert \right )\right ]\Big)^{q/p} {}\\ & & +\Big[\mathbb{E}\exp \left (\delta \Delta _{T}\right )\Big]^{\left (p-q\right )/p}\left [\mathbb{E}\ \left (\int _{ 0}^{T}e^{V _{s} }\left \vert \Phi \left (t,0,0\right )\right \vert \ \mathbf{1}_{(n,\infty )}\left (\gamma _{s}\right )\mathit{dQ}_{s}\right )^{p}\right ]^{q/p} {}\\ & & \rightarrow 0,\quad \text{as }n \rightarrow \infty; {}\\ \end{array}$$
-
(b)
$$\displaystyle\begin{array}{rcl} & & \mathbb{E}\ \left [\left (\int _{0}^{T}L_{ s}\mathbf{1}_{(n,\infty )}\left (\gamma _{s}\right )\mathit{ds}\right )^{q}\sup _{ s\in \left [0,T\right ]}e^{qV _{s} }\left \vert Y _{s}^{n}\right \vert ^{q}\right ] {}\\ & & \leq \left [\mathbb{E}\ \left (\int _{0}^{T}L_{ s}\mathbf{1}_{(n,\infty )}\left (\gamma _{s}\right )\mathit{ds}\right )^{ \frac{qp} {p-q} }\right ]^{\left (p-q\right )/p}\left (\mathbb{E}\sup _{s\in \left [0,T\right ]}e^{pV _{s}}\left \vert Y _{s}^{n}\right \vert ^{p}\right )^{q/p} {}\\ & & = \left [\mathbb{E}\ \left (\int _{0}^{T}L_{ s}\mathbf{1}_{(n,\infty )}\left (\gamma _{s}\right )\mathit{ds}\right )^{\delta }\right ]^{p/\left (p+\delta \right )}\left (\mathbb{E}\sup _{ s\in \left [0,T\right ]}e^{pV _{s} }\left \vert Y _{s}^{n}\right \vert ^{p}\right )^{\delta /\left (p+\delta \right )} {}\\ & & \rightarrow 0,\quad \text{as }n \rightarrow \infty; {}\\ \end{array}$$
-
(c)
$$\displaystyle\begin{array}{rcl} & & \mathbb{E}\left [\left (\int _{0}^{T}\ell_{ s}^{2}\mathbf{1}_{ (n,\infty )}\left (\gamma _{s}\right )\mathit{ds}\right )^{q/2}\left (\int _{ 0}^{T}e^{2V _{s} }\left \vert Z_{s}^{n}\right \vert ^{2}\mathit{ds}\right )^{q/2}\right ] {}\\ & & \leq \left [\mathbb{E}\ \left (\int _{0}^{T}\ell_{ s}^{2}\mathbf{1}_{ (n,\infty )}\left (\gamma _{s}\right )\mathit{ds}\right )^{\delta /2}\right ]^{p/\left (p+\delta \right )}\left [\mathbb{E}\ \left (\int _{ 0}^{T}e^{2V _{s} }\left \vert Z_{s}^{n}\right \vert ^{2}\mathit{ds}\right )^{p/2}\right ]^{\delta /\left (p+\delta \right )} {}\\ & & \rightarrow 0,\quad \text{as }n \rightarrow \infty. {}\\ \end{array}$$
Taking into account \(\left (a\right )\), \(\left (b\right )\) and \(\left (c\right )\) we deduce that there exists a pair \(\left (Y,Z\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) such that for \(q = \frac{p\delta } {p+\delta }\)
Now the inequality (5.48) clearly follows from (5.50) by Fatou’s Lemma.
Finally passing to the limit in (5.49) we deduce using Lemma 5.16 that \(\left (Y,Z\right )\) is a solution of BSDE (5.40). ■
5.3.3.3 BSDEs with Locally Lipschitz Coefficients
For a (forward) SDE, it is not hard to deduce from existence and uniqueness under global Lipschitz conditions an existence and uniqueness result under local Lipschitz conditions, at least until a possible explosion time. The reason is that one just needs to follow each path of the solution.
For BSDEs, the situation is dramatically different. Indeed, in a sense, solving a BSDE amounts to combining the flow of a backward ODE with the operation of taking continuously in time the conditional expectation, given the current σ-algebra \(\mathcal{F}_{t}\). A backward stochastic differential equation is not solved by following each individual path of the solution. Consequently one cannot a priori deduce an existence and uniqueness result under local Lipschitz conditions from the same result under global Lipschitz conditions. However, this is in fact possible, because we have an a priori bound on the solution. This is what we shall explain in this section.
We consider the BSDE
Assume that
-
\(\blacktriangle \) \(\eta: \Omega \rightarrow \mathbb{R}^{m}\) is an \(\mathcal{F}_{T}\) -measurable random vector;
-
\(\blacktriangle \) \(F: \Omega \times \left [0,T\right ] \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}\) satisfies
\(\left (\mathbf{BSDE}\mathbf{-}\mathbf{LL}\right )\):
-
\(\blacktriangle \) for all \(y \in \mathbb{R}^{m}\), \(z \in \mathbb{R}^{m\times k}\), the function \(F\left (\cdot,\cdot,y,z\right ): \left [0,T\right ] \rightarrow \mathbb{R}^{m}\) is \(\mathcal{P}\)-m.s.p.
-
\(\blacktriangle \) there exist measurable functions \(\ell,\kappa,\rho: \left [0,T\right ] \rightarrow \mathbb{R}_{+}\) and \(L: \left [0,T\right ] \times \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}\) satisfying:
-
∘ L is continuous and increasing in the second variable,
-
∘
$$\displaystyle{\int _{0}^{T}\left [L\left (t,q\right ) +\ell ^{2}\left (t\right ) +\kappa \left (t\right ) +\rho \left (t\right )\right ]dt < \infty,\quad \text{for all }q \in \mathbb{R}_{ +},}$$ -
∘ for all \(y,y^{{\prime}}\in \mathbb{R}^{m}\), \(z,z^{{\prime}}\in \mathbb{R}^{m\times k}\), dt-a. e.
$$\displaystyle{ \begin{array}{c@{\quad }l} \left (i\right )\quad \quad &\left \vert F(t,y^{{\prime}},z) - F(t,y,z)\right \vert \leq L\left (t,\left \vert y\right \vert \vee \left \vert y^{{\prime}}\right \vert \right )\vert y^{{\prime}}- y\vert, \\ \left (\mathit{ii}\right )\quad \quad &\vert F(t,y,z^{{\prime}}) - F(t,y,z)\vert \leq \ell\left (t\right )\vert z^{{\prime}}- z\vert, \\ \left (\mathit{iii}\right )\quad \quad &\left \vert F(t,y,0)\right \vert \leq \rho \left (t\right ) +\kappa \left (t\right )\left \vert y\right \vert.\end{array} }$$(5.52)
-
Let p > 1 and \(n_{p} = 1 \wedge \left (p - 1\right )\). Define
Observe that for all \(Y,Y ^{{\prime}}\in S_{m}^{0}\left [0,T\right ]\) satisfying \(\left \vert Y ^{{\prime}}\right \vert \leq \left \vert Y \right \vert\) and all \(Z \in \Lambda _{m\times k}^{0}\left (0,T\right )\),
Note that if \(\left (Y,Z\right ) \in S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) is a solution of (5.51), then by Proposition 5.2-A and the inequality (5.53) there exists a C p > 0 such that, \(\mathbb{P}\mbox{ -}a.s\)., for all \(t \in \left [0,T\right ]\)
which yields \(\mathbb{P}\mbox{ -}a.s\)., for all \(t \in \left [0,T\right ]\):
Define the continuous stochastic processes
and
Theorem 5.23.
Let p > 1 and the assumption \(\left (\mathbf{BSDE}\mathbf{-}\mathbf{LL}\right )\) be satisfied. If there exists a \(\delta > \frac{p} {p-1}\) such that for all λ ≥ 1
then the BSDE (5.51) has a unique solution \(\left (Y,Z\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) such that for all λ ≥ 1,
In particular \(\left (Y,Z\right ) \in S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\) and (5.54) holds; if η is a bounded random variable then there exists a constant C > 0 such that \(\mathbb{P}\text{-a.s.}\;\omega \in \Omega \)
Proof.
Consider the projection operator \(\pi: \Omega \times \left [0,T\right ] \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}\),
Note that for all \(y,y^{{\prime}}\in \mathbb{R}^{m}\), \(\pi \left (\cdot,\cdot,y\right )\) is a \(\mathcal{P}\)-m.c.s.p., \(\left \vert \pi _{t}\left (y\right )\right \vert \leq R_{t}\) and
The function \(\tilde{\Phi }\left (s,y,z\right )\mathop{ =}\limits^{ \mathit{def }}\Phi \left (s,\pi _{s}\left (y\right ),z\right )\) is globally Lipschitz with respect to \(\left (y,z\right )\):
and
Then by Theorem 5.21 the BSDE
has a unique solution \(\left (Y,Z\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) satisfying (5.58). Since by (5.53)
we infer by 5.54 that \(\left \vert Y _{t}\right \vert \leq R_{t}\) and consequently \(\tilde{\Phi }\left (t,Y _{t},Z_{t}\right ) = \Phi \left (t,Y _{t},Z_{t}\right )\), that is \(\left (Y,Z\right )\) is a solution of the Eq. (5.51). The solution is unique since any solution \(\left (Y,Z\right )\) of (5.51) satisfies \(\left \vert Y _{t}\right \vert \leq R_{t}\) and consequently it is a solution of (5.59). ■
5.3.4 BSDEs with Monotone Coefficients
5.3.4.1 The First BSDE: Monotone Coefficient \(\Phi \left (s,Y _{s}\right )\mathit{dQ}_{s}\)
We first consider the BSDE
We assume that
-
\(\blacktriangle \) \(\quad \eta: \Omega \rightarrow R^{m}\) is an \(\mathcal{F}_{T}\) -measurable random vector;
-
\(\blacktriangle \) Q is a \(\mathcal{P}\)-m.i.c.s.p.such that Q 0 = 0;
-
\(\blacktriangle \) \(\quad \Phi: \Omega \times \left [0,\infty \right [ \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}\) satisfies:
-
(a)
\(\forall \ y \in \mathbb{R}^{m},\;\Phi \left (\cdot,\cdot,y\right ): \Omega \times \left [0,T\right ] \rightarrow \mathbb{R}^{m}\) is \(\mathcal{P}\)-measurable;
-
(b)
the mapping \(y\longrightarrow \Phi \left (t,y\right ): \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}\) is continuous;
-
(c)
there exist a \(\mathcal{P}\)-m.s.p. \(\mu: \Omega \times \left [0,T\right ] \rightarrow \mathbb{R}\) such that
$$\displaystyle{\int _{0}^{T}\vert \mu _{ t}\vert \mathit{dQ}_{t} < \infty,\;\; \mathbb{P}\text{-a.s.},}$$and for all \(y,y^{{\prime}}\in \mathbb{R}^{m}\), \(\;d\mathbb{P} \otimes \mathit{dQ}_{t}\mbox{ -}a.e\).
$$\displaystyle{ \left \langle y^{{\prime}}- y,\Phi (t,y^{{\prime}}) - \Phi (t,y)\right \rangle \leq \mu _{ t}\left \vert y^{{\prime}}- y\right \vert ^{2}; }$$(5.62) -
(d)
for all ρ ≥ 0
$$\displaystyle{\int _{0}^{T}\Phi _{\rho }^{\#}\left (s\right )\mathit{dQ}_{ s} < \infty,\;\;a.s.}$$□
-
(a)
where
We recall the notations
and
Proposition 5.24.
Let p ≥ 1 and the assumptions ( 5.61 - BSDE-MH0 \(_{\Phi }\) ) be satisfied. If for all ρ > 0
then the BSDE (5.60) has a unique solution \(\left (Y,Z\right ) \in S_{m}^{1}\left (\left [0,T\right ];e^{\bar{\mu }}\right ) \times \Lambda _{m\times k}^{0}\left (0,T;e^{\bar{\mu }}\right )\) . Moreover
and for p > 1 there exists a positive C p (depending only on p) such that, \(\mathbb{P}\text{-a.s.},\;\) for all \(t \in \left [0,T\right ]\) :
Remark 5.25.
If \(\left (\bar{\mu }_{t}\right )_{t\geq 0}\) is a deterministic process then the assumption (5.63) is equivalent to
and the inequality (5.65) yields: for all \(t \in \left [0,T\right ]\)
Proof of Proposition 5.24.
\(\left (I\right )\;\;\) Uniqueness follows from Theorem 5.10 and Theorem 5.11. If \(\left (Y,Z\right ) \in S_{m}^{1}\left (\left [0,T\right ];e^{\bar{\mu }}\right ) \times \Lambda _{m\times k}^{0}\left (0,T;e^{\bar{\mu }}\right )\) is a solution, then by Proposition 5.2 and
the inequalities (5.64-j,jj) follow.
To prove the existence of the solution we write the equation in the form, \(\mathbb{P}\text{-a.s.}\;\)
where
We remark that \(\hat{\mu }_{t} -\hat{\mu }_{s} =\int _{ s}^{t}\mu _{r}^{+}\mathit{dQ}_{r} \geq \int _{s}^{t}\mu _{r}\mathit{dQ}_{r} =\bar{\mu } _{t} -\bar{\mu }_{s}\).
\(\left (\mathit{II}\text{-}a\right )\;\;\) Existence in the case: there exist b, c > 0such that for all \(t \in \left [0,T\right ]\)
and
- Step 1. :
-
Yosida approximation of − F.
Since \(y\mapsto - F\left (t,y\right ) =\mu _{t}y - \Phi (t,y): \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}\) is a monotone continuous operator (hence also a maximal monotone operator), it follows that for every \(\left (\omega,t,y\right ) \in \Omega \times [0,T] \times \mathbb{R}^{m}\) and \(\varepsilon > 0\) there exists a unique \(F_{\varepsilon } = F_{\varepsilon }\left (\omega,t,y\right ) \in \mathbb{R}^{m}\) such that
$$\displaystyle{F(\omega,t,y +\varepsilon F_{\varepsilon }) = F_{\varepsilon }.}$$From Annex B, Propositions 6.7 and 6.8, recall that \(F_{\varepsilon }\left (\cdot,\cdot,y\right ): \Omega \times \left [0,T\right ] \rightarrow \mathbb{R}^{m}\) is \(\mathcal{P}\)-m.s.p. for every \(y \in \mathbb{R}^{m}\) and
\(\forall \ \varepsilon,\delta > 0,\;\forall \ t \in \left [0,T\right ]\), \(\forall \ y,y^{{\prime}}\in \mathbb{R}^{m},\quad a.s\).
$$\displaystyle{\begin{array}{l@{\quad }l} \left (a\right )\;\;\quad &\left \langle F_{\varepsilon }\left (t,y\right ) - F_{\varepsilon }\left (t,y^{{\prime}}\right ),y - y^{{\prime}}\right \rangle \leq 0, \\ \left (b\right )\;\;\quad &\left \vert F_{\varepsilon }\left (t,y\right ) - F_{\varepsilon }\left (t,y^{{\prime}}\right )\right \vert \leq \dfrac{2} {\varepsilon } \vert y - y^{{\prime}}\vert, \\ \left (c\right )\;\;\quad &\left \vert F_{\varepsilon }\left (t,y\right )\right \vert \leq \left \vert F\left (t,y\right )\right \vert,\quad \;\quad \ \lim \limits _{\varepsilon \rightarrow 0}F_{\varepsilon }\left (t,y\right ) = F\left (t,y\right ),\end{array} }$$and
$$\displaystyle{\left \langle y - y^{{\prime}},F_{\varepsilon }(t,y) - F_{\delta }(t,y^{{\prime}})\right \rangle \leq \left (\varepsilon +\delta \right )\left \langle F_{\varepsilon }\left (t,y\right ),F_{\delta }(t,y^{{\prime}})\right \rangle.}$$Moreover, if \(\left \vert y\right \vert \leq b\) then
$$\displaystyle{ \left \vert F_{\varepsilon }(t,y)\right \vert \leq \left \vert \mu _{t}\right \vert b + \Phi _{b}^{\#}\left (t\right ). }$$(5.69) - Step 2. :
-
Approximating equation.
Let \(0 <\varepsilon \leq 1\). Since \(y\longmapsto F_{\varepsilon }\left (r,y\right ) +\mu _{r}y\) is a Lipschitz function with the Lipschitz constants \(L_{t} = \dfrac{2} {\varepsilon } + c\) and ℓ t = 0 we infer by Theorem 5.21 that the approximating equation
$$\displaystyle{ Y _{t}^{\varepsilon } =\eta +\int _{ t}^{T}\left [F_{\varepsilon }\left (r,Y _{ r}^{\varepsilon }\right ) +\mu _{ r}Y _{r}^{\varepsilon }\right ]\mathit{dQ}_{ r} -\int _{t}^{T}Z_{ r}^{\varepsilon }\mathit{dB}_{ r} }$$(5.70)has a unique solution \(\left (Y ^{\varepsilon },Z^{\varepsilon }\right ) \in S_{m}^{q}\left [0,T\right ] \times \Lambda _{m\times k}^{q}\left (0,T\right )\) for all q > 1.
- Step 3. :
-
Boundedness of \(\left (Y ^{\varepsilon },Z^{\varepsilon }\right )_{0<\varepsilon \leq 1}\) .
We denote by C, C ′ generic constants independent of \(\varepsilon,\delta \in ]0,1]\). Since
$$\displaystyle{\mathbb{E}\sup _{t\in \left [0,T\right ]}e^{q\bar{\mu }_{t} }\left \vert Y _{t}^{\varepsilon }\right \vert ^{q} \leq e^{qc}\mathbb{E}\sup _{ t\in \left [0,T\right ]}\left \vert Y _{t}^{\varepsilon }\right \vert ^{q} < \infty }$$and
$$\displaystyle\begin{array}{rcl} \left \langle Y _{s}^{\varepsilon },\left [F_{\varepsilon }\left (s,Y _{ s}^{\varepsilon }\right ) +\mu _{ s}Y _{s}^{\varepsilon }\right ]\mathit{dQ}_{ s}\right \rangle & \leq &\left \vert F_{\varepsilon }\left (s,0\right )\right \vert \left \vert Y _{s}^{\varepsilon }\right \vert \mathit{dQ}_{ s} +\mu _{s}\left \vert Y _{s}^{\varepsilon }\right \vert ^{2}\mathit{dQ}_{ s} {}\\ & \leq &\left \vert \Phi \left (s,0\right )\right \vert \left \vert Y _{s}^{\varepsilon }\right \vert \mathit{dQ}_{ s} +\mu _{s}\left \vert Y _{s}^{\varepsilon }\right \vert ^{2}\mathit{dQ}_{ s} {}\\ & \leq &\left \vert \Phi \left (s,0\right )\right \vert \left \vert Y _{s}^{\varepsilon }\right \vert \mathit{dQ}_{ s} +\mu _{ s}^{+}\left \vert Y _{ s}^{\varepsilon }\right \vert ^{2}\mathit{dQ}_{ s} {}\\ \end{array}$$we deduce, by Proposition 5.2, that for p > 1 and for all \(t \in \left [0,T\right ]\):
$$\displaystyle{ \begin{array}{l} \mathbb{E}^{\mathcal{F}_{t}}\sup \limits _{s\in \left [t,T\right ]}\left \vert e^{\hat{\mu }_{s}}Y _{s}^{\varepsilon }\right \vert ^{p} + \mathbb{E}^{\mathcal{F}_{t}}\left (\int _{t}^{T}e^{\hat{\mu }_{s}}\left \vert Z_{s}^{\varepsilon }\right \vert ^{2}\mathit{ds}\right )^{p/2} \\ \quad \quad \quad \leq C_{p}\ \mathbb{E}^{\mathcal{F}_{t}}\ \left [\left \vert e^{\hat{\mu }_{T}}\eta \right \vert ^{p} + \left (\int _{t}^{T}e^{\hat{\mu }_{s}}\left \vert \Phi \left (s,0\right )\right \vert \mathit{dQ}_{s}\right )^{p}\right ] \leq C_{p}b^{p},\end{array} }$$(5.71)and (5.65) with \(\left (Y,Z\right )\) replaced by \(\left (Y ^{\varepsilon },Z^{\varepsilon }\right )\).
By Corollary 6.81, for p = 1, we have \(\mathbb{P}\mbox{ -}a.s\)., for all \(t \in \left [0,T\right ]\):
$$\displaystyle{ \left \vert Y _{t}^{\varepsilon }\right \vert \leq e^{\hat{\mu }_{t} }\left \vert Y _{t}^{\varepsilon }\right \vert \leq \mathbb{E}^{\mathcal{F}_{t} }\ \left \vert e^{\hat{\mu }_{T}}\eta \right \vert + \mathbb{E}^{\mathcal{F}_{t}}\int _{t}^{T}e^{\hat{\mu }_{s}}\left \vert \Phi \left (s,0\right )\right \vert \mathit{dQ}_{s} \leq b. }$$(5.72)Now from (5.69) we deduce
$$\displaystyle\begin{array}{rcl} \left \vert F_{\varepsilon }\left (r,Y _{r}^{\varepsilon }\right ) +\mu _{ r}Y _{r}^{\varepsilon }\right \vert & \leq &\left \vert F_{\varepsilon }\left (r,Y _{ r}^{\varepsilon }\right )\right \vert + \left \vert \mu _{ r}Y _{r}^{\varepsilon }\right \vert {}\\ & \leq & \Phi _{b}^{\#}\left (t\right ) + 2\left \vert \mu _{ r}\right \vert b {}\\ & \leq & c + 2cb. {}\\ \end{array}$$ - Step 4. :
-
\(\left (Y ^{\varepsilon },Z^{\varepsilon }\right )\) is a Cauchy sequence in \(S_{m}^{q}\left [0,T\right ] \times \Lambda _{m\times k}^{q}\left (0,T\right )\), q > 0.
Let \(0 <\varepsilon,\delta \leq 1\). We have
$$\displaystyle{Y _{t}^{\varepsilon } - Y _{ t}^{\delta } =\int _{ t}^{T}\mathit{dK}_{ s}^{\varepsilon,\delta } -\int _{ t}^{T}\left (Z_{ s}^{\varepsilon } - Z_{ s}^{\delta }\right )\mathit{dB}_{ s},}$$where
$$\displaystyle{K_{t}^{\varepsilon,\delta } =\int _{ 0}^{t}\left (F_{\varepsilon }(s,Y _{ s}^{\varepsilon }) +\mu _{ s}Y _{s}^{\varepsilon } - F_{\delta }(s,Y _{ s}^{\delta }) -\mu _{ s}Y _{s}^{\delta })\right )\mathit{dQ}_{ s}.}$$Note that
$$\displaystyle\begin{array}{rcl} & & \left \langle Y _{s}^{\varepsilon } - Y _{ s}^{\delta },\mathit{dK}_{ s}^{\varepsilon,\delta }\right \rangle {}\\ & & = \left \langle Y _{s}^{\varepsilon } - Y _{ s}^{\delta },F_{\varepsilon }(s,Y _{ s}^{\varepsilon }) - F_{\delta }(s,Y _{ s}^{\delta })\right \rangle \mathit{dQ}_{ s} +\mu _{s}\left \vert Y _{s}^{\varepsilon } - Y _{ s}^{\delta }\right \vert ^{2}\mathit{dQ}_{ s} {}\\ & & \leq \left (\varepsilon +\delta \right )\left \langle F_{\varepsilon }(s,Y _{s}^{\varepsilon }),F_{\delta }(s,Y _{ s}^{\delta })\right \rangle \mathit{dQ}_{ s} +\mu _{s}\left \vert Y _{s}^{\varepsilon } - Y _{ s}^{\delta }\right \vert ^{2}\mathit{dQ}_{ s} {}\\ & & \leq \left (\varepsilon +\delta \right )c^{2}\mathit{dQ}_{ s} + c\left \vert Y _{s}^{\varepsilon } - Y _{ s}^{\delta }\right \vert ^{2}\mathit{dQ}_{ s}. {}\\ \end{array}$$Since 0 ≤ Q t ≤ c and for every q > 1
$$\displaystyle{\mathbb{E}\sup _{t\in \left [0,T\right ]}e^{qcQ_{t} }\left \vert Y _{t}^{\varepsilon } - Y _{ s}^{\delta }\right \vert ^{q} < \infty,}$$we infer from Proposition 5.2 with D = N = 0, λ = 0, that for q ≥ 2,
$$\displaystyle{\mathbb{E}\sup \limits _{s\in \left [0,T\right ]}\left \vert Y _{s}^{\varepsilon } - Y _{ s}^{\delta }\right \vert ^{q} + \mathbb{E}\left (\int _{ 0}^{T}\left \vert Z_{ s}^{\varepsilon } - Z_{ s}^{\delta }\right \vert ^{2}\mathit{ds}\right )^{q/2} \leq C\left (\varepsilon +\delta \right )^{q/2}.}$$For 0 < q < 2 we have
$$\displaystyle\begin{array}{rcl} & & \mathbb{E}\sup \limits _{s\in \left [0,T\right ]}\left \vert Y _{s}^{\varepsilon } - Y _{ s}^{\delta }\right \vert ^{q} + \mathbb{E}\left (\int _{ 0}^{T}\left \vert Z_{ s}^{\varepsilon } - Z_{ s}^{\delta }\right \vert ^{2}\mathit{ds}\right )^{q/2} {}\\ & & \qquad \leq \left (\mathbb{E}\sup \limits _{s\in \left [0,T\right ]}\left \vert Y _{s}^{\varepsilon }-Y _{ s}^{\delta }\right \vert ^{q+2}\right )^{q/\left (q+2\right )}+\left [\mathbb{E}\left (\int _{ 0}^{T}\left \vert Z_{ s}^{\varepsilon }-Z_{ s}^{\delta }\right \vert ^{2}\mathit{ds}\right )^{\frac{q+2} {2} }\right ]^{q/\left (q+2\right )} {}\\ & & \qquad \leq C^{{\prime}}\left (\varepsilon +\delta \right )^{q/2}. {}\\ \end{array}$$Hence there exists \(\left (Y,Z\right ) \in \bigcap _{q>0}S_{m}^{q}\left [0,T\right ] \times \Lambda _{m\times k}^{q}\left (0,T\right )\) such that
$$\displaystyle{\mathbb{E}\sup \limits _{s\in \left [0,T\right ]}\left \vert Y _{s}^{\varepsilon } - Y _{ s}\right \vert ^{q} + \mathbb{E}\left (\int _{ 0}^{T}\left \vert Z_{ s}^{\varepsilon } - Z_{ s}\right \vert ^{2}\mathit{ds}\right )^{q/2} \leq C\ \varepsilon ^{q/2}.}$$Note that
$$\displaystyle{F_{\varepsilon }\left (r,Y _{r}^{\varepsilon }\right ) +\mu _{ r}Y _{r}^{\varepsilon } = \Phi \left (r,Y _{ r}^{\varepsilon } +\varepsilon F_{\varepsilon }\left (r,Y _{ r}^{\varepsilon }\right )\right ) -\varepsilon \mu _{ r}F_{\varepsilon }\left (r,Y _{r}^{\varepsilon }\right )}$$and \(\left \vert F_{\varepsilon }\left (r,Y _{r}^{\varepsilon }\right )\right \vert + \left \vert \mu _{r}Y _{r}^{\varepsilon }\right \vert \leq C\).
Passing to the limit as \(\varepsilon \rightarrow 0_{+}\) in the approximating equation (5.70), we infer, by Lebesgue’s dominated convergence theorem, that \(\left (Y,Z\right )\) is a solution of the BSDE (5.60). Moreover passing to the limit on a subsequence, by Fatou’s Lemma we clearly infer that \(\left (Y,Z\right )\) satisfies (5.65), (5.64) and
since the same inequalities hold for \(\left (Y ^{\varepsilon },Z^{\varepsilon }\right )\).
\(\left (\mathit{II}\text{-}b\right )\;\;\) Existence under the assumption (5.67), but without (5.68).
Let
Let \(\zeta _{t} = Q_{t} + \left \vert \bar{\mu }_{t}\right \vert + \left \vert \mu _{t}\right \vert + \Phi _{b}^{\#}(t)\) and \(\hat{\mu }_{t} =\int _{ 0}^{t}\mu _{s}^{+}\mathit{dQ}_{s}\).
Since
by the step \(\left (\mathit{II}\text{-}a\right )\) the BSDE
has a unique solution \(\left (Y ^{n},Z^{n}\right ) \in \bigcap _{q>0}S_{m}^{q}\left [0,T\right ] \times \Lambda _{m\times k}^{q}\left (0,T\right )\). The solution \(\left (Y ^{n},Z^{n}\right )\) satisfies (5.65), (5.64) and (5.73) with \(\left (Y,Z\right )\) replaced by \(\left (Y ^{n},Z^{n}\right )\). Note that from (5.73) written for \(\left (Y ^{n},Z^{n}\right )\) we have for p ≥ 1,
Since
we conclude by Corollary 6.81, that for q = p if p > 1 and \(q \in \left (0,1\right )\) if p = 1 there exists a constant C q such that \(\mathbb{P}\mbox{ -}a.s\).,
Taking into account (5.63) we deduce that there exists a pair \(\left (Y,Z\right ) \in S_{m}^{q}\left (\left [0,T\right ];e^{\hat{\mu }}\right ) \times \Lambda _{m\times k}^{q}\left (0,T;e^{\hat{\mu }}\right )\) such that as n → ∞
Now using Lemma 5.16 we infer that \(\left (Y,Z\right )\) is a solution of the BSDE (5.76). We deduce by Fatou’s Lemma from the inequalities (5.65), (5.64) and (5.73) written for \(\left (Y ^{n},Z^{n}\right )\) that the same inequalities hold for the limit \(\left (Y,Z\right )\).
\(\left (\mathit{II}\text{-}c\right )\;\;\) Existence without the two assumptions (5.67) and (5.68).
Let
Define, for \(n \in \mathbb{N}^{{\ast}}\),
The condition (5.67) is satisfied:
and consequently by part \(\left (\mathit{II}\text{-}b\right )\) of this proof there exists a unique pair \(\left (Y ^{n},Z^{n}\right ) \in S_{m}^{p}\left (\left [0,T\right ];e^{\hat{\mu }}\right ) \times \Lambda _{m\times k}^{p}\left (0,T;e^{\hat{\mu }}\right )\) such that
and the inequalities (5.65), (5.64) for \(\left (Y ^{n},Z^{n}\right )\) in the place of \(\left (Y,Z\right )\) hold. Since
we deduce from Corollary 6.81 that in the case p > 1 we have
and in the case p = 1
for all 0 < q < 1.
Hence for every p ≥ 1 there exists \(\left (Y,Z\right ) \in S_{m}^{q}\left (\left [0,T\right ];e^{\bar{\mu }}\right ) \times \Lambda _{m\times k}^{q}\left (0,T;e^{\bar{\mu }}\right )\) (with q = p if p > 1, and 0 < q < 1 if p = 1) such that
Using Fatou’s Lemma, the inequalities (5.65) and (5.64) follow from the same inequalities written for \(\left (Y ^{n},Z^{n}\right )\). By Lemma 5.16 we infer that \(\left (Y,Z\right )\) is a solution of the BSDE (5.60). ■
Corollary 5.26.
Let p ≥ 1. If in Proposition 5.24 we replace the assumption (5.63) by
then the same conclusions follow.
Proof.
We remark that (Y, Z) solves the BSDE (5.60) if and only if \((\tilde{Y }_{t},\tilde{Z}_{t}):= (e^{\bar{\mu }_{t}}Y _{t},e^{\bar{\mu }_{t}}Z_{t})\) is solution of the BSDE
with
Note that \(\tilde{\eta }\) and \(\tilde{\Phi }\) satisfy the same assumptions (5.61-BSDE-MH0 \(_{\Phi }\)) as η and \(\Phi \), respectively, but with (5.62) replaced by
and consequently the corresponding \(\bar{\mu }\) and \(\hat{\mu }\) for \(\tilde{\Phi }\) are equal to 0. Therefore the condition (5.63) for \((\tilde{\eta },\tilde{\Phi })\) means precisely (5.75). ■
5.3.4.2 The Second BSDE: Monotone Coefficient \(F\left (t,Y _{t},Z_{t}\right )dt\)
In this subsection we study the BSDE
We shall assume:
-
\(\blacklozenge\) \(\eta: \Omega \rightarrow \mathbb{R}^{m}\) is an \(\mathcal{F}_{T}\) -measurable random vector;
-
\(\blacklozenge\) the function \(F\left (\cdot,\cdot,y,z\right ): \Omega \times \left [0,T\right ] \rightarrow \mathbb{R}^{m}\) is \(\mathcal{P}\)-measurable for every \(\left (y,z\right ) \in \mathbb{R}^{m} \times \mathbb{R}^{m\times k}\);
-
\(\blacklozenge\) there exist some deterministic functions \(\mu \in L^{1}\left (0,T; \mathbb{R}\right )\) and
\(\ell\in L^{2}\left (0,T; \mathbb{R}\right )\) such that
□
where
Theorem 5.27.
Let p > 1 and the assumptions ( 5.77 - BSDE-MH F ) be satisfied. If for all ρ ≥ 0:
then the BSDE (5.76):
has a unique solution \(\left (Y,Z\right ) \in S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\) . Moreover, uniqueness holds in \(S_{m}^{1+}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) , where
Proof.
The uniqueness is proved in Corollary 5.13. Let us prove existence.
We use again a contraction argument, which is slightly different from that in the proof of Theorem 5.17.
Note that a solution of the Eq. (5.76) is a fixed point of the mapping \(\Gamma: S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right ) \rightarrow S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\) defined by
where
By Proposition 5.24 and Remark 5.25, with \(\Phi \left (\omega,t,y\right ) = F\left (\omega,t,y,U_{t}\left (\omega \right )\right )\), the mapping \(\Gamma \) is well defined since for all ρ ≥ 0
Let \(M \in \mathbb{N}^{{\ast}}\) and 0 = T 0 < T 1 < ⋯ < T M = T, with \(T_{i} = \dfrac{iT} {M}\). Since the function \(t\longmapsto \int _{0}^{t}\ell^{2}\left (r\right )\mathit{dr}: \left [0,T\right ] \rightarrow \mathbb{R}_{ +}\) is uniformly continuous, we see that
First, we show that the Eq. (5.76) has a unique solution on \(\left [T_{M-1},T\right ]\) in the Banach space \(S_{m}^{p}\left [T_{M-1},T\right ] \times \Lambda _{m\times k}^{p}\left (T_{M-1},T\right )\).
Let
To this end it is sufficient to prove that \(\Gamma \) is a strict contraction on the space \(S_{m}^{p}\left [T_{M-1},T\right ] \times \Lambda _{m\times k}^{p}\left (T_{M-1},T\right )\) with respect to the (equivalent) norm \(\left \vert \!\left \vert \!\left \vert \left (Y,Z\right )\right \vert \!\right \vert \!\right \vert _{M}\)
for M large enough.
Let \(\left (X,U\right ),\left (X^{{\prime}},U^{{\prime}}\right ) \in S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\). Then
where
Since
and
we have, by Proposition 5.2 with \(\left [t,T\right ]\) replaced by \(\left [T_{M-1},T\right ]\) and D = R = 0, λ = 0,
Let \(M_{0} \in \mathbb{N}^{{\ast}}\) be such that
Then
Hence the Eq. (5.76) has a unique solution in the space \(S_{m}^{p}\left [T_{M_{0}-1},T\right ] \times \Lambda _{m\times k}^{p}\left (T_{M_{0}-1},T\right )\). The next step is to solve the equation on the interval \(\left [T_{M_{0}-2},T_{M_{0}-1}\right ]\) with the final value \(Y \left (T_{M_{0}-1}\right )\). Repeating the same arguments, the proof is completed in M 0 steps. ■
Corollary 5.28.
Consider the BSDE: \(\forall \ t \in \left [0,T\right ],\ \; \mathbb{P}\text{-a.s.}\;\)
If p > 1, \(S \in S_{m}^{p}\left [0,T\right ]\), \(\eta \in L^{p}\left (\Omega,\mathcal{F}_{T}, \mathbb{P}; \mathbb{R}^{m}\right )\) , F satisfies the assumptions ( MH F ), and for all ρ ≥ 0
then the Eq. (5.79) has a unique solution \(\left (Y,Z\right ) \in S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\) .
Proof.
By the substitutions \(\hat{Y }_{t} = Y _{t} + S_{t}\), \(\hat{\eta }=\eta +S_{T}\) and \(\hat{F}\left (t,y,z\right ) = F\left (t,y - S_{t},z\right )\) the Eq. (5.34) is transformed into
which satisfies the assumptions of Theorem 5.27. ■
5.3.4.3 The Third BSDE: Monotone Coefficient \(\Phi \left (s,Y _{s},Z_{s}\right )\mathit{dQ}_{s}\)
We now generalize Theorem 5.27 to the case of the general BSDE (5.12) which we recall here:
The assumptions will be those from the beginning of Sect. 5.3.1.
Let p, a > 1 and \(n_{p} = 1 \wedge \left (p - 1\right )\). Define
We say that \(Y \in S_{m}^{p}\left (\left [0,T\right ];e^{\bar{\mu }}\right )\) if \(Y \in S_{m}^{0}\left [0,T\right ]\) and
In the same manner \(Z \in \Lambda _{m\times k}^{p}\left (0,T;e^{\bar{\mu }}\right )\) if \(Z \in \Lambda _{m\times k}^{0}\left (0,T\right )\) and
We first prove the following:
Lemma 5.29.
Let p > 1 and the assumptions ( 5.13 - BSDE-H \(_{\Phi }\) ) (i.e. (5.14) and (5.15) ) from Sect. 5.3.1 be satisfied. Moreover assume
If in addition
-
\(\left (h_{1}\right )\) \(\mathbb{E}\ \Big(\int _{0}^{T}\sup \limits _{ \left \vert y\right \vert \leq \rho }\left \vert e^{\bar{\mu }_{t}}\Phi \left (s,e^{-\bar{\mu }_{t}}y,0\right ) -\mu _{t}y\right \vert \mathit{dQ}_{t}\Big)^{p} < \infty \) , for all ρ ≥ 0, or
-
\(\left (h_{2}\right )\) μ ≥ 0 and \(\mathbb{E}\ \Big(\int _{0}^{T}e^{\bar{\mu }_{t} }\sup \limits _{\left \vert y\right \vert \leq \rho }\left \vert \Phi \left (t,y,0\right )\right \vert \mathit{dQ}_{s}\Big)^{p} < \infty \) , for all ρ ≥ 0,
then the BSDE (5.80) has a unique solution
Proof.
Uniqueness follows from Theorem 5.10. To prove the existence we shall use the Banach fixed point theorem. Let \(\Gamma: S_{m}^{p}\left (\left [0,T\right ];e^{\bar{\mu }}\right ) \times \Lambda _{m\times k}^{p}\left (0,T;e^{\bar{\mu }}\right ) \rightarrow S_{m}^{p}\left (\left [0,T\right ];e^{\bar{\mu }}\right ) \times \Lambda _{m\times k}^{p}\left (0,T;e^{\bar{\mu }}\right )\) be defined by \(\left (Y,Z\right ) = \Gamma \left (X,U\right )\), where
-
(A)
\(\Gamma \) is well defined.
Let \(\left (X,U\right ) \in S_{m}^{p}\left (\left [0,T\right ];e^{\bar{\mu }}\right ) \times \Lambda _{m\times k}^{p}\left (0,T;e^{\bar{\mu }}\right )\). The function \(\tilde{\Phi }\left (\omega,t,y\right ) = \Phi \left (\omega,t,y,U_{t}\left (\omega \right )\right )\) is monotone
$$\displaystyle{\left \langle y - y^{{\prime}},\Phi \left (r,y,U_{ r}\right ) - \Phi \left (r,y^{{\prime}},U_{ r}\right )\right \rangle \leq \mu _{r}\left \vert y - y^{{\prime}}\right \vert ^{2}.}$$Under \(\left (h_{1}\right )\) the assumptions of Corollary 5.26 are satisfied, because we have
$$\displaystyle\begin{array}{rcl} & & \mathbb{E}\ \left (\int _{0}^{T}\sup \limits _{ \left \vert y\right \vert \leq \rho }\left \vert e^{\bar{\mu }_{t} }\Phi (t,e^{\bar{\mu }_{t}}y,U_{t}) -\mu _{t}y\right \vert \mathit{dQ}_{t}\right )^{p} {}\\ & & \qquad \leq \mathbb{E}\ \left (\int _{0}^{T}\sup \limits _{ \left \vert y\right \vert \leq \rho }\left \vert e^{\bar{\mu }_{t} }\Phi (t,e^{\bar{\mu }_{t}}y,0) -\mu _{t}y\right \vert \mathit{dQ}_{t} + \int _{0}^{T}e^{\bar{\mu }_{t}}\ell\left (t\right )\left \vert U_{t}\right \vert dt\right )^{p} {}\\ \end{array}$$and
$$\displaystyle\begin{array}{rcl} & & \mathbb{E}\ \left (\int _{0}^{T}e^{\bar{\mu }_{t} }\left \vert U_{t}\right \vert \ell\left (t\right )dt\right )^{p} {}\\ & & \qquad \leq \left (\int _{0}^{T}\ell^{2}\left (t\right )dt\right )^{p/2}\mathbb{E}\ \left (\int _{ 0}^{T}e^{2\bar{\mu }_{t} }\left \vert U_{t}\right \vert ^{2}dt\right )^{p/2} {}\\ & & \qquad < \infty. {}\\ \end{array}$$Under \(\left (h_{2}\right )\) the assumptions of Proposition 5.24 are satisfied because for μ ≥ 0 we have \(\hat{\mu }_{t} = \int _{0}^{t}\mu _{ s}^{+}\mathit{ds} = \int _{ 0}^{t}\mu _{ s}\mathit{ds} =\bar{\mu } _{t}\) and
$$\displaystyle\begin{array}{rcl} & & \mathbb{E}\ \left (\int _{0}^{T}e^{\hat{\mu }_{t} }\sup \limits _{\left \vert y\right \vert \leq \rho }\Phi (t,y,U_{t})\mathit{dQ}_{t}\right )^{p} {}\\ & & \qquad \leq \mathbb{E}\ \left (\int _{0}^{T}e^{\bar{\mu }_{t} }\sup \limits _{\left \vert y\right \vert \leq \rho }\Phi (t,y,0)\mathit{dQ}_{t} + \int _{0}^{T}e^{\bar{\mu }_{t} }\ell\left (t\right )\left \vert U_{t}\right \vert dt\right )^{p} < \infty. {}\\ \end{array}$$ -
(B)
\(\Gamma \) is a strict contraction. Let \(M \in \mathbb{N}^{{\ast}}\) and 0 = T 0 < T 1 < ⋯ < T M = T, with \(T_{i} = \dfrac{iT} {M}\). To prove the existence on \(\left [T_{M-1},T\right ]\) of the solution it is sufficient to prove that \(\Gamma \) is a strict contraction on the space \(S_{m}^{p}\left (\left [T_{M-1},T\right ];e^{\bar{\mu }}\right ) \times \Lambda _{m\times k}^{p}\left (T_{M-1},T;e^{\bar{\mu }}\right )\) with respect to the norm \(\left \vert \!\left \vert \!\left \vert \left (Y,Z\right )\right \vert \!\right \vert \!\right \vert _{M}\)
$$\displaystyle{\left \vert \!\left \vert \!\left \vert \left (Y,Z\right )\right \vert \!\right \vert \!\right \vert _{M}^{p}\mathop{ =}\limits^{ \mathit{def }}\ \mathbb{E}\ \left [\left (\sup \limits _{ r\in \left [T_{M-1},T\right ]}e^{p\bar{\mu }_{r} }\left \vert Y _{r}\right \vert ^{p}\right ) + \left (\int _{ T_{M-1}}^{T}e^{2\bar{\mu }_{r} }\left \vert Z_{r}\right \vert ^{2}\mathit{dr}\right )^{p/2}\right ],}$$for M large enough. The proof continues exactly as in Theorem 5.27. Iteratively the existence follows on every interval \(\left [T_{i-1},T_{i}\right ]\), for i = M, M − 1, …, 2, 1, and finally we get the existence on \(\left [0,T\right ]\). ■
Theorem 5.30.
Let the assumptions ( 5.13 - BSDE-H \(_{\Phi }\) ) (i.e. (5.14) and (5.15) ) from Sect. 5.3.1 be satisfied. Let p,a > 1 be fixed, \(n_{p} = 1 \wedge \left (p - 1\right )\)
Assume there exists a \(\delta > \frac{p} {p-1}\) such that for \(q = \frac{p\delta } {p+\delta }\)
If in addition
-
\(\left (h_{1}\right )\) \(\mathbb{E}\ \Big(\int _{0}^{T}\sup \limits _{ \left \vert y\right \vert \leq \rho }\left \vert e^{\bar{\mu }_{t}}\Phi \left (s,e^{-\bar{\mu }_{t}}y,0\right ) -\mu _{t}y\right \vert \mathit{dQ}_{t}\Big)^{p} < \infty \) , for all ρ ≥ 0, or
-
\(\left (h_{2}\right )\) μ ≥ 0 and \(\mathbb{E}\ \Big(\int _{0}^{T}e^{\bar{\mu }_{t} }\sup \limits _{\left \vert y\right \vert \leq \rho }\left \vert \Phi \left (t,y,0\right )\right \vert \mathit{dQ}_{s}\Big)^{p} < \infty \) , for all ρ ≥ 0,
then the BSDE (5.80) has a unique solution \(\left (Y,Z\right ) \in S_{m}^{0}\left (\left [0,T\right ]\right ) \times \Lambda _{m\times k}^{0}\left (0,T\right )\) such that
$$\displaystyle{\mathbb{E}\sup \limits _{t\in \left [0,T\right ]}e^{pV _{s} }\left \vert Y _{s}\right \vert ^{p} + \mathbb{E}\ \left (\int _{ 0}^{T}e^{2V _{s} }\left \vert Z_{s}\right \vert ^{2}\mathit{ds}\right )^{p/2} < \infty.}$$Moreover, for all \(t \in \left [0,T\right ]\) :
$$\displaystyle{ \begin{array}{l} \mathbb{E}^{\mathcal{F}_{t}}\sup \limits _{s\geq t}e^{pV _{s}}\left \vert Y _{s}\right \vert ^{p} + \mathbb{E}^{\mathcal{F}_{t}}\ \left (\int _{t}^{T}e^{2V _{s}}\left \vert Z_{s}\right \vert ^{2}\mathit{ds}\right )^{p/2} \\ \quad \leq C_{p}\ \mathbb{E}^{\mathcal{F}_{t}}\bigg[e^{pV _{T}}\left \vert \eta \right \vert ^{p} +\Big (\int _{t}^{T}e^{V _{s}}\left \vert \Phi \left (s,0,0\right )\right \vert \mathit{dQ}_{s}\Big)^{p}\bigg].\end{array} }$$(5.84)
Proof.
Uniqueness follows from Theorem 5.10.
Existence. By Lemma 5.29 we infer that the approximating BSDE
has a unique solution \(\left (Y ^{n},Z^{n}\right ) \in S_{m}^{p}\left (\left [0,T\right ];e^{\bar{\mu }}\right ) \times \Lambda _{m\times k}^{p}\left (0,T;e^{\bar{\mu }}\right )\).
Let \(\ell_{s}^{n} =\ell _{s}\mathbf{1}_{\left [0,n\right ]}\left (\ell_{s}\right )\) and
We have for all \(n,i \in \mathbb{N}\)
Therefore
Since
we obtain, by Proposition 5.2-A, that
By Beppo Levi’s monotone convergence Theorem 1.9 it follows for i → ∞ that for all \(t \in \left [0,T\right ]\),
Consequently by (5.83-i) for all \(n \in \mathbb{N}^{{\ast}}\),
Let \(\delta > \frac{p} {p-1}\), \(q = \frac{p\delta } {p+\delta }\), \(n_{q}\mathop{ =}\limits^{ \mathit{def }}1 \wedge \left (q - 1\right )\) and \(n_{p}\mathop{ =}\limits^{ \mathit{def }}1 \wedge \left (p - 1\right )\) satisfy (5.83-ii,iii). Clearly 1 < q < p and 0 < n q ≤ n p . If we define
we have, for all \(n \in \mathbb{N}^{{\ast}}\),
Hence for all \(n,i \in \mathbb{N}^{{\ast}}\)
Since
by Proposition 5.2-A, we infer that
as n → ∞.
We deduce that there exists a pair \(\left (Y,Z\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) such that for \(q = \frac{p\delta } {p+\delta }\)
Now the inequality (5.84) clearly follows from (5.86) by Fatou’s Lemma.
Finally passing to the limit in (5.85) we deduce via Lemma 5.16 that \(\left (Y,Z\right )\) is a solution of BSDE (5.80). ■
5.3.5 Linear BSDEs
Let m = 1 and consider the BSDE
where
-
η is an \(\mathcal{F}_{T}\)-measurable random variable;
-
Q is a \(\mathcal{P}\)-m.i.c.s.p. such that Q 0 = 0;
-
\(\left (a_{t}\right )_{t\geq 0}\), \(\left (b_{t}\right )_{t\geq 0}\) are \(\mathbb{R}\)-valued \(\mathcal{P}\)-m.s.p. and \(\left (c_{t}\right )_{t\geq 0}\) is an \(\mathbb{R}^{k}\)-valued \(\mathcal{P}\)-m.s.p.;
-
for some p > 1 and for all λ ≥ 0,
$$\displaystyle{ \begin{array}{r@{\quad }l} \left (j\right )\quad \quad &\mathbb{E}\left [\left (1 + \left \vert \eta \right \vert ^{p}\right )\exp \left (\lambda V _{T}\right )\right ] < \infty, \\ \left (\,\mathit{jj}\right )\quad \quad &\mathbb{E}\Big(\int _{0}^{T}\left \vert b_{ s}\right \vert \exp \left (\lambda V _{s}\right )\mathit{dQ}_{s}\Big)^{p} < \infty,\end{array} }$$(5.88)where
$$\displaystyle{V _{t} = \int _{0}^{t}\left \vert a_{ s}\right \vert \mathit{dQ}_{s} + \frac{1} {n_{p}}\int _{0}^{t}\left \vert c_{ s}\right \vert ^{2}\mathit{ds}.}$$
By Theorem 5.21 the BSDE (5.87) has a unique solution satisfying
Let
Then
Since for all δ > 0, \(\mathbb{E}\left [\exp \left (\delta \Lambda _{T}\right )\right ] < \infty \) we have \(\mathbb{E}\sup \limits _{s\in \left [0,T\right ]}\left \vert \Gamma _{s}\right \vert ^{\delta } < \infty \) for all δ > 0. Consequently there exists 1 < q < p such that
By the representation Theorem 2.42 there exists a unique stochastic process \(R \in \Lambda _{1\times k}^{q}\left (0,T\right )\) such that
Proposition 5.31.
Let the assumption (5.88) be satisfied. Then the solution of the BSDE (5.87) is given by
Proof.
It is sufficient to verify that \(\left (Y,Z\right )\) given by (5.90) is a solution of (5.87). We have
Consequently, from Itô’s formula,
Since, moreover, Y T = η, we conclude that \(\left (Y,Z\right )\) is a solution of the BSDE (5.87). ■
5.3.6 Comparison Results
In this section we again restrict ourselves to the case m = 1.
5.3.6.1 Lipschitz Case
Let \(\left (Y,Z\right ) \in S^{0}\left [0,T\right ] \times \Lambda _{k}^{0}\left (0,T\right )\) be a solution of the BSDE
and \(\left (\tilde{Y },\tilde{Z}\right ) \in S^{0}\left [0,T\right ] \times \Lambda _{k}^{0}\left (0,T\right )\) a solution of the BSDE
Assume that the functions \(\Phi,\tilde{\Phi }: \Omega \times \left [0,\infty \right [ \times \mathbb{R} \times \mathbb{R}^{k} \rightarrow \mathbb{R}\) are \(\left (\mathcal{P}, \mathbb{R} \times \mathbb{R}^{k}\right )\)-Carathéodory functions (\(\mathcal{P}\)-m.s.p. with respect to \(\left (\omega,t\right )\) and continuous with respect to \(\left (x,z\right ) \in \mathbb{R} \times \mathbb{R}^{k}\)) such that
We give a comparison result in the case when one of the two functions \(\Phi \) and \(\tilde{\Phi }\) satisfies some Lipschitz conditions.
Let p > 1. Without loss of generality we assume that \(\Phi \) satisfies the assumptions of Theorem 5.21. Then the Eq. (5.91) has a unique solution \(\left (Y,Z\right )\) satisfying
Proposition 5.32.
Let p > 1 and the assumptions of Theorem 5.21 be satisfied. Assume that \(\left (\tilde{Y },\tilde{Z}\right )\) is a solution of the BSDE (5.92) and for all δ ≥ 0,
If
-
(a)
Then \(\mathbb{P}\) -a.s \(.\;\omega \in \Omega \), \(Y _{t}\left (\omega \right ) \geq \tilde{ Y }_{t}\left (\omega \right )\) , for all \(t \in \left [0,T\right ]\) .
-
(b)
If moreover there exists a \(t_{0} \in [0,T[\) such that \(\mathbb{P}\mbox{ -}a.s\) .
$$\displaystyle{\left (\eta -\tilde{\eta }\right ) + \int _{t_{0}}^{T}\left [\Phi (s,\tilde{Y }_{ s},\tilde{Z}_{s}) -\tilde{ \Phi }(s,\tilde{Y }_{s},\tilde{Z}_{s})\right ]\mathit{dQ}_{s} > 0}$$then \(Y _{t_{0}} >\tilde{ Y }_{t_{0}}\) \(\mathbb{P}\mbox{ -}a.s\) . In particular if \(\eta >\tilde{\eta }\), \(\mathbb{P}\mbox{ -}a.s\) ., then \(Y _{t}\left (\omega \right ) >\tilde{ Y }_{t}\left (\omega \right )\) , for all \(t \in \left [0,T\right ]\), \(\mathbb{P}\text{-a.s.}\;\omega \in \Omega \) .
Proof.
Observe that \(Y _{t} -\tilde{ Y }_{t}\) can be written in the form
where
\(b_{s} = \Phi (s,\tilde{Y }_{s},\tilde{Z}_{s}) -\tilde{ \Phi }(s,\tilde{Y }_{s},\tilde{Z}_{s})\), and
(recall that α is a \(\mathcal{P}\)-m.s.p. such that α s dQ s = ds).
From | a s | ≤ L s , | c s | ≤ ℓ s , the assumption of the Proposition, and the argument of the preceding section, we deduce that
for all δ ≥ 0. Hence by Proposition 5.31
which clearly yields the conclusions of Proposition 5.32. ■
5.3.6.2 Monotone Case
We now give a comparison result for the solutions of the Eqs. (5.91) and (5.92) in the case when one of the two functions \(\Phi \) and \(\tilde{\Phi }\) satisfies a monotonicity condition. To be precise we assume without loss of generality that \(\Phi \) satisfies the assumptions (5.13-BSDE-H \(_{\Phi }\)). Let
Then for a, p > 1 and \(n_{p} = \left (p - 1\right ) \wedge 1\),
Proposition 5.33.
Let the assumptions ( 5.13 - BSDE-H \(_{\Phi }\) ) be satisfied. Let \(\left (Y,Z\right ) \in S^{0}\left [0,T\right ] \times \Lambda _{k}^{0}\left (0,T\right )\) be a solution of (5.91) and \(\left (\tilde{Y },\tilde{Z}\right ) \in S^{0}\left [0,T\right ] \times \Lambda _{k}^{0}\left (0,T\right )\) be a solution of (5.92) , such that (5.93) and the condition
are satisfied. Assume that \(\mathbb{P}\text{-a.s.}\) :
Then \(\mathbb{P}\) -a.s., \(Y _{t}\left (\omega \right ) \geq \tilde{ Y }_{t}\left (\omega \right )\) , for all \(t \in \left [0,T\right ]\) .
Proof.
Recall from Proposition 2.33 that if
then
where
and \(\left \{P_{t}: t \geq 0\right \}\), P 0 = 0, is an increasing continuous stochastic process defined by (2.33).
We have
and therefore
with
and (see (2.33))
Since for a, p > 1 and \(n_{p} = \left (p - 1\right ) \wedge 1\),
we obtain, by the inequality (6.107) from Proposition 6.80, that for all 0 ≤ t ≤ T:
Consequently for all 0 ≤ t ≤ T:
■
We now give a strict comparison result in the case of monotone coefficients. Namely, we consider a solution \(\left (Y,Z\right ) \in S^{0}\left [0,T\right ] \times \Lambda _{k}^{0}\left (0,T\right )\) of the BSDE
and a solution \(\left (\tilde{Y },\tilde{Z}\right ) \in S^{0}\left [0,T\right ] \times \Lambda _{k}^{0}\left (0,T\right )\) of the BSDE
where \(\Phi,\tilde{\Phi }: \Omega \times \left [0,T\right ] \times \mathbb{R} \times \mathbb{R}^{k} \rightarrow \mathbb{R}\) satisfy:
\(\left (\mathbf{CR1}\right )\)
-
\(\Phi \left (\cdot,t,y,z\right ),\tilde{\Phi }\left (\cdot,t,y,z\right )\) are \(\mathcal{F}_{t}\) -measurable for all \(\left (t,y,z\right ) \in \left [0,T\right ] \times \mathbb{R} \times \mathbb{R}^{k}\),
-
\(\Phi \left (\omega,\cdot,\cdot,\cdot \right ),\tilde{\Phi }\left (\omega,\cdot,\cdot,\cdot \right )\) are continuous \(\mathbb{P}\text{-a.s.}\;\omega \in \Omega \),
-
\(\Phi \) satisfies the assumption (5.13-BSDE-H \(_{\Phi }\)).
We have
with \(b_{s} = \Phi (s,Y _{s},\tilde{Z}_{s}) -\tilde{ \Phi }(s,\tilde{Y }_{s},\tilde{Z}_{s})\) and
(recall that α is a \(\mathcal{P}\)-m.s.p. such that α s dQ s = ds). Note that \(\left \vert c_{s}\right \vert \leq \ell_{s}\).
Assume that
\(\left (\mathbf{CR2}\right )\)
-
η and \(\tilde{\eta }\) are \(\mathcal{F}_{T}\)-measurable random variables;
-
for some p > 1 and for all δ ≥ 0,
$$\displaystyle{\begin{array}{r@{\quad }l} \left (j\right )\quad \quad &\mathbb{E}\left [\left (1 + \left \vert \eta -\tilde{\eta }\right \vert ^{p}\right )\exp \left (\delta \int _{ 0}^{T}\left (\ell_{ r}\right )^{2}\mathit{dr}\right )\right ] < \infty, \\ \left (\,\mathit{jj}\right )\quad \quad &\mathbb{E}\ \Big[\int _{0}^{T}\left \vert \Phi (s,Y _{ s},\tilde{Z}_{s}) -\tilde{ \Phi }(s,\tilde{Y }_{s},\tilde{Z}_{s})\right \vert \exp \left (\delta \int _{0}^{s}\left (\ell_{ r}\right )^{2}\mathit{dr}\right )\mathit{dQ}_{ s}\Big]^{p} < \infty.\end{array} }$$
Then by Proposition 5.31, for all δ ≥ 0
and for any stopping times 0 ≤ θ ≤ σ ≤ T
where
Proposition 5.34.
Let \(\left (Y,Z\right ) \in S^{0}\left [0,T\right ] \times \Lambda _{k}^{0}\left (0,T\right )\) be a solution for (5.94) and \(\left (\tilde{Y },\tilde{Z}\right ) \in S^{0}\left [0,T\right ] \times \Lambda _{k}^{0}\left (0,T\right )\) be a solution for (5.95) , such that
Assume that the assumptions \(\left (\mathbf{CR1}\right )\), \(\left (\mathbf{CR2}\right )\) are satisfied and \(\tilde{Z}\) is a continuous stochastic process.
If 0 ≤ t 0 < T, \(A \in \mathcal{F}_{t_{0}}\) and
then
Proof.
By Theorem 5.33 we have
Assume that \(\mathbb{P}\left (\{Y _{t_{0}} =\tilde{ Y }_{t_{0}}\} \cap A\right ) > 0\). Let the stopping time
if the set under inf is non-empty and τ = T if the set is empty. Clearly τ > t 0 a.s. on \(\{Y _{t_{0}} =\tilde{ Y }_{t_{0}}\}\). Setting in (5.96) θ = t 0 and σ = τ we obtain
which is a contradiction. Hence \(\mathbb{P}\left (\{Y _{t_{0}} =\tilde{ Y }_{t_{0}}\} \cap A\right ) = 0\) and the conclusion \(\left (\,\mathit{jj}\right )\) follows. ■
Unlike in the Lipschitz case \(\eta >\tilde{\eta }\) does not imply that \(Y _{t} >\tilde{ Y }_{t}\) for all \(t \in \left [0,T\right ]\), as the following example will show. Let \(F\left (x\right ) =\tilde{ F}\left (x\right ) = -\sqrt{x^{+}}\).
Clearly
is the unique solution of the BSDE
and (\(\tilde{Y }_{t},\tilde{Z}_{t}) = \left (0,0\right ),t \geq 0\), is the unique solution of
We have \(Y _{1} = 1 > 0 =\tilde{ Y }_{1}\), but \(Y _{0} =\tilde{ Y }_{0}\).
5.4 Semilinear Parabolic PDEs
We need to put our BSDE into a Markovian framework: the final condition η and the coefficient F of the BSDE will be functionals of B as “explicit” functions of the solution of a forward SDE driven by {B t }.
Let \(f: [0,T] \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}\) be continuous and globally monotone in x, uniformly with respect to t, \(g\,:\, [0,T] \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d\times d}\) be continuous and globally Lipschitz in x uniformly with respect to t. Let {X s t, x ; t ≤ s ≤ T} denote the solution of the SDE
and consider the backward SDE
where \(\kappa \,:\, \mathbb{R}^{d} \rightarrow \mathbb{R}^{m}\) and \(F\,:\, [0,T] \times \mathbb{R}^{d} \times \mathbb{R}^{m} \times \mathbb{R}^{m\times d} \rightarrow \mathbb{R}^{m}\) are continuous and such that for some K, μ, p > 0,
where for each ρ > 0, there exists a K ρ > 0 such that γ(ρ, x) ≤ K ρ (1 + | x | p) and one of the two following conditions hold:
-
| f(t, x) | + | g(t, x) | ≤ K(1 + | x | ) and | μ(t, x) | + ℓ 2(t, x) ≤ K;
-
| f(t, x) | + | μ(t, x) | + ℓ 2(t, x) ≤ K(1 + | x | ) and | g(t, x) | ≤ K.
In the case m > 1 we reinforce one of the above conditions into
This is necessary for our uniqueness proof of the viscosity solution of systems of PDEs, see Theorem 6.106 in Annex D.
Finally the following additional assumption is needed again for the uniqueness of viscosity solutions
for all \(x,y \in \mathbb{R}^{d}\) such that | x | ≤ R, | y | ≤ R, \(r \in \mathbb{R}^{m}\), \(p \in \mathbb{R}^{d}\), where for each R > 0, \(\mathbf{m}_{R} \in C(\mathbb{R}_{+})\) is increasing and m R (0) = 0.
Remark 5.35.
-
(i)
Clearly, for each t ≤ s ≤ T, Y s t, x is \(\mathcal{F}_{s}^{t} = g\{B_{r} - B_{t},\,t \leq r \leq s\} \vee \mathcal{N}\) measurable, where \(\mathcal{N}\) is the class of the \(\mathbb{P}\)-null sets of \(\mathcal{F}\). Hence Y t t, x is a.s. constant (i.e. deterministic).
-
(ii)
It is not hard to see, using uniqueness for BSDEs, that \(Y _{t+h}^{t,x} = Y _{ t+h}^{t+h,X_{t+h}^{t,x} }\), h > 0.
We shall denote by
the infinitesimal generator of the Markov process {X s t, x ; t ≤ s ≤ T}.
5.4.1 Parabolic Systems in the Whole Space
We first consider the following system of backward semilinear parabolic PDEs
where \(F \in C([0,T] \times \mathbb{R}^{d} \times \mathbb{R}^{m} \times \mathbb{R}^{m\times d}; \mathbb{R}^{m})\), and \(\kappa \in C(\mathbb{R}^{d}, \mathbb{R}^{m})\) grows at most polynomially at infinity.
We can first establish the following:
Theorem 5.36.
Let \(u \in C^{1,2}([0,T] \times \mathbb{R}^{d}; \mathbb{R}^{m})\) be a classical solution of (5.99) . Then for each \((t,x) \in [0,T] \times \mathbb{R}^{d}\), \(\{(u(s,X_{s}^{t,x}),(\nabla ug)(s,X_{s}^{t,x}));t \leq s \leq T\}\) is the solution of the BSDE (5.98) . In particular, u(t,x) = Y t t,x .
Proof.
The result follows by applying Itô’s formula to u(s, X s t, x). ■
We now want to connect (5.97)–(5.98) with (5.99) in the other direction, i.e. prove that (5.97)–(5.98) provides a solution of (5.99). In order to avoid restrictive assumptions on the coefficients in (5.97)–(5.98), we will consider (5.99) in the viscosity sense. This imposes just one restriction. Indeed for the notion of viscosity solution of the system of PDEs (5.99) to make sense, we need to make the following restriction: for 0 ≤ i ≤ k, the i -th coordinate of F depends only on the i -th row of the matrix z. Then the first line in (5.99) reads
which we rewrite in the form
where
is defined by
for all 1 ≤ i ≤ m, \((t,x) \in [0,T] \times \mathbb{R}^{d}\), \(r \in \mathbb{R}^{m}\), \(p \in \mathbb{R}^{d}\), \(X \in \mathbb{S}^{d}\).
We add the following assumptions. For each ρ > 0, there exists a K ρ such that for some p > 1, all \((t,x) \in [0,T] \times \mathbb{R}^{d}\), ρ > 0,
and there exists a K > 0 such that for all \((t,x) \in [0,T] \times \mathbb{R}^{d}\), \(y,y^{{\prime}}\in \mathbb{R}^{m}\), \(z,z^{{\prime}}\in \mathbb{R}^{d}\),
The definition of the viscosity solution of a system of elliptic PDEs is given in Definition 6.94 in Annex D. The adaptation to systems of parabolic PDEs is obvious.
We now establish the main result of this section.
Theorem 5.37.
Under the above assumptions, \(u(t,x)\mathop{ =}\limits^{ \mathit{def }}Y _{t}^{t,x}\) is a continuous function of (t,x) and it is the unique viscosity solution of (5.99) which grows at most polynomially at infinity.
Proof.
Uniqueness follows from Theorem 6.106 in Annex D.
The continuity follows from the mean-square continuity of \(\{Y _{s}^{t,x},\,x \in \mathbb{R}^{d},\,0 \leq t \leq s \leq T\}\), which in turn follows from the continuity of X ⋅ t, x with respect to t, x and Theorem 5.10. The polynomial growth follows from classical moment estimates for X ⋅ t, x, the assumptions on the growth of f and g, and Proposition 5.7.
To prove that u is a viscosity sub-solution, take any 1 ≤ i ≤ k, \(\varphi \in C^{1,2}([0,T] \times \mathbb{R}^{d})\) and \((t,x) \in [0,T) \times \mathbb{R}^{d}\) such that \(u_{i}-\varphi\) has a local maximum at (t, x). We assume without loss of generality that
We suppose that
and we will find a contradiction.
Let 0 < α ≤ T − t be such that for all t ≤ s ≤ t +α, | y − x | ≤ α,
and define
Let now
It follows from the statement in Remark 5.35(ii) that
We hence have that (first approximating τ by a sequence of stopping times taking at most finitely many values)
Consequently \((\overline{Y },\overline{Z})\) solves the one-dimensional BSDE
On the other hand, from Itô’s formula,
solves the one-dimensional BSDE, for all \(s \in \left [t,t+\alpha \right ]\)
From \(u_{i}(\tau,X_{\tau }^{t,x}) \leq \varphi (\tau,X_{\tau }^{t,x})\) and the choices of α and τ, we deduce from Proposition 5.34 that \(\overline{Y }_{t} <\hat{ Y }_{t}\), i.e. \(u_{i}(t,x) <\varphi (t,x)\), which contradicts our standing assumption. ■
Remark 5.38.
Suppose that k = 1 and F has the special form:
In that case, the BSDE is linear:
hence it has an explicit solution (see Proposition 5.31):
Now \(Y _{t}^{t,x} = \mathbb{E}(Y _{t}^{t,x})\), so that
which is the well-known Feynman–Kac formula.
Clearly, Theorem 5.37 can be considered as a nonlinear extension of the Feynman–Kac formula.
Remark 5.39.
We have proved that a certain function of (t, x), defined via the solution of a probabilistic problem, is the solution of a system of backward parabolic partial differential equations. Suppose that b, g and f do not depend on t, and let
Then v solves the system of forward parabolic PDEs:
On the other hand, we have that
where \(\{(\bar{Y }_{s}^{t,x},\bar{Z}_{s}^{t,x});\,0 \leq s \leq t\}\), solves the BSDE
So we have a probabilistic representation for a system of forward parabolic PDEs, which is valid on \(\mathbb{R}_{+} \times \mathbb{R}^{d}\).
5.4.2 Parabolic Dirichlet Problem
We now combine the situation of the preceding subsection with that of Sect. 3.8.3, and we consider the following system of parabolic semilinear PDEs with Dirichlet boundary condition
where in addition to the situation in the previous subsection, we give ourselves a function \(\chi \in C([0,T] \times \partial D)\). We assume that
Now, together with the SDE (5.97), for each \((t,x) \in [0,T] \times \overline{D}\) we consider the BSDE, for all \(s \in \left [t,T\right ]\)
Again Itô’s formula allows us to establish the following:
Theorem 5.40.
Suppose that the above conditions on the coefficients and the domain D are satisfied. Let \(u \in C^{1,2}([0,T] \times D; \mathbb{R}^{m}) \cap C([0,T] \times \overline{D}; \mathbb{R}^{m})\) be a classical solution of ( 5.100 ). Then for each (t,x) ∈ [0,T] × D,
is the solution of the BSDE (5.101) .
We now wish to prove that u(t, x): = Y t t, x is a viscosity solution of (5.100). From the discussion in Sect. 3.8.3, we deduce that the condition (3.111) is necessary for u to be continuous.
We now prove the following:
Theorem 5.41.
Under the above conditions, including those of Theorem 5.37 and (3.111) , u(t,x):= Y t t,x is a continuous function from \([0,T] \times \overline{D}\) into \(\mathbb{R}^{m}\) , and it is the unique viscosity solution of (5.100) .
Proof.
Uniqueness follows from the arguments developed in Annex D. The continuity of u follows from Proposition 3.45 and the argument at the beginning of the proof of Theorem 5.37.
Let us prove that u is a viscosity sub-solution. Let 1 ≤ i ≤ k, \(\varphi \in C^{1,2}([0,T] \times \mathbb{R}^{d})\) and \((t,x) \in [0,T) \times \overline{D}\) be such that \(u_{i}-\varphi\) has a local maximum at (t, x), and \(u_{i}(t,x) =\varphi (t,x)\). If x ∈ D, then the argument in the proof of Theorem 5.37 (with \(\alpha < d(x,\partial D)\)) establishes the required inequality. The same is true if \((t,x) \in [0,T] \times \partial D\setminus \Lambda \) (this time choosing \(\alpha < d((t,x),\Lambda )\)). Finally if \((t,x) \in \Lambda \), then τ t, x = t, a.s., hence u(t, x) = χ(t, x). The result follows. ■
5.4.3 Parabolic Neumann Problem
We use again the notations from Sect. 5.4.1, and we add a nonlinear Neumann condition on the boundary of the bounded open connected subset D of \(\mathbb{R}^{d}\), whose boundary ∂ D is assumed to be of class C 2.
Let
be such that for some ρ > 0,
for all \((t,x) \in [0,T] \times \partial D\), \(y,y^{{\prime}}\in \mathbb{R}^{m}\).
We now consider the following system of semilinear parabolic PDEs with nonlinear Neumann boundary condition:
Let X t, x be the process solution of the reflected stochastic differential equation, for all \(s \in \left [t,T,\right ]\), \(\mathbb{P}\mbox{ -}a.s\).
To each \((t,x) \in [0,T] \times \overline{D}\) we associate the BSDE
Itô’s formula again allows us to establish the following:
Theorem 5.42.
Under the above assumptions on the coefficients and the domain D, if \(u \in C^{1,2}([0,T] \times D; \mathbb{R}^{m}) \cap C^{0,1}([0,T] \times \overline{D}; \mathbb{R}^{m})\) is a classical solution of (5.103) , then for each \((t,x) \in [0,T] \times \mathbb{R}^{d}\), \(\{(u(s,X_{s}^{t,x}),(\nabla ug)(s,X_{s}^{t,x}));t \leq s \leq T\}\) is the solution of the BSDE (5.104) .
Inspired by [59] we now prove:
Theorem 5.43.
Under the above conditions, including those of Theorem 5.37 , if in addition either G does not depend upon its third argument, or else the additional assumptions from Proposition 5.83 below are satisfied, then u(t,x):= Y t t,x is a continuous function from \([0,T] \times \overline{D}\) into \(\mathbb{R}^{m}\) , and it is the unique viscosity solution of (5.103) .
Proof.
Uniqueness follows from a combination of the arguments in the proofs of Theorems 6.106 and 6.112. If G does not depend upon its third argument, then the continuity of u follows from Corollary 4.56 combined with the argument at the beginning of the proof of Theorem 5.37. In the other case, we refer to [46] for the proof of the continuity of u. We now prove that u is a viscosity sub-solution. Let 1 ≤ i ≤ k, \(\varphi \in C^{1,2}([0,T] \times \mathbb{R}^{d})\) and \((t,x) \in [0,T) \times \overline{D}\) be such that \(u_{i}-\varphi\) has a local maximum at (t, x), and \(u_{i}(t,x) =\varphi (t,x)\). The case where x ∈ D is treated as in the proof of Theorem 5.37. Suppose now that \(x \in \partial D\). As usual we argue by contradiction. Suppose that for some α > 0, all \((s,y) \in B((t,x),\alpha ) \cap \overline{D}\) satisfy
The contradiction can now be established as in the proof of Theorem 5.37, making use of the strict comparison result from Proposition 5.34. ■
5.5 BSDEs with a Subdifferential Coefficient
5.5.1 Uniqueness
We extend the estimates and the uniqueness result in the case of the multivalued BSDE
where again T > 0 is a fixed deterministic time and \(\partial \varphi\) and \(\partial \psi\) are subdifferential operators attached to the convex lower semicontinuous functions \(\varphi,\psi: \mathbb{R}^{m} \rightarrow \left ]-\infty,+\infty \right ]\).
Such multivalued backward stochastic differential equations are also called backward stochastic variational inequalities (BSVI).
It is natural here to assume there exists a \(u_{0} \in \mathbb{R}^{m}\) such that \(\partial \varphi \left (u_{0}\right )\neq \varnothing \) and \(\partial \psi \left (u_{0}\right )\neq \varnothing \).
If \(Q_{t}\left (\omega \right )\mathop{ =}\limits^{ \mathit{def }}t + A_{t}\left (\omega \right )\) and \(\left \{\alpha _{t}: t \in \left [0,T\right ]\right \}\) is a real positive \(\mathcal{P}\)-m.s.p. (given by the Radon–Nikodym representation theorem) such that 0 ≤ α ≤ 1 and
then the Eq. (5.106) becomes
where
(we use the convention 0 ⋅ ∞ = 0 and write \(\partial \Psi \) for \(\partial _{y}\Psi \)).
We also remark that if \(u_{0} \in \mathrm{ Dom}\left (\partial \varphi \right )\bigcap \mathrm{Dom}\left (\partial \psi \right )\), \(\hat{u}_{01} \in \partial \varphi \left (u_{0}\right )\) and \(\hat{u}_{02} \in \partial \psi \left (u_{0}\right )\), then
We shall assume that the following assumptions hold:
-
(i)
\(\eta: \Omega \rightarrow \mathbb{R}^{m}\) is an \(\mathcal{F}_{T}\) -measurable random vector;
-
(ii)
Q is a \(\mathcal{P}\)-m.i.c.s.p. such that Q 0 = 0;
-
(iii)
\(\left (\omega,t\right )\longmapsto \alpha _{t}\left (\omega \right ): \Omega \times \left [0,T\right ] \rightarrow \left [0,1\right ]\) is \(\mathcal{P}\)-m.s.p. such that α t dQ t = dt;
-
(iv)
\(\Phi: \Omega \times \left [0,T\right ] \times \mathbb{R}^{m} \times \mathbb{R}^{m\times k} \rightarrow \mathbb{R}^{m}\) satisfies the assumptions (5.13-BSDE-H \(_{\Phi }\));
-
(v)
\(\Psi: \Omega \times \left [0,T\right ] \times \mathbb{R}^{m} \rightarrow ] -\infty,+\infty ]\) satisfies
-
\(\blacktriangle \) \(\Psi \left (\cdot,\cdot,y\right )\) is \(\mathcal{P}\)-m.s.p. for all \(y \in \mathbb{R}^{m}\),
-
\(\blacktriangle \) \(y\longmapsto \Psi \left (\omega,t,y\right ): \mathbb{R}^{m} \rightarrow ] -\infty,+\infty ]\) is a proper convex l.s.c. function,
-
\(\blacktriangle \) \(\exists \ u_{0} \in \mathbb{R}^{m}\) and an \(\mathbb{R}^{m}\)-valued \(\mathcal{P}\)-m.s.p. \(\left (\hat{u}_{t}\right )_{t\in \left [0,T\right ]}\) such that
$$\displaystyle{\left (u_{0},\hat{u}_{t}\right ) \in \partial _{y}\Psi \left (\omega,t,\cdot \right ),\quad d\mathbb{P} \otimes dt\mbox{ -}a.e.\;\left (\omega,t\right ) \in \Omega \times \left [0,T\right ].}$$□
-
Definition 5.44.
A pair \(\left (Y,Z\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) of stochastic processes is a solution of the backward stochastic variational inequality (5.107) if there exist \(K \in S_{m}^{0}\left [0,T\right ]\), K 0 = 0, such that
and \(\mathbb{P}\)-a.s., for all \(t \in \left [0,T\right ]\):
(we also say that the triple \(\left (Y,Z,K\right )\) is a solution of the Eq. (5.107)).
Remark 5.45.
If K is absolutely continuous with respect to dQ t , i.e. there exists a progressively measurable stochastic process U such that
then \(\mathit{dK}_{t} \in \partial \Psi \left (t,Y _{t}\right )\mathit{dQ}_{t}\) means, \(\mathbb{P}\text{-a.s.}\;\omega \in \Omega \),
In this case we also say that the triple \(\left (Y,Z,U\right )\) is a solution of the Eq. (5.107).
If \(\mathit{dK}_{t} \in \partial \Psi _{y}\left (t,Y _{t}\right )\mathit{dQ}_{t}\), \(d\tilde{K}_{t} \in \partial \Psi _{y}(t,\tilde{Y }_{t})\mathit{dQ}_{t}\) and
then, using the subdifferential inequalities
we infer that, for all 0 ≤ t ≤ s ≤ T
Let a, p > 1 and
Recall the notations
and
Note that if μ is a determinist process then \(S_{m}^{p}\left (\left [0,T\right ];e^{\bar{\mu }}\right ) = S_{m}^{p}\left (\left [0,T\right ]\right )\).
Proposition 5.46.
Let the assumptions \(\left (\mathbf{BSVI}\mbox{ -}\mathbf{H}_{\eta,\Psi,\Phi }\right )\) be satisfied. Then for every a,p > 1 there exists a constant C a,p such that for all solutions \(\left (Y,Z\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) of the BSDE (5.107) satisfying
the following inequality holds \(\mathbb{P}\mbox{ -}a.s\) ., for all \(t \in \left [0,T\right ]\) :
Proof.
We have
Note that
where \(n_{p} = \left (p - 1\right ) \wedge 1.\)
From the subdifferential inequalities we have
so
Hence
Now (5.111) follows from Proposition 5.2. ■
Corollary 5.47.
Let p = 1. Let the assumptions \(\left (\mathbf{BSVI}\mbox{ -}\mathbf{H}_{\eta,\Psi,\Phi }\right )\) be satisfied and \(\Phi (t,y,z) \equiv \Phi (t,y)\) for all \(t \in \left [0,T\right ]\), \(y \in \mathbb{R}^{m}\) and \(z \in \mathbb{R}^{m\times k}\) ( \(\Phi \) is independent of z; \(\ell_{t} \equiv 0\) and \(V _{t} =\bar{\mu } _{t} =\int _{ 0}^{t}\mu _{s}\mathit{dQ}_{s}\) ). Let
If \(\left (Y,Z\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) is a solution of the BSDE (5.107) satisfying
then the following inequality holds \(\mathbb{P}\mbox{ -}a.s\) ., for all \(t \in \left [0,T\right ]\) :
Moreover for every \(q \in \left (0,1\right )\) there exists a constant C q such that
Proof.
From the proof of Proposition 5.46 we have
and the conclusions follow by Corollary 6.81. ■
Remark 5.48.
A consequence of (5.111) is the following. Denoting
then for all \(t \in \left [0,T\right ]\):
Corollary 5.49.
Let p ≥ 2, r 0 > 0 and
Then
Proof.
Let \(v \in C\left (\left [0,T\right ]; \mathbb{R}^{m}\right )\) be arbitrary. From the subdifferential inequality
we deduce
Since
we see that
Therefore
(5.113) now follows by Proposition 5.2. ■
Proposition 5.50 (Uniqueness).
Let a,p > 1. Let the assumptions ( 5.108 - BSVI-H \(_{\eta,\Psi,\Phi }\) ) be satisfied. If \(\left (Y,Z\right ),(\hat{Y },\hat{Z}) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) are two solutions of the BSDE (5.107) corresponding respectively to η and \(\hat{\eta }\) such that
then \(\mathbb{P}\mbox{ -}a.s\) ., for all \(t \in \left [0,T\right ]\) :
and there exists a constant C a,p such that, \(\mathbb{P}\) -a.s., for all \(t \in \left [0,T\right ]\) :
Uniqueness in the space \(S_{m}^{p}\left (\left [0,T\right ];e^{V }\right ) \times \Lambda _{m\times k}^{0}\left (0,T\right )\) follows. Moreover, if \(\left (\ell_{t}\right )_{t\in \left [0,T\right ]}\) is a deterministic process, uniqueness of the solution \(\left (Y,Z\right )\) of the BSDE (5.107) holds in \(S_{m}^{1+}\left (\left [0,T\right ];e^{\bar{\mu }}\right ) \times \Lambda _{m\times k}^{0}\left (0,T\right )\) .
Proof.
Let \(\left (Y,Z\right )\), \((\hat{Y },\hat{Z}) \in S_{m}^{0}\left (\left [0,T\right ]0\right ) \times \Lambda _{m\times k}^{0}\left (0,T\right )\) be two solutions corresponding to η and \(\hat{\eta }\) respectively. Then
where
Since by (5.110) \(\left \langle Y _{s} -\hat{ Y }_{s},\mathit{dK}_{s} -\mathit{d\hat{K}}_{s}\right \rangle \geq 0\), we have for all a > 1:
where \(n_{p} = \left (p - 1\right ) \wedge 1\). (5.114) and (5.115) follow from Proposition 5.2 and, consequently, uniqueness follows, too.
Let now \(\left (\ell_{t}\right )_{t\in \left [0,T\right ]}\) be a deterministic process. If (Y, Z), \((\hat{Y },\hat{Z}) \in S_{m}^{1+}(\left [0,T\right ];e^{\bar{\mu }}) \times \Lambda _{m\times k}^{0}\left (0,T\right )\), then there exists a p > 1 such that \(Y,\hat{Y } \in S_{m}^{p}\left (\left [0,T\right ];e^{\bar{\mu }}\right )\) and the uniqueness follows from the first step. ■
Proposition 5.51 (Uniqueness).
Let p = 1. Let the assumptions ( 5.108 - BSVI-H \(_{\eta,\Psi,\Phi }\) ) be satisfied and \(\Phi \) be independent of \(z \in \mathbb{R}^{m\times k}\) ( \(\ell_{t} \equiv 0\) and \(V _{t} =\bar{\mu } _{t} =\int _{ 0}^{t}\mu _{s}\mathit{dQ}_{s}\) ). If \(\left (Y,Z\right ),(\hat{Y },\hat{Z}) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) are two solutions of the BSDE (5.107) corresponding respectively to η and \(\hat{\eta }\) such that
then \(\mathbb{P}\mbox{ -}a.s\) ., for all \(t \in \left [0,T\right ]\) :
and for every \(q \in \left (0,1\right )\) there exists a constant C q such that
Proof.
Following the proof of Proposition 5.50 we now have
and the conclusions follow by Corollary 6.81. ■
5.5.2 Existence
We consider the following backward stochastic variational inequality (BSVI)
and we suppose that the following assumptions hold:
-
(A1) \(\eta: \Omega \rightarrow \mathbb{R}^{m}\) is an \(\mathcal{F}_{T}\)-measurable random vector.
-
(A2) \(F: \Omega \times \left [0,T\right ] \times \mathbb{R}^{m} \times \mathbb{R}^{m\times k} \rightarrow \mathbb{R}^{m}\) satisfies the assumptions (5.77-BSDE-MH F ) (from Sect. 5.3.4).
-
(A3) \(\varphi: \mathbb{R}^{m} \rightarrow (-\infty,+\infty ]\) is a proper, convex l.s.c. function.
Recall that the subdifferential of \(\varphi\) is given by
and by \(\left (y,\hat{y}\right ) \in \partial \varphi\) we understand that \(y \in \mathrm{ Dom}\left (\partial \varphi \right )\) and \(\hat{y} \in \partial \varphi \left (y\right )\).
We define
Let \(\varepsilon > 0\) and denote the Moreau regularization of \(\varphi\) by
where \(J_{\varepsilon }\left (y\right ) = \left (I_{m\times m} +\varepsilon \partial \varphi \right )^{-1}\left (y\right )\). Note that \(\varphi _{\varepsilon }\) is a C 1 convex function and \(J_{\varepsilon }\) is a 1-Lipschitz function.
We mention some properties (see Annex B: Convex Functions): for all \(x,y \in \mathbb{R}^{m}\)
Throughout this subsection we fix a pair \(\left (u_{0},\hat{u}_{0}\right ) \in \partial \varphi\). Then by (6.26) from Annex B we have
We will make the following assumption:
-
(A4) There exist p ≥ 2,a positive stochastic process \(\beta \in L^{1}\left (\Omega \times \left (0,T\right )\right )\) , a positive function \(b \in L^{1}\left (0,T\right )\) and real numbers κ ≥ 0, λ ∈ ]0, 1[such that
$$\displaystyle{ \begin{array}{l} \mathit{for\ all}\ \left (u,\hat{u}\right ) \in \partial \varphi \mathit{\ }\mathit{and\ }z \in \mathbb{R}^{m\times k}: \\ \;\;\left \langle \hat{u},F\left (t,u,z\right )\right \rangle \leq \lambda \left \vert \hat{u}\right \vert ^{2} +\beta _{t} + b\left (t\right )\left \vert u\right \vert ^{p} +\kappa \left \vert z\right \vert ^{2} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;d\mathbb{P} \otimes dt\text{-a.e., }\left (\omega,t\right ) \in \Omega \times \left [0,T\right ].\end{array} }$$(5.120)
We note that if \(\left \langle \hat{u},F\left (t,u,z\right )\right \rangle \leq 0\) for all \(\left (u,\hat{u}\right ) \in \partial \varphi\), then the condition (5.120) is satisfied with \(\beta _{t} = b\left (t\right ) =\kappa = 0\). If for example \(\varphi = I_{\overline{D}}\) (the convex indicator of the closed convex set \(\overline{D}\)) and n y denotes any unit outward normal vector to \(\overline{D}\) at \(y \in \mathrm{ Bd}\left (\overline{D}\right )\), then the condition \(\left \langle \mathbf{n}_{y},F\left (t,y,z\right )\right \rangle \leq 0\) for all \(y \in \mathrm{ Bd}\left (\overline{D}\right )\) yields (5.120) with \(\beta _{t} = b\left (t\right ) =\kappa = 0\) (for example). In this last case by Itô’s formula for \(\psi \left (\tilde{Y }\right ) = \left [\mathrm{dist}_{\overline{D}}\left (\tilde{Y }\right )\right ]^{2}\), where
and by the uniqueness of the triple \(\left (Y,Z,U\right )\) satisfying (5.107) we infer that \(\left (Y,Z,U\right ) = \left (\tilde{Y },\tilde{Z},0\right )\).
Theorem 5.52 (Existence - Uniqueness).
Let p ≥ 2 and assumptions (A1 − A4 ) be satisfied with this p. Suppose moreover that, for all ρ ≥ 0,
Then there exists a unique pair \(\left (Y,Z\right ) \in S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\) and a unique stochastic process \(U \in \Lambda _{m}^{2}\left (0,T\right )\) such that
and for all \(t \in \left [0,T\right ]\) :
Moreover, uniqueness holds in \(S_{m}^{1+}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) , where
Proof.
Let \(\left (Y,Z\right )\), \((\tilde{Y },\tilde{Z}) \in S_{m}^{1+}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) be two solutions. Then \(Y,\tilde{Y } \in S_{m}^{p}\left [0,T\right ]\), for some p. Uniqueness follows from Proposition 5.50.
The proof of the existence will be split into several steps.
- Step 1. :
-
Approximating problem.
For \(\varepsilon \in (0,1]\) consider the approximating equation: \(\mathbb{P}\mbox{ -}a.s\)., for all \(t \in \left [0,T\right ]\),
where \(\nabla \varphi _{\varepsilon }\) is the gradient of the Moreau regularization \(\varphi _{\varepsilon }\) of \(\varphi\). It follows (without assumption (A4 )) from Theorem 5.27 that Eq. (5.122) has a unique solution \(\;\left (Y ^{\varepsilon },Z^{\varepsilon }\right ) \in S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\).
- Step 2. :
-
Boundedness of \(Y ^{\varepsilon }\) and \(Z^{\varepsilon }\).
Let \(\left (u_{0},\hat{u}_{0}\right ) \in \partial \varphi\), a > 1 and
$$\displaystyle{V \left (t\right ) = V _{t}^{a,p}\mathop{ =}\limits^{ \mathit{def }}\int _{ 0}^{t}\left [\mu \left (s\right ) + \dfrac{a} {2n_{p}}\ell^{2}\left (s\right )\right ]\mathit{ds} =\int _{ 0}^{t}\left [\mu \left (s\right ) + \dfrac{a} {2}\ell^{2}\left (s\right )\right ]\mathit{ds}}$$(p ≥ 2 yields \(n_{p} = 1 \wedge \left (p - 1\right ) = 1\)).
Let \(\left (u_{0},\hat{u}_{0}\right ) \in \partial \varphi\) be fixed. From Proposition 5.46 with \(\Psi \) replaced by \(\varphi _{\varepsilon }\) and dQ s by ds, there exists a constant C a, p (depending only on a and p) such that the following inequality holds \(\mathbb{P}\mbox{ -}a.s\)., for all \(t \in \left [0,T\right ]\):
Note that \(\left \vert \nabla \varphi _{\varepsilon }\left (u_{0}\right )\right \vert \leq \left \vert \hat{u}_{0}\right \vert\) and \(\left \vert \varphi _{\varepsilon }\left (u_{0}\right )\right \vert \leq \varphi \left (u_{0}\right ) + \left \vert \hat{u}_{0}\right \vert ^{2}\). Hence there exists a constant C independent of \(\varepsilon\) such that
Throughout the proof we shall fix a = 2 and therefore
- Step 3. :
-
Boundedness of \(\nabla \varphi _{\varepsilon }(Y ^{\varepsilon })\).
Using the following stochastic subdifferential inequality given by Lemma 2.38
$$\displaystyle{\varphi _{\varepsilon }(Y _{t}^{\varepsilon }) +\int _{ t}^{T}\left \langle \nabla \varphi _{\varepsilon }(Y _{ s}^{\varepsilon }),\,{\mathit{dY}}_{ s}^{\varepsilon }\right \rangle \leq \varphi _{\varepsilon }(Y _{ T}^{\varepsilon }) =\varphi _{\varepsilon }(\eta ) \leq \varphi (\eta ),}$$we deduce that, for all \(t \in \left [0,T\right ]\),
$$\displaystyle{ \begin{array}{r} \varphi _{\varepsilon }(Y _{t}^{\varepsilon }) + \int _{t}^{T}\left \vert \nabla \varphi _{\varepsilon }(Y _{ s}^{\varepsilon })\right \vert ^{2}\mathit{ds} \leq \varphi (\eta ) + \int _{ t}^{T}\left \langle \nabla \varphi _{\varepsilon }(Y _{ s}^{\varepsilon }),F\left (s,Y _{ s}^{\varepsilon },Z_{ s}^{\varepsilon }\right )\right \rangle \mathit{ds} \\ -\int _{t}^{T}\left \langle \nabla \varphi _{\varepsilon }(Y _{ s}^{\varepsilon }),\,Z_{ s}^{\varepsilon }\mathit{dB}_{ s}\right \rangle.\end{array} }$$(5.125)
Since \(\left \vert \nabla \varphi _{\varepsilon }\left (y\right )\right \vert \leq \vert \nabla \varphi _{\varepsilon }\left (y\right ) -\nabla \varphi _{\varepsilon }\left (u_{0}\right )\vert + \left \vert \nabla \varphi _{\varepsilon }\left (u_{0}\right )\right \vert \leq \dfrac{1} {\varepsilon } \left \vert y - u_{0}\right \vert + \left \vert \hat{u}_{0}\right \vert\) and
we have
Under assumption (A4 ), since \(\nabla \varphi _{\varepsilon }(Y _{s}^{\varepsilon }) \in \partial \varphi \left (J_{\varepsilon }\left (Y _{s}^{\varepsilon }\right )\right )\), it follows that
Using here the inequalities (5.119), then from (5.125) we infer that for all \(t \in \left [0,T\right ]\),
which yields, via estimates (5.124) and the backward Gronwall inequality (Corollary 6.62), that there exists a constant C > 0 independent of \(\varepsilon \in (0,1]\) such that
- Step 4. :
-
Cauchy sequence and convergence.
Let \(\varepsilon,\delta \in (0,1]\).
We can write
where
Then
and by Proposition 5.2, with a = p = 2,
Hence there exist \(\left (Y,Z,U\right ) \in S_{m}^{2}\left [0,T\right ] \times \Lambda _{m\times k}^{2}\left (0,T\right ) \times \Lambda _{m}^{2}\left (0,T\right )\) and a sequence \(\varepsilon _{n} \searrow 0\) such that
Passing to the limit in (5.122) we conclude that
Since \(\nabla \varphi _{\varepsilon }(Y _{s}^{\varepsilon }) \in \partial \varphi \left (J_{\varepsilon }\left (Y _{s}^{\varepsilon }\right )\right )\) it follows that for all \(A \in \mathcal{F}\), 0 ≤ s ≤ t ≤ T and \(v \in S_{m}^{2}\left [0,T\right ]\),
Passing to \(\liminf\) for \(\varepsilon =\varepsilon _{n} \searrow 0\) in the above inequality we obtain that \(U_{s} \in \partial \varphi \left (Y _{s}\right )\). Hence \(\left (Y,Z,U\right ) \in S_{m}^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right ) \times \Lambda _{m}^{2}\left (0,T\right )\) and \(\left (Y,Z,K\right )\), with \(K_{t} = \int _{0}^{t}U_{ s}\mathit{ds}\), is the solution of BSVI (5.116). The proof is complete. ■
Remark 5.53.
The existence Theorem 5.52 is well adapted to the Hilbert space setting, since we do not impose an assumption of the form
which is very restrictive for infinite dimensional spaces. In the context of Hilbert spaces Theorem 5.52 holds in the same form (see [57] where some examples of partial differential backward stochastic variational inequalities are given too).
Let \(\left (u_{0},\hat{u}_{0}\right ) \in \partial \varphi\) be fixed. From the inequality (5.123) we have for a = p = 2
and consequently if \(\left \vert \eta \right \vert + \int _{0}^{T}\left \vert F\left (s,u_{ 0},0\right )\right \vert \mathit{ds} \leq M_{0}\), then a.s. for all \(t \in \left [0,T\right ]\),
Corollary 5.54.
If in Theorem 5.52 we replace the assumption (A4 ) by
-
(A5) There exist M 0 ,L > 0 such that:
$$\displaystyle\begin{array}{rcl} \left (i\right )& & \quad 0 \leq \ell_{t} \leq L,\ \text{ a.e., }t \in \left [0,T\right ], {}\\ \left ({\mathit{ii}}\right )& & \quad \left \vert \eta \right \vert + \int _{0}^{T}\left \vert F\left (s,u_{ 0},0\right )\right \vert \mathit{ds} \leq M_{0},\ \text{ a.s.,}\omega \in \Omega, {}\\ \left ({\mathit{iii}}\right )& & \quad \exists R_{0} > 0\text{ sufficient large such that} {}\\ & & \qquad \qquad \quad \mathbb{E}\int _{0}^{T}\left (F_{ R_{0}}^{\#}(s)\right )^{2}\mathit{ds} < \infty, {}\\ \end{array}$$
(in the proof R 0 is defined by (5.128) ) the conclusions of Theorem 5.52 hold.
Proof.
Let R 0 be defined by (5.128). The proof follows the same steps and computations as in Theorem 5.52 with the modification of Step 3: the estimate (5.126) now takes the following form (considering (5.128)),
Using this inequality in (5.125) we directly obtain (5.127). ■
Remark 5.55.
We note that if \(F\left (\omega,t,y,z\right ) = F\left (y,z\right )\), then the assumption (A5 ) becomes \(\left \vert \eta \right \vert \leq M_{0},\;\) a.s., \(\omega \in \Omega \).
Remark 5.56.
In the particular case where \(\varphi\) is the convex indicator of a convex subset \(D \subset \mathbb{R}^{m}\), the BSDE (5.116) is a reflected BSDE. As first noted in [34], the process K which maintains the solution inside D is absolutely continuous, unlike in the case of forward SDEs. The intuitive reason for this is that K does not need to fight against the martingale term. The situation is probably quite different in the case of nonconvex sets, but reflecting BSDEs at the boundary of nonconvex sets remains an open problem. The theory of reflected BSDEs was initiated in [25], where reflection in \(\mathbb{R}\) above a given continuous adapted process was considered.
5.6 BSDEs with Random Final Time
5.6.1 BSDEs with a Monotone Coefficient
Let us now discuss the existence and uniqueness of a solution to an equation which we would like to write as
In most cases the above integrals will not make sense. For this reason we shall give below a weaker formulation of the above BSDE.
We formulate the following assumptions:
-
(i)
p, a > 1, \(n_{p} = 1 \wedge \left (p - 1\right )\),
-
(ii)
\(\eta \in L^{p}\left (\Omega,\mathcal{F}_{\infty }, \mathbb{P}; \mathbb{R}^{m}\right )\) and \(\left (\xi,\zeta \right ) \in S_{d}^{p} \times \Lambda _{d\times k}^{p}\left (0,\infty \right )\) is the unique pair such that
$$\displaystyle{\xi _{t} =\eta -\int _{t}^{\infty }\zeta _{ s}\mathit{dB}_{s},\;\;t \geq 0,\;a.s.,}$$(in particular \(\left (\xi _{t}\right )_{t\geq 0}\) is given by \(\xi _{t} = \mathbb{E}^{\mathcal{F}_{t}}\eta\) ).
-
(iii)
\(\left (\omega,t\right )\longmapsto Q_{t}\left (\omega \right ): \Omega \times [0,\infty [\rightarrow \mathbb{R}\) is a \(\mathcal{P}\)-m.i.c.s.p. such that Q 0 = 0.
-
\(\blacklozenge\) \(\forall \ y \in \mathbb{R}^{m}\), \(z \in \mathbb{R}^{m\times k}\), the function \(\Phi \left (\cdot,\cdot,y,z\right ): \Omega \times [0,\infty [\rightarrow \mathbb{R}^{m}\) is \(\mathcal{P}\)-measurable;
-
\(\blacklozenge\) there exist \(\ell\in L_{loc}^{2}\left (\mathbb{R}_{+}; \mathbb{R}_{+}\right )\) (a deterministic function) and two \(\mathcal{P}\)-m.s.p \(\mu,\alpha: \Omega \times [0,\infty [\rightarrow \mathbb{R}\), α ≥ 0, such that α t dQ t = dt and
$$\displaystyle{\int _{0}^{T}\vert \mu _{ t}\vert \mathit{dQ}_{t} < \infty,\quad \mathit{for\ all}\mathit{\ }T > 0, \mathbb{P}\text{-a.s.};}$$ -
\(\blacklozenge\) for all \(y,y^{{\prime}}\in \mathbb{R}^{m}\) and \(\;z,z^{{\prime}}\in \mathbb{R}^{m\times k},\;\;d\mathbb{P} \otimes \mathit{dQ}_{t}\text{-}a.e\).:
$$\displaystyle{ \begin{array}{ll} &\quad \mathit{Continuity:} \\ \left (C_{y}\right ) &\quad y\longrightarrow \Phi \left (t,y,z\right ): \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}\,\,\mathit{is\ continuous}; \\ &\quad \mathit{Monotonicity\ condition:} \\ \left (M_{y}\right )&\quad \left \langle y^{{\prime}}- y,\Phi (t,y^{{\prime}},z) - \Phi (t,y,z)\right \rangle \leq \mu _{t}\vert y^{{\prime}}- y\vert ^{2}\mathit{\ }; \\ &\quad \mathit{Lipschitz\ condition:} \\ \left (L_{z}\right ) &\quad \vert \Phi (t,y,z^{{\prime}}) - \Phi (t,y,z)\vert \leq \alpha _{t}\ \ell\left (t\right )\vert z^{{\prime}}- z\vert; \\ &\quad \mathit{Boundedness\ condition:} \\ \left (B_{y}\right ) &\quad \int _{0}^{T}\Phi _{\rho }^{\#}\left (s\right )\mathit{dQ}_{ s} < \infty,\;\;\forall \ \rho,T \geq 0,\end{array} }$$(5.131)
□
-
where \(\Phi _{\rho }^{\#}\left (t\right ) =\sup \left \{\left \vert \Phi (t,y,0)\right \vert: \left \vert y\right \vert \leq \rho \right \}\).
Define
and
Note that
Finally we recall the usual notation
Theorem 5.57.
Let p,a > 1 and V be defined by (5.132) . Let the assumptions \(\left (\mathbf{BSDE}\mbox{ -}\mathbf{H}_{\infty }\right )\) be satisfied and
If, moreover, for all ρ ≥ 0
-
\(\left (h_{1}\right )\) \(\quad \mathbb{E}\ \Big(\int _{0}^{T}\sup \limits _{ \left \vert y\right \vert \leq \rho }\left \vert e^{\bar{\mu }_{t}}\Phi \left (s,e^{-\bar{\mu }_{t}}y,0\right ) -\mu _{t}y\right \vert \mathit{dQ}_{t}\Big)^{p} < \infty \) , or
-
\(\left (h_{2}\right )\) μ ≥ 0 and \(\mathbb{E}\ \Big(\int _{0}^{T}e^{\bar{\mu }_{t} }\sup \limits _{\left \vert y\right \vert \leq \rho }\left \vert \Phi \left (t,y,0\right )\right \vert \mathit{dQ}_{s}\Big)^{p} < \infty \),
then there exists a unique solution \(\left (Y _{t},Z_{t}\right )_{t\geq 0} \in S_{m}^{0} \times \Lambda _{m\times k}^{0}\) of the BSDE (5.129) in the sense that (here ∀0 ≤ t ≤ T means for all t and all T such that 0 ≤ t ≤ T)
Moreover
and there exists a constant C a,p depending only on \(\left (a,p\right )\) such that for all t ≥ 0,
Proof.
Uniqueness. If \(\left (Y,Z\right )\) and \(\left (\hat{Y },\hat{Z}\right )\) are two solutions of (5.133) in the space \(S_{m}^{p}\left (e^{\bar{\mu }}\right ) \times \Lambda _{m\times k}^{0} = S_{m}^{p}\left (e^{V }\right ) \times \Lambda _{m\times k}^{0}\). Then from (5.24) there exists a positive constant C a, p depending only on \(\left (a,p\right )\), such that
where we have used ((5. 133)(jj)). Uniqueness follows.
Existence. Note that
and since \(\mathbb{E}\left (\sup _{t\in \left [0,T\right ]}e^{p\bar{\mu }_{t}}\left \vert \eta \right \vert ^{p}\right ) < \infty \), by Corollary 6.83
Hence
For any fixed \(n \in \mathbb{N}^{{\ast}}\), we consider the approximating equation
By Lemma 5.29, this equation has a unique solution \(\left (Y ^{n},Z^{n}\right ) \in S_{m}^{p}\left (\left [0,n\right ];e^{\bar{\mu }}\right ) \times \Lambda _{m\times k}^{p}\left (0,n;e^{\bar{\mu }}\right )\). We set Y s n = ξ s and Z s n = ζ s for s > n.
Since the approximating equation can be written in the form: \(\mathbb{P}\text{-a.s.}\), for all \(t \in \left [0,n\right ]\),
we deduce from (5.19) that \(\mathbb{P}\text{-a.s.}\), for all 0 ≤ t ≤ n,
where C a, p is a constant depending only upon \(\left (a,p\right )\). In particular for \(i \in \mathbb{N}^{{\ast}}\):
Note that by uniqueness
Using the inequality (5.24) in this context we infer that
Hence
This shows there exist progressively measurable stochastic processes \(Y: \mathbb{R}_{+} \times \Omega \rightarrow \mathbb{R}^{m}\) and \(Z: \mathbb{R}_{+} \times \Omega \rightarrow \mathbb{R}^{m\times k}\) such that
From (5.136) we deduce by letting n → ∞ that for all t ≥ 0, \(\mathbb{P}\text{-a.s.}\),
Since \(\left (\xi,\zeta \right ) \in S_{m}^{p}\left (\left [0,T\right ];e^{V }\right ) \times \Lambda _{m\times k}^{p}\left (0,T;e^{V }\right )\), for all T > 0, it clearly follows from (5.139) that
Let 0 ≤ t ≤ T ≤ n. Now by Lemma 5.5 we can pass to the limit in
and taking into account (5.139) we deduce that \(\left (Y,Z\right )\) satisfies (5.133). The proof is complete.
Remark 5.58.
If, moreover, there exists a constant b such that \(\sup _{t\geq 0}V _{t} \leq b\), \(\mathbb{P}\text{-a.s.}\), then the conditions (5.143-\(\left (\mathit{jjj}\right )\)) can be replaced by the stronger statement than (5.133):
Indeed using the backward Burkholder–Davis–Gundy inequality (2.51) we have
□
Let \(\tau: \Omega \rightarrow [0,\infty ]\) be a stopping time and \(\eta \in L^{p}\left (\Omega,\mathcal{F}_{\tau }, \mathbb{P}; \mathbb{R}^{m}\right )\), p > 1. We now consider the BSDE
in the sense which will be made precise in the next theorem. Plainly the BSDE (5.142) a particular case of Eq. (5.129) where \(\Phi \) is of the form \(\mathbf{1}_{\left [0,\tau \right ]}\Phi \), since by Lemma 2.43 Z t = 0 for all t > τ.
Recall that the unique pair \(\left (\xi,\zeta \right ) \in S_{d}^{p} \times \Lambda _{d\times k}^{p}\left (0,\infty \right )\) such that
satisfies \(\xi _{t} = \mathbb{E}^{\mathcal{F}_{t\wedge \tau }}\eta\) and \(\zeta _{t} = \mathbf{1}_{\left [0,\tau \right ]}\left (t\right )\zeta _{t}\).
Define
We deduce from Theorem 5.57:
Corollary 5.59.
Let a,p > 1 and \(\tau: \Omega \rightarrow [0,\infty ]\) be a stopping time. Let the assumptions \(\left (\mathbf{BSDE}\mbox{ -}\mathbf{H}_{\infty }\right )\) with \(\Phi \left (s,y,z\right ) = \mathbf{1}_{\left [0,\tau \right ]}\left (s\right )\Phi \left (s,y,z\right )\) be satisfied and \(\eta \in L^{p}\left (\Omega,\mathcal{F}_{\tau }, \mathbb{P}; \mathbb{R}^{m}\right )\) . Assume moreover
and for all \(\rho \geq 0\left (h_{1}\right )\quad \mathbb{E}\ \Big(\int _{0}^{T\wedge \tau }\sup \limits _{ \left \vert y\right \vert \leq \rho }\left \vert e^{\bar{\mu }_{t}}\Phi \left (s,e^{-\bar{\mu }_{t}}y,0\right ) -\mu _{t}y\right \vert \mathit{dQ}_{t}\Big)^{p} < \infty \) , or \(\left (h_{1}\right )\quad \mu \geq 0\) and \(\mathbb{E}\ \Big(\int _{0}^{T\wedge \tau }e^{\bar{\mu }_{t} }\sup \limits _{\left \vert y\right \vert \leq \rho }\left \vert \Phi \left (t,y,0\right )\right \vert \mathit{dQ}_{s}\Big)^{p} < \infty \) .
-
(A)
Then there exists a unique solution \(\left (Y _{t},Z_{t}\right )_{t\geq 0} \in S_{m}^{0} \times \Lambda _{m\times k}^{0}\), \(\left (Y _{t},Z_{t}\right ) = \left (\eta,0\right )\) if t > τ, of the BSDE (5.142) in the sense that
$$\displaystyle{ \left \{\begin{array}{rr} \left (j\right )&\quad Y _{t} = Y _{T} + \int _{t}^{T}\Phi \left (s,Y _{ s},Z_{s}\right )\mathit{dQ}_{s} -\int _{t}^{T}Z_{ s}\mathit{dB}_{s},\;\;a.s., \\ & \text{ for all }0 \leq t \leq T, \\ \left (\,\mathit{jj}\right )\quad &{l}{\mathbb{E}\sup \limits _{s\in \left [0,T\right ]}e^{pV _{s}}\left \vert Y _{s}\right \vert ^{p} < \infty,\quad \text{for all }T \geq 0,} \\ \left (\,\mathit{jjj}\right )\quad &{l}{\lim \limits _{T\rightarrow \infty }\ \mathbb{E}\ e^{pV _{T\wedge \tau }}\left \vert Y _{T\wedge \tau }-\xi _{T\wedge \tau }\right \vert ^{p} = 0.}\end{array} \right. }$$(5.143)Moreover
$$\displaystyle{\mathbb{E}\ \left (\int _{0}^{\tau }e^{2V _{s} }\left \vert Z_{s}\right \vert ^{2}\mathit{ds}\right )^{p/2} < \infty }$$and there exists a constant C a,p depending only on \(\left (a,p\right )\) such that for all t ≥ 0,
$$\displaystyle{ \begin{array}{r} \mathbb{E}\ \sup \limits _{t\wedge \tau \leq s\leq \tau }\left \vert e^{V _{s}}\left (Y _{s} -\xi _{s}\right )\right \vert ^{p} + \mathbb{E}\ \left (\int _{t\wedge \tau }^{\tau }e^{2V _{s}}\left \vert Z_{s} -\zeta _{s}\right \vert ^{2}\mathit{ds}\right )^{p/2} \\ \leq C_{a,p}\mathbb{E}\ \left (\int _{t\wedge \tau }^{\tau }e^{V _{s} }\left \vert \Phi \left (s,\xi _{s},\zeta _{s}\right )\right \vert \mathit{dQ}_{s}\right )^{p}.\end{array} }$$(5.144) -
(B)
If, moreover, there exists a constant b such that \(\sup _{0\leq t\leq \tau }V _{t} \leq b\), \(\mathbb{P}\text{-a.s.}\) , then the conditions ( 5.143 - \(\left (\mathit{jjj}\right )\) ) can be replaced by
$$\displaystyle{ \begin{array}{rl} \left (\mathit{jjj}^{{\prime}}\right )&\quad \lim \limits _{T\rightarrow \infty }\ \mathbb{E}\ e^{pV _{T\wedge \tau }}\left \vert Y _{T\wedge \tau }-\eta \right \vert ^{p} = 0.\end{array} }$$(5.145)
□
5.6.2 BSVIs with Random Final Time
In this section we are interested in the following generalized backward stochastic variational inequality (BSVI for short):
where \(\partial \varphi\), ∂ ψ are the subdifferentials of the convex lower semicontinuous functions \(\varphi\), ψ, \(\left \{A_{t}: t \geq 0\right \}\) is a progressively measurable increasing continuous stochastic process, and τ is a stopping time.
In fact we will define and prove the existence of the solution for an equivalent form of (5.146):
with Q, \(\Phi \) and \(\Psi \) adequately defined.
We mention that the presence of the process A is justified by the possible applications of Eq. (5.146) in obtaining a probabilistic interpretation for the solution of PDEs with Neumann boundary conditions; since τ is a stopping time the BSVI (5.146) can be used for elliptic PDEs.
Because (5.146) is quite a complicated equation, in order to simplify the presentation we shall restrict ourselves to p = 2. The case p ≥ 2 can be found in [47].
We begin to give the main assumptions for this section.
-
(A1) The random variable \(\tau: \Omega \rightarrow \left [0,\infty \right ]\) is a stopping time.
-
(A2) The random variable \(\eta: \Omega \rightarrow \mathbb{R}^{m}\) is \(\mathcal{F}_{\tau }\) -measurable, \(\mathbb{E}\left \vert \eta \right \vert ^{2} < \infty \) and the stochastic process \(\left (\xi,\zeta \right ) \in S_{m}^{2} \times \Lambda _{m\times k}^{2}\left (0,\infty \right )\) is the unique pair associated to η given by the martingale representation formula (Corollary 2.44)
$$\displaystyle{\left \{\begin{array}{l} \xi _{t} =\eta -\int _{t}^{\infty }\zeta _{ s}\mathit{dB}_{s},\;t \geq 0,\;\mathit{a.s.,} \\ \xi _{t} = \mathbb{E}^{\mathcal{F}_{t}}\eta = \mathbb{E}^{\mathcal{F}_{t\wedge \tau }}\eta \quad \mathit{and}\quad \zeta _{t} = \mathbf{1}_{\left [0,\tau \right ]}\left (t\right )\zeta _{t}.\end{array} \right.}$$ -
(A3) The process \(\left \{A_{t}: t \geq 0\right \}\) is a progressively measurable increasing continuous stochastic process such that A 0 = 0,
$$\displaystyle{Q_{t}\left (\omega \right ) = t + A_{t}\left (\omega \right ),}$$and \(\left \{\alpha _{t}: t \geq 0\right \}\) is a real positive p.m.s.p. (given by the Radon–Nikodym representation theorem) such that \(\alpha \in \left [0,1\right ]\) and
$$\displaystyle{dt =\alpha _{t}\mathit{dQ}_{t}\quad \mathit{and}\quad \mathit{dA}_{t} = \left (1 -\alpha _{t}\right )\mathit{dQ}_{t}.}$$ -
(A4) The functions \(F: \Omega \times \mathbb{R}_{+} \times \mathbb{R}^{m} \times \mathbb{R}^{m\times k} \rightarrow \mathbb{R}^{m}\) and \(G: \Omega \times \mathbb{R}_{+} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}\) are such that
$$\displaystyle{\left \{\begin{array}{l} F\left (\cdot,\cdot,y,z\right ),\ G\left (\cdot,\cdot,y\right )\mathit{\ are\ p.m.s.p.,\ for\ all}\left (y,z\right ) \in \mathbb{R}^{m} \times \mathbb{R}^{m\times k}\,\text{,} \\ F\left (\omega,t,\cdot,\cdot \right ),\ G\left (\omega,t,\cdot \right )\mathit{\ {are continuous functions, }}d\mathbb{P} \otimes dt\mathit{-a.e.,}\end{array} \right.}$$and \(\mathbb{P}\) -a.s.,
$$\displaystyle{\int _{0}^{T}F_{\rho }^{\#}\left (s\right )\mathit{ds} +\int _{ 0}^{T}G_{\rho }^{\#}\left (s\right )\mathit{dA}_{ s} < \infty,\quad \forall \,\rho,T \geq 0,}$$where
$$\displaystyle{F_{\rho }^{\#}\left (\omega,s\right ):=\sup \limits _{\left \vert y\right \vert \leq \rho }\left \vert F\left (\omega,s,y,0\right )\right \vert,\quad G_{\rho }^{\#}\left (\omega,s\right ):=\sup \limits _{\left \vert y\right \vert \leq \rho }\left \vert G\left (\omega,s,y\right )\right \vert.}$$ -
(A5) Assume that there exist three progressively measurable positive stochastic processes \(\mu,\nu,\ell: \Omega \times \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}\) such that
$$\displaystyle{\int _{0}^{T}\left (\mu _{ s} + \left (\ell_{s}\right )^{2}\right )\mathit{ds} + \int _{ 0}^{T}\nu _{ s}\mathit{dA}_{s} < \infty,\;\mathit{for\ all}\mathit{\ }T > 0,\ \mathbb{P}\mathit{-a.s.,}}$$and \(\mathbb{P}\text{-a.s.}\;\omega \in \Omega \), for all \(t \in \left [0,\tau \left (\omega \right )\right ]\), \(y,y^{{\prime}}\in \mathbb{R}^{m}\), \(z,z^{{\prime}}\in \mathbb{R}^{m\times k}\),
$$\displaystyle{ \begin{array}{rl} \left (i\right )&\quad \left \langle y^{{\prime}}- y,F(t,y^{{\prime}},z) - F(t,y,z)\right \rangle \leq \mu _{t}\,\left \vert y^{{\prime}}- y\right \vert ^{2}, \\ \left (\mathit{ii}\right )&\quad \left \langle y^{{\prime}}- y,G(t,y^{{\prime}}) - G(t,y)\right \rangle \leq \nu _{t}\,\left \vert y^{{\prime}}- y\right \vert ^{2}, \\ \left (\mathit{iii}\right )&\quad \left \vert F(t,y,z^{{\prime}}) - F(t,y,z)\right \vert \leq \ell_{t}\,\left \vert z^{{\prime}}- z\right \vert.\end{array} }$$(5.148)
Let us introduce the functions
The relations (5.148) yield
Let
Concerning \(\varphi\) and ψ we shall assume:
-
(A6) \(\varphi,\psi: \mathbb{R}^{m} \rightarrow \left [0,+\infty \right ]\) are proper convex lower semicontinuous (l.s.c.) functions, \(\partial \varphi\) and ∂ ψ are the subdifferentials of \(\varphi\) and ψ, respectively, and there exists a \(u_{0} \in \mathbb{R}^{m}\) such that \(0 \in \partial \varphi \left (u_{0}\right ) \cap \partial \psi \left (u_{0}\right )\) (which is equivalent to \(\varphi \left (u_{0}\right ) \leq \varphi \left (y\right )\) and \(\psi \left (u_{0}\right ) \leq \psi \left (y\right )\) for all \(y \in \mathbb{R}^{m}\)). Define
$$\displaystyle{\Psi \left (\omega,t,y\right ) = \mathbf{1}_{\left [0,\tau \left (\omega \right )\right ]}\left (t\right )\left [\alpha _{t}\left (\omega \right )\varphi \left (y\right ) + \left (1 -\alpha _{t}\left (\omega \right )\right )\psi \left (y\right )\right ].}$$ -
(A7) If \(\mathbb{P}\left (\tau > N\right ) > 0\), for all \(N \in \mathbb{N}^{{\ast}}\), then for every \(\bar{\eta }\in \overline{\eta,u_{0}} = \mathit{conv}\left \{\eta,u_{0}\right \}\) there exist two progressively measurable stochastic processes \(\bar{\xi }^{\left (1\right )},\bar{\xi }^{\left (2\right )}\) such that \(\bar{\xi }_{t}^{\left (1\right )} \in \partial \varphi \left (\mathbb{E}^{\mathcal{F}_{t}}\bar{\eta }\right )\), \(\bar{\xi }_{t}^{\left (2\right )} \in \partial \psi \left (\mathbb{E}^{\mathcal{F}_{t}}\bar{\eta }\right )\) a.e. t ≥ 0 and
$$\displaystyle{\mathbb{E}\ \left (\int _{0}^{\tau }e^{V _{s} }\vert \bar{\xi }_{s}^{\left (1\right )}\vert \mathit{ds}\right )^{2} + \mathbb{E}\ \left (\int _{ 0}^{\tau }e^{V _{s} }\vert \bar{\xi }_{s}^{\left (2\right )}\vert \mathit{dA}_{ s}\right )^{2} < \infty;}$$if \(\bar{\xi }_{s} = \mathbf{1}_{\left [0,\tau \right ]}\left (s\right )[\bar{\xi }_{s}^{\left (1\right )}\alpha _{s} +\bar{\xi }_{ s}^{\left (2\right )}\left (1 -\alpha _{s}\right )]\), then \(\bar{\xi }_{s}\left (\omega \right ) \in \partial \Psi \left (\omega,s, \mathbb{E}^{\mathcal{F}_{s}}\bar{\eta }\right )\) and
$$\displaystyle{\mathbb{E}\ \left (\int _{0}^{\tau }e^{V _{s} }\left \vert \bar{\xi }_{s}\right \vert \mathit{dQ}_{s}\right )^{2} < \infty }$$(in the case \(\bar{\eta }=\eta\) we define \(\xi _{t} = \mathbb{E}^{\mathcal{F}_{t}}\eta\) and in the place of \((\bar{\xi }^{\left (1\right )},\bar{\xi }^{\left (2\right )},\bar{\xi })\) we shall use the notation \((\hat{\xi }^{\left (1\right )},\hat{\xi }^{\left (2\right )},\hat{\xi })\)).
Remark 5.60.
In place of the assumption (A7) we can consider two particular cases:
-
(A7 ′) \(\eta: \Omega \rightarrow \mathbb{O}\), where \(\mathbb{O}\) is the closed convex set defined by
$$\displaystyle{\mathbb{O}\mathop{ =}\limits^{ \mathit{def }}\left \{y \in \mathbb{R}^{m}:\varphi \left (y\right ) =\varphi \left (u_{ 0}\right )\text{ and }\psi \left (y\right ) =\psi \left (u_{0}\right )\right \};}$$or
-
(A7 ′ ′) there exist r 0 > 0 and \(v_{0} \in \mathrm{Dom}\left (\varphi \right ) \cap \mathrm{Dom}\left (\psi \right )\) such that
$$\displaystyle{\begin{array}{rl} \left (i\right )&\quad \eta: \Omega \rightarrow \overline{B\left (v_{0},r_{0}\right )} \subset \mathrm{int}\left (\mathrm{Dom}\left (\varphi \right )\right ) \cap \mathrm{int}\left (\mathrm{Dom}\left (\psi \right )\right ), \\ \left (\mathit{ii}\right )&\quad \mathbb{E}\left (e^{V _{\tau }}\left (\tau +A_{\tau }\right )\right ) < \infty.\end{array} }$$
Indeed:
-
\(\mathrm{(A}_{7}^{{\prime}}\mathrm{) \Rightarrow }\text{(A}_{7}\text{): }\) since \(\bar{\xi }_{t} = \mathbb{E}^{\mathcal{F}_{t}}\bar{\eta } \in \mathbb{O}\) for all t ≥ 0 we can set \(\bar{\xi }^{\left (1\right )} =\bar{\xi } ^{\left (2\right )} =\bar{\xi }= 0\) for every \(\bar{\eta }\in \overline{\eta,u_{0}};\)
-
\(\mathrm{(A}_{7}^{{\prime\prime}}\mathrm{)} \Rightarrow \text{(A}_{7}\text{): }\) by Proposition 6.2-\(\left (d\right )\) there exists an M 0 > 0 such that \(\partial \varphi \left (u\right ) \subset B\left (0,M_{0}\right )\) and \(\partial \psi \left (u\right ) \subset B\left (0,M_{0}\right )\) for all \(u \in \overline{B\left (v_{0},r_{0}\right )}\) and consequently \(\big\vert \bar{\xi }_{t}^{\left (1\right )}\big\vert +\big \vert \bar{\xi }_{t}^{\left (2\right )}\big\vert \leq 2M_{0}\) for all \(\bar{\eta }\in \overline{\eta,u_{0}}\) because \(\mathbb{E}^{\mathcal{F}_{t}}\bar{\eta } \in \overline{B\left (v_{0},r_{0}\right )}\) for all t ≥ 0. Observe that from \(\left (\mathrm{A}_{7}^{{\prime\prime}}\text{-ii}\right )\) it follows that \(\mathbb{P}\left (\tau = \infty \right ) = 0\).
Definition 5.61.
By the notation \(\mathit{dK}_{t} \in \partial \psi \left (Y _{t}\right )\mathit{dA}_{t}\) we shall understand that:
-
K is an \(\mathbb{R}^{m}\)-valued locally bounded variation stochastic process;
-
Y is an \(\mathbb{R}^{m}\)-valued continuous stochastic process such that \(\int _{0}^{T}\psi \left (Y _{ t}\right )\mathit{dA}_{t} < \infty \), a.s. \(\forall \,T \geq 0\); and
-
\(\mathbb{P}\text{-a.s.}\), for all 0 ≤ t ≤ s
$$\displaystyle{\int _{t}^{s}\left \langle y\left (r\right ) - Y _{ r},\mathit{dK}_{r}\right \rangle + \int _{t}^{s}\psi \left (Y _{ r}\right )\mathit{dA}_{r} \leq \int _{t}^{s}\varphi \left (y\left (r\right )\right )\mathit{dA}_{ r}\text{, }\forall y \in C\left (\mathbb{R}_{+}; \mathbb{R}^{m}\right ),}$$(we have an analogous definition for \(\mathit{dK}_{t} \in \partial \varphi \left (Y _{t}\right )dt\)).
Remark 5.62.
The condition \(0 \in \partial \varphi \left (u_{0}\right ) \cap \partial \psi \left (u_{0}\right )\) does not restrict the generality of the problem because from \(\mathrm{Dom}\left (\partial \varphi \right ) \cap \mathrm{Dom}\left (\partial \psi \right )\neq \varnothing \) it follows that there exists \(u_{0} \in \mathrm{Dom}\left (\partial \varphi \right ) \cap \mathrm{Dom}\left (\partial \psi \right )\) and \(\hat{u}_{01} \in \partial \varphi \left (u_{0}\right )\), \(\hat{u}_{02} \in \partial \psi \left (u_{0}\right );\) in this case equation (5.146) is equivalent to
where \(\hat{F}\left (s,y,z\right ) = F\left (t,y,z\right ) -\hat{ u}_{01}\), \(\hat{G}\left (s,y,z\right ) = G\left (t,y\right ) -\hat{ u}_{02}\), \(\hat{\varphi }\left (y\right ) =\varphi \left (y\right ) -\left \langle \hat{u}_{01},y - u_{0}\right \rangle\), \(\hat{\psi }\left (y\right ) =\psi \left (y\right ) -\left \langle \hat{u}_{02},y - u_{0}\right \rangle\), \(\partial \hat{\varphi }\left (y\right ) = \partial \varphi \left (y\right ) -\hat{ u}_{01}\), \(\partial \hat{\psi }\left (y\right ) = \partial \psi \left (y\right ) -\hat{ u}_{02}\) and \(d\hat{K}_{t} = \mathit{dK}_{t} -\hat{ u}_{01}dt -\hat{ u}_{02}\mathit{dA}_{t}\).
Let \(\varepsilon > 0\) and define the Moreau–Yosida regularization of \(\varphi\) by
which is a C 1 convex function and \(\nabla \varphi _{\varepsilon }(x) = \partial \varphi _{\varepsilon }\left (x\right ) \in \partial \varphi \left (J_{\varepsilon }x\right )\), where \(J_{\varepsilon }x = x -\varepsilon \nabla \varphi _{\varepsilon }(x)\). (For further properties see Annex B, Section “Convex Functions”.) Since \(0 \in \partial \varphi \left (u_{0}\right )\) we deduce that \(\varphi \left (u_{0}\right ) =\varphi _{\varepsilon }\left (u_{0}\right ) \leq \varphi _{\varepsilon }\left (u\right ) \leq \varphi \left (u\right )\), \(J_{\varepsilon }\left (u_{0}\right ) = u_{0}\) and \(\nabla \varphi _{\varepsilon }(u_{0}) = 0\).
We introduce the compatibility conditions between \(\varphi,\psi\) and F, G:
-
(A8) There exists a c > 0 and two progressively measurable stochastic processes f, g: \(\Omega \times \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}\) satisfying
$$\displaystyle{\mathbb{E}\int _{0}^{\tau }e^{2V _{s} }\left \vert f_{s}\right \vert ^{2}\mathit{ds} + \mathbb{E}\int _{ 0}^{\tau }e^{2V _{s} }\left \vert g_{s}\right \vert ^{2}\mathit{dA}_{ s} < \infty,}$$such that for all \(\varepsilon > 0\), t ≥ 0, \(y \in \mathbb{R}^{m}\), \(z \in \mathbb{R}^{m\times k}\), \(\mathbb{P}\text{-a.s.}\)
$$\displaystyle{ \begin{array}{rl} \left (i\right )&\quad \left \langle \nabla \varphi _{\varepsilon }\left (y\right ),\nabla \psi _{\varepsilon }\left (y\right )\right \rangle \geq 0, \\ \left (\mathit{ii}\right )&\quad \left \langle \nabla \varphi _{\varepsilon }\left (y\right ),G\left (t,y\right )\right \rangle \leq c\left \vert \nabla \psi _{\varepsilon }\left (y\right )\right \vert \left [\left \vert G\left (t,y\right )\right \vert + g_{t}\right ], \\ \left (\mathit{iii}\right )&\quad \left \langle \nabla \psi _{\varepsilon }\left (y\right ),F\left (t,y,z\right )\right \rangle \leq c\left \vert \nabla \varphi _{\varepsilon }\left (y\right )\right \vert \left [\left \vert F\left (t,y,z\right )\right \vert + f_{t}\right ],\end{array} }$$(5.152)and
$$\displaystyle{ \begin{array}{cc} \left (\mathit{iv}\right )& \quad \left \langle \nabla \varphi _{\varepsilon }\left (y\right ),-G\left (t,u_{0}\right )\right \rangle \leq c\left \vert \nabla \psi _{\varepsilon }\left (y\right )\right \vert \left [\left \vert G\left (t,u_{0}\right )\right \vert + g_{t}\right ], \\ \left (\mathit{v}\right ) &\quad \left \langle \nabla \psi _{\varepsilon }\left (y\right ),-F\left (t,u_{0},0\right )\right \rangle \leq c\left \vert \nabla \varphi _{\varepsilon }\left (y\right )\right \vert \left [\left \vert F\left (t,u_{0},0\right )\right \vert + f_{t}\right ].\end{array} }$$(5.153)
Example 5.63.
-
\(\left (e_{1}\right )\) If \(\varphi =\psi\) then the compatibility assumptions (5.152) and (5.153) are clearly satisfied.
-
\(\left (e_{2}\right )\) Let m = 1. Since \(\nabla \varphi _{\varepsilon }\) and \(\nabla \psi _{\varepsilon }\) are increasing monotone functions on \(\mathbb{R}\), we see that, if \(G\left (t,u_{0}\right ) = F\left (t,u_{0},0\right ) = 0\) and
$$\displaystyle{\left (y\, - u_{0}\right )G\left (t,y\right ) \leq 0\quad \text{and}\quad \left (y\, - u_{0}\right )F\left (t,y,z\right ) \leq 0,\;\forall \,t,y,z,}$$then the compatibility assumptions (5.152) and (5.153) are satisfied.
-
\(\left (e_{3}\right )\) Let m = 1. If \(\varphi,\psi: \mathbb{R} \rightarrow (-\infty,+\infty ]\) are the convex indicator functions
$$\displaystyle{\varphi \left (y\right ) = \left \{\begin{array}{rl} 0,&\text{if}\ y \in \left [a,b\right ],\\ + \infty, &\text{if} \ y\notin \left [a, b \right ], \end{array} \right.\quad \text{and}\quad \psi \left (y\right ) = \left \{\begin{array}{rl} 0,&\text{if}\ y \in \left [c,d\right ],\\ + \infty, &\text{if} \ y\notin \left [c, d \right ], \end{array} \right.}$$where \(-\infty \leq a \leq b \leq +\infty \) and \(-\infty \leq c \leq d \leq +\infty \) are such that \(\left [a,b\right ] \cap \left [c,d\right ]\neq \varnothing \) (see the assumption (A6)), then
$$\displaystyle\begin{array}{rcl} \nabla \varphi _{\varepsilon }\left (y\right )& =& \frac{1} {\varepsilon } \left [\left (y - b\right )^{+} -\left (a - y\right )^{+}\right ],\quad \text{and} {}\\ \nabla \psi _{\varepsilon }\left (y\right )& =& \frac{1} {\varepsilon } \left [\left (y - d\right )^{+} -\left (c - y\right )^{+}\right ]. {}\\ \end{array}$$The assumption (A8-i) is clearly fulfilled; the next compatibility assumptions (A8-ii, iii, iv, v) are satisfied if for example \(G\left (t,u_{0}\right ) = F\left (t,u_{0},0\right ) = 0\) and
$$\displaystyle{G\left (t,y\right ) \geq 0,\mathit{\;}\mathit{for}y \leq a,\quad G\left (t,y\right ) \leq 0,\mathit{\;}\mathit{for}y \geq b,}$$and, respectively,
$$\displaystyle{F\left (t,y,z\right ) \geq 0,\mathit{\;}\mathit{for}y \leq c,\quad F\left (t,y,z\right ) \leq 0,\mathit{\;}\mathit{for}y \geq d.}$$
We complete the assumptions with some general boundedness conditions
-
(A9) For all ρ > 0
$$\displaystyle{ \begin{array}{rl} \left (i\right )\;&\mathbb{E}\ e^{2V _{\tau }}\left (\left \vert \eta -u_{0}\right \vert ^{2} +\varphi (\eta ) -\varphi (u_{0}) +\psi (\eta ) -\psi (u_{0})\right ) < \infty, \\ \left (\mathit{ii}\right )\;&\mathbb{E}\ \left (\int _{0}^{\tau }e^{V _{s} }F_{\rho }^{\#}\left (s\right )\mathit{ds}\right )^{2} + \mathbb{E}\ \left (\int _{0}^{\tau }e^{V _{s} }G_{\rho }^{\#}\left (s\right )\mathit{dA}_{s}\right )^{2} < \infty, \\ \left (\mathit{iii}\right )\;&\mathbb{E}\ \left [\int _{0}^{\tau }e^{2V _{s} }\left \vert F_{\rho }^{\#}\left (s\right )\right \vert ^{2}\mathit{ds} + \int _{0}^{\tau }e^{2V _{s} }\left \vert G_{\rho }^{\#}\left (s\right )\right \vert ^{2}\mathit{dA}_{s}\right ] < \infty, \\ \left (\mathit{iv}\right )\;&\mathbb{E}\ \left (\int _{0}^{T}e^{V _{s} }\mathit{dQ}_{s}\right )^{2} < \infty \text{, for all }T \geq 0;\end{array} }$$(5.154)
and some special boundedness conditions
-
(A10) There exist L, b > 0 such that for all 0 ≤ t ≤ τ, \(\mathbb{P}\text{-a.s.}\)
$$\displaystyle{ \begin{array}{ll} \left (a\right )&\quad \ell_{t} + \int _{0}^{\tau }\left (\ell_{ s}\right )^{2}\mathit{ds} \leq L, \\ {r} {\left (b\right )\quad }&{r}{e^{V _{\tau }}\left \vert \eta -u_{0}\right \vert + \left \vert H\left (t,u_{0},0\right )\right \vert + \int _{0}^{\tau }e^{V _{s} }\left \vert H\left (s,u_{0},0\right )\right \vert \mathit{dQ}_{s} \leq b},\end{array} }$$(5.155)where again H is defined by
$$\displaystyle{H(t,y,z) =\alpha _{t}F(t,y,z) + (1 -\alpha _{t})G(t,y).}$$We also recall the definition of
$$\displaystyle{\Psi (t,y) =\alpha _{t}\varphi (y) + (1 -\alpha _{t})\psi (y).}$$
Since V ≥ 0, we remark that under (A10) we have
and for all t ≥ 0,
Therefore by Proposition 6.80-A, for all q > 0
Using the definition of Q, H and \(\Psi \) we can rewrite (5.146) in the form
Definition 5.64.
We call \(\left (Y _{t},Z_{t}\right )_{t\geq 0}\) a solution of (5.157) if
-
\(\left (d_{1}\right )\) \(\left (Y,Z\right ) \in S_{m}^{0} \times \Lambda _{m\times k}^{0}\),
-
\(\left (d_{2}\right )\) \(\left (Y _{t},Z_{t}\right ) = \left (\xi _{t},\zeta _{t}\right ) = \left (\eta,0\right )\), if t > τ,
-
\(\left (d_{3}\right )\) \(\mathbb{P}\)-a.s., for all T ≥ 0,
$$\displaystyle{\int _{0}^{T}\left [\left \vert F\left (s,Y _{ s},Z_{s}\right )\right \vert + \left \vert \varphi \left (Y _{s}\right )\right \vert \right ]\mathit{ds} + \int _{0}^{T}\left [\left \vert G\left (s,Y _{ s}\right )\right \vert + \left \vert \psi \left (Y _{s}\right )\right \vert \right ]\mathit{dA}_{s} < \infty,}$$ -
\(\left (d_{4}\right )\) there exists a K ∈ S m 0 such that \(\mathbb{P}\text{-a.s.}\)
$$\displaystyle{\begin{array}{rl} \left (i\right )&\quad \left \updownarrow K\right \updownarrow _{T} < \infty,\;\;\forall T \geq 0, \\ \left (\mathit{ii}\right )&\quad \mathit{dK}_{t} \in \partial _{y}\Psi \left (t,Y _{t}\right )\mathit{dQ}_{t},\end{array} }$$ -
\(\left (d_{5}\right )\) \(e^{2V _{T}}\left \vert Y _{T} -\xi _{T}\right \vert ^{2} + \int _{T}^{\infty }e^{2V _{s}}\left \vert Z_{s} -\zeta _{s}\right \vert ^{2}\mathit{ds}\;\;\mathop{\longrightarrow}\limits_{}^{\;\;prob.\;\;}0,\;\) as \(T \rightarrow \infty \),
-
\(\left (d_{6}\right )\) \(\mathbb{P}\text{-a.s.}\), for all 0 ≤ t ≤ T,
$$\displaystyle{ Y _{t} + K_{T} - K_{t} = Y _{T} + \int _{t}^{T}H\left (s,Y _{ s},Z_{s}\right )\mathit{dQ}_{s} -\int _{t}^{T}Z_{ s}\mathit{dB}_{s} }$$(5.158)(we also say that the triplet \(\left (Y,Z,K\right )\) is a solution of (5.157)).
Remark 5.65.
If there exists a constant C such that \(\sup _{t\in \left [0,\tau \right ]}\left \vert V _{t}\left (\omega \right )\right \vert \leq C\), \(\mathbb{P}\text{-a.s.}\;\omega \in \Omega \), then the condition \(\left (d_{5}\right )\) from Definition 5.64 is equivalent to
In the rest of this book, a constant depending upon p > 0 is denoted by C p ; in this section since we are only considering the case p = 2 we will denote the corresponding constant by C 2.
We now give the main result.
Theorem 5.66.
Let the assumptions (A 1 –A 10 ) be satisfied. Then the backward stochastic variational inequality (5.157) has a unique solution \(\left (Y,Z,K\right ) \in S_{m}^{0} \times \Lambda _{m\times k}^{0} \times S_{m}^{0}\) such that
Moreover there exists \(U^{(1)},U^{(2)} \in \Lambda _{m}^{0}\) , with \(U^{(1)} \in \partial \varphi (Y _{t})\) \(d\mathbb{P} \otimes dt\) a.e. and \(U_{t}^{(2)} \in \partial \psi (Y _{t})\) \(d\mathbb{P} \otimes \mathit{dA}_{t}\) a.e., so that with \(U_{t} = \mathbf{1}_{[0,\tau ]}(t)[\alpha _{t}U_{t}^{(1)} + (1 -\alpha _{t})U_{t}^{(2)}]\), \(\mathit{dK}_{t} = U_{t}{\mathit{dQ}}_{t} \in \partial _{y}\Psi (t,Y _{t}){\mathit{dQ}}_{t}\),
and for all 0 ≤ t ≤ T,
The solution also satisfies for some positive constants:
-
(A)
for all t ≥ 0 and all q > 0,
$$\displaystyle{ \begin{array}{rl} \left (j\right )&\quad \vert Y _{t} - u_{0}\vert \leq e^{V _{t}}\vert Y _{t} - u_{0}\vert \leq C_{b}, \\ \left (\,\mathit{jj}\right )&\quad \mathbb{E}\ \left (\int _{0}^{\infty }e^{2V _{r} }\left \vert Z_{r}\right \vert ^{2}\mathit{dr}\right )^{q/2} \leq C_{q,b};\end{array} }$$(5.161) -
(B)
for all t ≥ 0,
$$\displaystyle{ \begin{array}{l} \mathbb{E}^{\mathcal{F}_{t}}\sup \limits _{s\geq t}\left \vert e^{V _{s}}\left (Y _{s} - u_{0}\right )\right \vert ^{2} + \mathbb{E}^{\mathcal{F}_{t}}\ \left (\int _{t}^{\infty }e^{2V _{s}}\left \vert Z_{s}\right \vert ^{2}\mathit{ds}\right ) \\ \quad \quad + \mathbb{E}^{\mathcal{F}_{t}}\ \int _{t}^{\infty }e^{2V _{s}}\mathbf{1}_{\left [0,\tau \right ]}\left (s\right )\left [\left \vert \varphi \left (Y _{s}\right ) -\varphi \left (u_{0}\right )\right \vert \mathit{ds} + \left \vert \psi \left (Y _{s}\right ) -\psi \left (u_{0}\right )\right \vert \mathit{dA}_{s}\right ] \\ \leq C_{2}\ \mathbb{E}^{\mathcal{F}_{t}}\bigg[e^{2V _{\tau }}\left \vert \eta -u_{0}\right \vert ^{2} +\Big (\int _{t}^{\tau }e^{V _{s}}\left (\left \vert F\left (s,u_{0},0\right )\right \vert \mathit{ds} + \left \vert G\left (s,u_{0}\right )\right \vert \mathit{dA}_{s}\right )\Big)^{2}\bigg];\end{array} }$$(5.162) -
(C)
for all t ≥ 0,
$$\displaystyle{ \begin{array}{l} \mathbb{E}\sup \limits _{s\geq t}e^{2V _{s}}\left \vert Y _{s} -\xi _{s}\right \vert ^{2} + \mathbb{E}\int _{t}^{\infty }e^{2V _{s}}\vert Z_{s} -\zeta _{s}\vert ^{2}\mathit{ds} \\ \quad \quad \quad \quad \quad \quad + \mathbb{E}\int _{t}^{\infty }e^{2V _{s} }\left \vert \Psi \left (s,Y _{s}\right ) - \Psi \left (s,\xi _{s}\right )\right \vert \mathit{dQ}_{s} \\ \leq C_{2}\mathbb{E}\ \Big(\int _{t}^{\infty }\mathbf{1}_{\left [0,\tau \right ]}\left (s\right )e^{V _{s}}\big[\vert \hat{\xi }_{ s}\vert \mathit{dQ}_{s} + \left (\vert F(s,\xi _{s},0)\vert +\ell _{s}\left \vert \zeta _{s}\right \vert \right )dsx \\ \quad \quad \quad \quad \quad \quad + \left \vert G(s,\xi _{s})\right \vert \mathit{dA}_{s}\big]\Big)^{2};\end{array} }$$(5.163) -
(D)
for all t ≥ 0
$$\displaystyle{ \begin{array}{l} \mathbb{E}\left [e^{2V _{t}}\left (\varphi (Y _{t}) -\varphi (u_{0}) +\psi (Y _{t}) -\psi (u_{0})\right )\right ] \\ \quad \quad + \dfrac{1} {2}\mathbb{E}\int _{t}^{\infty }\mathbf{1}_{\left [0,\tau \right ]}\left (s\right )e^{2V _{s}}\Big(\left \vert U_{ s}^{\left (1\right )}\right \vert ^{2}\mathit{ds} + \left \vert U_{ s}^{\left (2\right )}\right \vert ^{2}\mathit{dA}_{ s}\Big) \\ \leq \mathbb{E}\left [e^{2V _{\tau }}\left (\varphi (\eta ) -\varphi (u_{0}) +\psi (\eta ) -\psi (u_{0})\right )\right ] \\ \quad \quad + \left (1 + c\right )^{2}\mathbb{E}\int _{t}^{\infty }\mathbf{1}_{\left [0,\tau \right ]}\left (s\right )e^{2V _{s}}\left (\vert F(s,Y _{ s},Z_{s})\vert ^{2} + \vert f_{ s}\vert ^{2}\right )\mathit{ds} \\ \quad \quad + \left (1 + c\right )^{2}\mathbb{E}\int _{t}^{\infty }\mathbf{1}_{\left [0,\tau \right ]}\left (s\right )e^{2V _{s}}\left (\vert G(s,Y _{ s})\vert ^{2} + \vert g_{ s}\vert ^{2}\right )\mathit{dA}_{ s}.\end{array} }$$(5.164)
Proof.
Uniqueness. If \(\left (Y,Z,K\right )\), \(\left (Y ^{{\prime}},Z^{{\prime}},K^{{\prime}}\right )\) are two solutions, in the sense of Definition 5.64, that satisfy (5.160), then
Applying the monotonicity and Lipschitz property of the function H and taking into account that
for \(\;\mathit{dK}_{s} \in \partial _{y}\Psi (s,Y _{s})\mathit{dQ}_{s}\) and \(\mathit{dK}_{s}^{{\prime}}\in \partial _{y}\Psi (s,Y _{s}^{{\prime}})\mathit{dQ}_{s}\), then
Using Corollary 6.82 from Annex C, it follows that
which yields the uniqueness.
The proof of the existence will be split into several steps.
-
A.
Approximating problem. Let \(n \in \mathbb{N}^{{\ast}}\) and \(\varepsilon = \dfrac{1} {n}\).
Let
and
We note that
and
We consider the approximating stochastic equation: for all t ≥ 0,
or equivalently
To show the existence of a solution \(\left (Y ^{n},Z^{n}\right )\) of (5.166) we intend to use Lemma 5.29-\(\left (h_{2}\right )\).
Since \(\left \langle y^{{\prime}}- y,\nabla \varphi _{\varepsilon }\left (y^{{\prime}}\right ) -\nabla \varphi _{\varepsilon }\left (y\right )\right \rangle \geq 0\) (and similarly for \(\nabla \psi _{\varepsilon }\)) we notice that \(\Phi _{n}\) satisfies the inequalities
Consequently the corresponding assumptions (5.13-BSDE-H \(_{\Phi }\)) for \(\Phi _{n}\) are satisfied.
We have
For the assumption \(\left (h_{2}\right )\) from Lemma 5.29 we have for all ρ > 0,
because \(\nabla _{y}\Psi ^{n}(s,u_{0}) = 0\) and
By Lemma 5.29-\(\left (h_{2}\right )\) Eq. (5.166) has a unique solution \((Y ^{n},Z^{n}) \in S_{m}^{0} \times \Lambda _{m\times k}^{0}\) such that
Consequently for all T > n,
Now we remark that
and therefore
By Proposition 6.80 we have for all q, T > 0
and
-
B.
Boundedness of Y n and Z n .
If n ≤ T, then
Passing to the limit as T → ∞ in (5.169) we infer (by the Beppo Levi monotone convergence theorem) that for all t ≥ 0
In particular, using the assumption (5.155), we deduce that for all t ≥ 0
Moreover from (5.168) for all q > 0
-
C.
Estimates on \(\left \vert Y _{t}^{n} -\xi _{t}\right \vert\) and \(\left \vert Z_{t}^{n} -\zeta _{t}\right \vert\) for large t ≥ 0.
If there exists an N 0 > 0 such that \(\tau \left (\omega \right ) \leq N_{0}\), \(\mathbb{P}\text{-a.s.}\;\omega \in \Omega \), then Y t n = ξ t = η and Z t n = ζ t = 0 for all t ≥ N 0.
We next consider the case where \(\mathbb{P}\left (\tau > N\right ) > 0\) for all \(N \in \mathbb{N}^{{\ast}}\).
Since
we infer, from (5.166), that (Y n, Z n) satisfies for all \(t \in \left [0,n\right ]\) the equality
We have
From \(\left \langle \nabla _{y}\Psi ^{n}\left (t,\xi _{t}\right ),y -\xi _{t}\right \rangle \leq \Psi ^{n}\left (t,y\right ) - \Psi ^{n}\left (t,\xi _{t}\right ) \leq \langle y -\xi _{t},\nabla _{y}\Psi ^{n}\left (t,y\right )\rangle\) we infer that
where \(\hat{\xi }_{t} \in \partial _{y}\Psi \left (t,\xi _{t}\right )\) is given by the assumption (A7). Using the inequality (5.167)) it follows that, as signed measures on ℝ +,
Since
by Proposition 5.2 we deduce that for 0 ≤ t ≤ T,
Recall that \((Y _{s}^{n},Z_{s}^{n}) = (\xi _{s},\zeta _{s}),\;\forall \,s > n\). Passing to the limit T → ∞ in (5.174), we obtain by the Beppo Levi monotone convergence theorem
-
D.
Boundedness of \(\nabla \varphi _{\varepsilon }(Y _{t}^{n})\) and \(\nabla \psi _{\varepsilon }(Y _{t}^{n})\).
By the stochastic subdifferential inequality from Lemma 2.38 and Remark 2.39, for all 0 ≤ t ≤ T
(and a similar inequality for \(\psi _{\varepsilon }\)). We infer that
Using the definition of the function H(t, y, z) given in (5.149), the compatibility assumptions (5.152) yield
and respectively
Recall that \(\varphi \left (y\right ) \geq \varphi _{\varepsilon }\left (u_{0}\right ) =\varphi \left (u_{0}\right )\) and \(\psi \left (y\right ) \geq \psi _{\varepsilon }\left (u_{0}\right ) =\psi \left (u_{0}\right )\). From (5.152), (5.176–5.178) and the inequality \(a\left (x + y\right ) \leq \frac{1} {2}a^{2} + x^{2} + y^{2}\) we obtain
The stochastic integral from this last inequality has the property
because by \(\nabla \varphi _{\varepsilon }\left (u_{0}\right ) = \nabla \psi _{\varepsilon }\left (u_{0}\right ) = 0\) we have
Let T ≥ n. By Jensen’s inequality it follows that
Now from inequality (5.179) we infer by Beppo Levi’s monotone convergence theorem for T → ∞
By (5.171), (5.168) and the assumption (5.154(iii)) we deduce that there exists a constant C independent of n such that
and
Therefore from (5.180) we have
and
Since
we see from (5.181) that, for all t ≥ 0,
(recall that \(\varepsilon = 1/n\)).
-
E.
Cauchy sequences and convergence.
Note that by assumption (A10) and (5.156) we have \(\left \vert \xi _{s} - u_{0}\right \vert \leq b\) and
Hence by assumption (5.154ii)
and by (5.175),
By uniqueness it follows that, for all \(t \in \left [0,n\right ]\),
where on \(\left [0,n\right ]\)
By (6.28, with a = 0, \(\varepsilon = 1/n\) and \(\delta = 1/\left (n + i\right )\))
and using (5.150) we have on \(\left [0,n\right ]\)
Since by (5.171),
we obtain by Proposition 5.2 that
The estimates (5.182) and (5.184) give us
Hence
and
-
F.
Passage to the limit.
Consequently there exists \((Y,Z) \in S_{m}^{0} \times \Lambda _{m\times k}^{0}\) such that
We have that \((Y _{t},Z_{t}) = \left (\eta,0\right )\) for t > τ, since Y t n = ξ t = η and Z t n = ζ t = 0 for t > τ.
Taking into account (5.183) and
the inequality (5.162) follows from (5.170) by Fatou’s Lemma.
Also by Fatou’s Lemma from (5.175) we obtain (5.163) and from (5.171) and (5.172) we deduce (5.161).
From (5.182) there exist two p.m.s.p. \(U^{\left (1\right )}\) and \(U^{\left (2\right )}\), such that along a subsequence still indexed by n, we have for \(\varepsilon = \frac{1} {n} \rightarrow 0\)
Using (5.183) and applying Fatou’s Lemma we have
and similarly for ψ. Passing to \(\liminf _{n\rightarrow +\infty }\) in (5.180) we obtain (5.164).
From (5.165) we have for all 0 ≤ t ≤ T ≤ n, \(\mathbb{P}\)-a.s.
and passing to the limit we conclude that
with
By (5.118 − b), we see that, for all \(E \in \mathcal{F}\), 0 ≤ s ≤ t and X ∈ S m 2,
Passing to \(\liminf\) for \(n \rightarrow \infty \) in the above inequality we obtain \(U_{s}^{\left (1\right )} \in \partial \varphi (Y _{s}),d\mathbb{P} \otimes \mathit{ds}\)-a.e. and, with similar arguments, \(U_{s}^{\left (2\right )} \in \partial \psi (Y _{s}),d\mathbb{P} \otimes \mathit{dA}_{s}\)-a.e. Summarizing the above conclusions we conclude that (Y, Z, U) is a solution of the BSVI (5.157).
We want to highlight the fact that the assumption (A10-b) is too strong for many applications. The next two results are concerned with the existence of a solution for the backward stochastic variational inequality (5.146) recalled here for convenience:
without the boundedness conditions from (A10).
Consider the closed convex sets
Since every point of \(\mathbb{O}\) is a minimum point for \(\varphi\) and ψ, it follows that \(\nabla \varphi _{\varepsilon }\left (u\right ) = \nabla \psi _{\varepsilon }\left (u\right ) = 0\) for all \(u \in \mathbb{O}\).
Theorem 5.67.
Let the assumptions \(\left (\mathrm{A}_{1}\right ),\ldots,\left (\mathrm{A}_{9}\right )\) be satisfied and assume that there exists a δ 0 > 0 such that
Assume moreover there exists δ ∈ (0,∞] and \(q = 1 + \frac{\delta } {2+\delta }\) (with q = 2 if δ = ∞) such that
Then the BSVI (5.188) has a unique solution \(\left (Y,Z,K\right ) \in S_{m}^{0} \times \Lambda _{m\times k}^{0} \times S_{m}^{0}\) in the sense of Definition 5.64 such that for \(q = 1 + \frac{\delta } {2+\delta }\) and q = 2 if δ = ∞,
Moreover the inequalities (5.162) and (5.163) hold.
Proof.
\(\left (\boldsymbol{I}\right )\boldsymbol{Uniqueness}\). The proof of uniqueness is similar to that given for Theorem 5.66 except that now by Corollary 6.82 from Annex C, we have
\(\left (\mathbf{\mathit{II}}\right )\mathbf{\mathit{Existence.}}\) Step 1. Approximation of the problem’s data to satisfy \(\left (\mathrm{A}_{10}\right )\). Let
Define, for \(n \in \mathbb{N}^{{\ast}}\),
and
Let \(\left (\xi ^{n},\zeta ^{n}\right )\) be given by the martingale representation theorem (Corollary 2.44): for all t ≥ 0, \(\xi _{t}^{n} = \mathbb{E}^{\mathcal{F}_{t}}\eta _{n}\) and
or equivalently, for all T > 0
It is easy to verify that
Applying Corollary 6.83, first on \(\left [t,T\right ]\) and then letting T → ∞, we infer that for all t ≥ 0
Since the assumptions (A1, …, A9) are satisfied by (η, F, G, \(\varphi\), ψ, V, μ, ν, ℓ) it follows that the same assumptions are satisfied replacing (η, F, G, \(\varphi\), ψ, V, μ, ν, ℓ) by (η n , \(\hat{F}_{n}\), \(\hat{G}_{n}\), \(\varphi\), ψ, V n, μ, ν, ℓ n).
With respect to (A10) we have
and
Hence \(\left (\eta _{n},\hat{F}_{n},\hat{G}_{n},\mu,\nu,\ell^{n},V ^{n}\right )\) satisfies (A10).
Step 2. Approximating equation and estimates. By Step 1 we are in the conditions of Theorem 5.66 and therefore the approximating equation
has a unique solution \(\left (Y ^{n},Z^{n},K^{n}\right ) \in S_{m}^{0} \times \Lambda _{m\times k}^{0} \times S_{m}^{0}\), \(\left (Y _{s}^{n},Z_{s}^{n}\right ) = \left (\xi _{s}^{n},0\right )\) for s > τ and \(U_{s}^{n} =\alpha _{s}U_{s}^{1,n} + \left (1 -\alpha _{s}\right )U_{s}^{2,n}\) such that
Moreover the inequalities (5.161), (5.162), (5.163) and (5.164) hold with (η, F, G, \(\varphi\), ψ, V, μ, ν, ℓ, C b , C q, b ) by (η n , \(\hat{F}_{n}\), \(\hat{G}_{n}\), \(\varphi\), ψ, V n, μ, ν, ℓ n, C n , C q, n ).
Using in (5.193) \(V _{t}^{n+i} -\left (n + i\right )^{3} \leq V _{t}^{n}\) we get for all \(i \in \mathbb{N}\)
Since \(\left \langle Y _{s}^{n} - u_{0},U_{s}^{n} - 0\right \rangle \geq 0\) and
we infer by Corollary 6.82 for p = 2 and 0 ≤ t ≤ T,
But by (5.192-b) and V T n+i ≤ V T we have
Using Beppo Levi’s monotone convergence theorem and (5.194-jj ′ ′) we can pass to the limit in (5.195), first \(\limsup _{T\rightarrow \infty }\) and then \(\lim _{i\rightarrow \infty }\). We obtain
and, in particular,
Step 3. Cauchy sequence and convergences. Let
We have for any j > i > 1
then for all T > 0, by Proposition 5.2 with \(q = 1 + \frac{\delta } {2+\delta } \in \left (1,2\right )\) and q = 2 if δ = ∞,
But
with
the last inequality is obtained by Hölder’s inequality since \(\frac{1} {2+\delta } + \frac{1} {2/q} = 1\). Thus
Here we have
and as T → ∞ we infer
Using (5.196) for the boundedness of \(\mathbb{E}\int _{0}^{\infty }e^{2V _{s} }\left \vert Z_{s}^{n}\right \vert ^{2}\mathit{ds}\) we get from (5.198) as T → ∞ and then passing to the limit as j → ∞:
which yields by (5.190) the existence of a pair \(\left (Y,Z\right ) \in S_{m}^{0} \times \Lambda _{m\times k}^{0}\) such that
Now by Fatou’s lemma from (5.196) we obtain (5.162) and consequently (5.191-j).
To verify (5.191-jj), following the proof of Theorem 5.66, we have
By (5.192-a) and (5.197) we have
Furthermore, using (5.194-jj ′ ′) and (5.192-c), we have
In the same manner as above when we proved (5.196) we obtain, by Corollary 6.82 for p ∈ (1, 2], similar inequalities with \(\left (\xi _{s},\zeta _{s}\right )\) in place of \(\left (u_{0},0\right )\) and passing successively to the limit T → ∞ and i → ∞ we get that for all t ≥ 0
Since \(\vert \hat{H}_{n}(s,\xi _{s},\zeta _{s})\vert \leq \left \vert H(s,\xi _{s},0)\right \vert +\ell _{s}\left \vert \zeta _{s}\right \vert + \vert H(s,u_{0},0)\vert \mathbf{1}_{\left (n,\infty \right )}\left (\gamma _{s}\right )\), from (5.200) with Fatou’s Lemma applied to the left-hand side and the Lebesgue dominated convergence theorem applied to the right-hand side we infer by taking the limit as n → ∞
which yields (5.163) if we choose p = 2.
In the case \(p = q = 1 + \frac{\delta } {2+\delta } \in \left (1,2\right )\), by Hölder’s inequality, we have
In the case p = q = 2, δ = ∞ and ℓ is a deterministic process
Using the assumptions (5.190) and (5.154-ii), from (5.201) we infer (5.191-jj).
Step 4. Estimates on the subdifferential term \(\mathit{dK}_{s}^{n} = U_{s}^{n}dQ_{s} \in \partial _{y}\Psi \left (s,Y _{s}^{n}\right )dQ_{s}\) . We now use the assumption (5.189) on the interior of \(\mathrm{Dom}\left (\varphi \right )\). From the proof of Corollary 5.49 we have
where
and \(\hat{u}_{t} = 0 \in \partial _{y}\Psi \left (\omega,t,u_{0}\right )\). Hence
By Proposition 6.80-B we obtain
From the convergence (5.199) and the equality
it follows, via Lemma 5.16, that there exists a K ∈ S m 0 such that
As in Proposition 1.20 and Corollary 1.22 we obtain
and
Finally passing to the limit in
we complete the proof. ■
Remark 5.68.
In this last theorem, in contrast to the results in Theorem 5.66, we have not been able to show that the process K is absolutely continuous.
To end this section we discuss a particular case of BSVI (5.146) that we recall here for the convenience of the reader:
where the assumptions (A1),…, (A10) from the beginning of this section will to be replaced by
- \(\left (L_{1}\right )\)::
-
(A1) + (A2) + (A3) are satisfied;
- \(\left (L_{2}\right )\)::
-
the functions \(F: \Omega \times \mathbb{R}_{+} \times \mathbb{R}^{m} \times \mathbb{R}^{m\times k} \rightarrow \mathbb{R}^{m}\) and \(G: \Omega \times \mathbb{R}_{+} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}\) satisfy
$$\displaystyle{F\left (\cdot,\cdot,y,z\right ),\ G\left (\cdot,\cdot,y\right )\mathit{\ is\ p.m.s.p.,\ for\ each}\left (y,z\right ) \in \mathbb{R}^{m} \times \mathbb{R}^{m\times k}\,\text{,}}$$and there exists an L > 0 such that, \(\mathbb{P}\text{-a.s.}\;\omega \in \Omega \), a.e. t ≥ 0, for all y, y ′, z, z ′
$$\displaystyle{ \begin{array}{rl} \left ({\mathit{i}}\right )&\quad \left \langle y^{{\prime}}- y,F(t,y^{{\prime}},z) - F(t,y,z)\right \rangle \leq \frac{L} {2} \,\left \vert y^{{\prime}}- y\right \vert ^{2}, \\ \left ({\mathit{ii}}\right )&\quad \left \vert F(t,y,z^{{\prime}}) - F(t,y,z)\right \vert \leq \sqrt{\frac{L} {2}} \,\left \vert z^{{\prime}}- z\right \vert, \\ \left ({\mathit{iii}}\right )&\quad \left \vert F\left (\omega,t,y,0\right )\right \vert \leq \frac{L} {2} \left (1 + \left \vert y\right \vert \right ), \\ \left ({\mathit{iv}}\right )&\quad \left \langle y^{{\prime}}- y,G(t,y^{{\prime}}) - G(t,y)\right \rangle \leq L\,\left \vert y^{{\prime}}- y\right \vert ^{2}, \\ \left ({\mathit{v}}\right )&\quad \left \vert G\left (\omega,t,y\right )\right \vert \leq L\left (1 + \left \vert y\right \vert \right ).\end{array} }$$(5.203)
We remark that in this case \(\mu _{t} = \frac{L} {2} \mathbf{1}_{\left [0,\tau \right ]}\left (t\right )\), \(\ell_{t} = \sqrt{\frac{L} {2}} \mathbf{1}_{\left [0,\tau \right ]}\left (t\right )\), \(\nu _{t} = L\mathbf{1}_{\left [0,\tau \right ]}\left (t\right )\) and
- \(\left (L_{3}\right )\)::
-
(A6)+(A7)+(A8) are satisfied;
- \(\left (L_{5}\right )\)::
-
assume that
$$\displaystyle{ \mathbb{E}\ e^{2LQ_{\tau }}\left (1 + \left \vert \eta \right \vert ^{2} + \left \vert \varphi (\eta )\right \vert + \left \vert \psi (\eta )\right \vert \right ) < \infty. }$$(5.204)
We highlight that under \(\left (L_{1}\right ),\ldots,\left (L_{5}\right )\), the assumptions \(\text{(A}_{1}\text{),}\ldots,\text{ (A}_{9}\text{)}\) are satisfied. Also from \(\left (L_{5}\right )\) we have
and consequently τ < ∞, \(\mathbb{P}\text{-a.s.}\) Moreover it is not difficult to verify that by \(\left (L_{2}\right )\), \(\left (L_{5}\right )\) and (5.192-a) the condition (5.190) is satisfied for all δ ∈ (0, ∞) and for all \(q = 1 + \frac{\delta } {2+\delta } \in \left (1,2\right )\). Hence with the exception of assumption (5.189) on the interior of \(\mathrm{Dom}\left (\varphi \right )\) all other assumptions of Theorem 5.67 are satisfied.
Theorem 5.69.
Under the assumptions \(\left (L_{1}\right ),\ldots,\left (L_{5}\right )\) the BSVI (5.202) has a unique solution \(\left (Y,Z,K\right ) \in S_{m}^{0} \times \Lambda _{m\times k}^{0} \times S_{m}^{0}\) in the sense of Definition 5.64 which satisfies for all \(q \in \left (1,2\right )\) :
Moreover there exist \(U^{\left (1\right )},U^{\left (2\right )} \in \Lambda _{m}^{0}\), \(U_{t}^{\left (1\right )} \in \partial \varphi \left (Y _{t}\right )\) and \(U_{t}^{\left (2\right )} \in \partial \psi \left (Y _{t}\right )\), \(\ d\mathbb{P}\otimes dt\) -a.e. such that \(\mathit{dK}_{t} = U_{t}dQ_{t} \in \partial _{y}\Psi \left (t,Y _{t}\right )dQ_{t}\) , where
and
The inequalities (5.162), (5.163) and (5.164) hold with V t = LQ t .
Proof.
The proof is similar to that of Theorem 5.67: the Steps 1–3 are exactly the same. To pass to the limit in the approximating equation
we need a new argument for Step 4 since now the interior condition (5.189) is not satisfied.
Step 4 ′ . Estimates on subdifferential terms U 1, n and U 2, n. By Theorem 5.66 we have
Note that V t n = V t −θ t n, where \(\theta _{t}^{n} = \int _{0}^{t\wedge \tau }\frac{L} {2} \,\mathbf{1}_{\left (n,\infty \right )}\left (\lambda _{s}\right )\mathit{ds}\). Since
we obtain
and
Consequently there exist two p.m.s.p. \(U^{\left (1\right )}\) and \(U^{\left (2\right )}\), such that along a subsequence still indexed by n,
Passing to the limit in (5.205) the result follows in a standard manner (see the proof of Theorem 5.66). ■
Remark 5.70.
If τ = T < ∞ is a deterministic final time, then the assertions of Theorem 5.69 are also true with q = 2 (and δ = ∞) by setting \(\ell_{s} = \sqrt{\frac{L} {2}} \mathbf{1}_{\left [0,T\right ]}\left (s\right )\).
5.6.3 Weak Variational Solutions
In this subsection we discuss again the existence and the uniqueness of a solution \(\left (Y,Z\right )\) of BSVI (5.146) that we recall here:
under the assumptions \(\text{(A}_{1}\text{)},\ldots,\text{(A}_{9}\text{)}\) presented in Sect. 5.6.2. Adding the assumption (A10) we have Theorem 5.66. Replacing the assumption (A10) by (5.190) and the interior condition (5.189) we have Theorem 5.67. Furthermore if the stochastic processes \(\left (\mu _{t}\right )_{t\geq 0}\), \(\left (\nu _{t}\right )_{t\varepsilon 0}\), \(\left (\ell_{t}\right )_{t\geq 0}\) are constants and some boundedness assumptions (5.203-iii, v) and (5.204) are satisfied then we can renounce assumption (A10), and obtain the existence and uniqueness of a solution \(\left (Y,Z\right )\) for (5.206): see Theorem 5.69.
The aim of this subsection is to obtain existence and uniqueness under the assumptions (A1), …, (A9) and (5.190), i.e. to see what happens in Theorem 5.67 without the interior condition (5.189). It is not clear how we can obtain some estimates on the subdifferential term \(\mathit{dK}_{s}^{n} = U_{s}^{n}dQ_{s} \in \partial _{y}\Psi \left (s,Y _{s}^{n}\right )dQ_{s}\), except for the particular case treated in Theorem 5.69. For this reason we shall give a weak variational formulation for the solution as in [47]. The stochastic variational formulation for forward SDEs was introduced by Răşcanu in [62].
Let us define the space \(\mathcal{L}_{m}^{p}\), p ≥ 0, of continuous semimartingales M of the form
where \(\gamma \in \mathbb{R}^{m}\), \(\Lambda \) and \(\Theta \) are two p.m.s.p. such that on every interval \(\left [0,T\right ] \subset \mathbb{R}_{+}\), \(\Lambda \in L^{p}\left (\Omega;L^{1}\left (0,T; \mathbb{R}^{m}\right )\right )\), \(\Theta \in L^{p}\left (\Omega;L^{2}\left (0,T; \mathbb{R}^{m\times k}\right )\right )\).
For an intuitive introduction, let \(M \in \mathcal{L}_{m}^{0}\) and \(\left (Y,Z,K\right )\) be a solution of (5.157), in the sense of Definition 5.64. By Itô’s formula for \(\dfrac{1} {2}\left \vert M_{t} - Y _{t}\right \vert ^{2}\) and the subdifferential inequality
we obtain the inequality
Therefore, we propose the following weak formulation for the solution.
Definition 5.71.
We call \(\left (Y _{t},Z_{t}\right )_{t\geq 0}\) a weak variational solution of (5.206) if \(\left (Y,Z\right ) \in S_{m}^{0} \times \Lambda _{m\times k}^{0}\), \(\left (Y _{t},Z_{t}\right ) = \left (\xi _{t},\zeta _{t}\right ) = \left (\eta,0\right )\) for t > τ and
Theorem 5.72.
Let the assumptions (A 1 , …,A 9 ) and ( 5.190 -(i ′ ) and (ii, with q = 2)) be satisfied. Then the BSVI (5.206) has a unique weak variational solution \(\left (Y,Z\right ) \in S_{m}^{0} \times \Lambda _{m\times k}^{0}\) in the sense of Definition 5.71 such that
Moreover the inequalities (5.162) and (5.163) hold.
Proof.
Existence We remark that we are in the conditions of Theorem 5.67 without the interior condition (5.189). Therefore we start with the same approximating equation as in the proof of Theorem 5.67
and we follow exactly the same Steps 1–3 as there.
We obtain the existence of \((Y,Z) \in S_{m}^{0} \times \Lambda _{m\times k}^{0}\) such that
\((Y _{t},Z_{t}) = \left (\eta,0\right )\) for t > τ and (Y, Z) satisfies (5.208), the inequalities (5.162) and (5.163), and (5.207-i, iii).
Let \(M_{\cdot } =\gamma -\int _{0}^{\cdot }\Lambda _{r}dQ_{r} + \int _{0}^{\cdot }\Theta _{r}\mathit{dB}_{r} \in \mathcal{L}_{m}^{0}\). By Itô’s formula for \(\frac{1} {2}\left \vert M_{t} - Y _{t}^{n}\right \vert ^{2}\) we deduce that, for all 0 ≤ t ≤ s,
Passing to the \(\lim \inf\) it follows that the pair \(\left (Y,Z\right )\) satisfies the inequality (5.207-ii).
Uniqueness. In order to prove the uniqueness of the solution, let \(\left (\hat{Y },\hat{Z}\right ) \in S_{m}^{0} \times \Lambda _{m\times k}^{0}\) and \(\left (\tilde{Y },\tilde{Z}\right ) \in S_{m}^{0} \times \Lambda _{m\times k}^{0}\) be two weak variational solutions of (5.206) corresponding to \(\hat{\eta }\) and \(\tilde{\eta }\), respectively. Therefore for all \(M_{\cdot } =\gamma -\int _{0}^{\cdot }\Lambda _{r}dQ_{r} + \int _{0}^{\cdot }\Theta _{r}\mathit{dB}_{r} \in \mathcal{L}_{m}^{0}\),
Let \(Y = \frac{\hat{Y }+\tilde{Y }} {2}\), \(Z = \frac{\hat{Z}+\tilde{Z}} {2}\) and \(h_{r} = \dfrac{1} {2}\left [H(r,\hat{Y }_{r},\hat{Z}_{r}) + H(r,\tilde{Y }_{r},\tilde{Z}_{r})\right ]\).
From the convexity of \(\varphi\) we see that
and using the identity
we obtain
and
Therefore, since
we have for all \(M_{\cdot } =\gamma -\int _{0}^{\cdot }\Lambda _{r}dQ_{r} + \int _{0}^{\cdot }\Theta _{r}\mathit{dB}_{r} \in \mathcal{L}_{m}^{0}\)
where
Let
Clearly, \(M^{\varepsilon } \in \mathcal{L}_{m}^{0}\) since \(M_{t}^{\varepsilon } = M_{0}^{\varepsilon } + \int _{0}^{t}dM_{r}^{\varepsilon } = Y _{0} + \int _{0}^{t}\frac{Y _{r}-M_{r}^{\varepsilon }} {Q_{\varepsilon }} dQ_{r} + \int _{0}^{t}0\mathit{dB}_{ r}\). By Lemma 6.21 it follows that for all 0 ≤ t ≤ s ≤ T
and consequently
because \(\Psi \left (r,M_{r}^{\varepsilon }\right )dQ_{r} = \mathbf{1}_{\left [0,\tau \right ]}\left (r\right )\left [\varphi \left (M_{r}^{\varepsilon }\right )\mathit{dr} +\psi \left (M_{r}^{\varepsilon }\right )\mathit{dA}_{r}\right ]\).
Using the inequality
from (5.210) with \(M = M^{\varepsilon }\), \(\varepsilon \rightarrow 0_{+}\), we obtain that for all 0 ≤ t ≤ s,
which yields, by Proposition 6.69
Taking the expectation and then passing to the limit as s → ∞ uniqueness follows (see the properties of the solutions given in (5.208)).
5.7 Semilinear Elliptic PDEs
5.7.1 Elliptic Equations in the Whole Space
We will first consider elliptic PDEs in \(\mathbb{R}^{d}\), and then in a bounded open subset of \(\mathbb{R}^{d}\), with Dirichlet boundary condition.
Let {X t x; t ≥ 0} denote the solution of the forward SDE:
where \(f\,:\, \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}\) is continuous and globally monotone, \(g\,:\, \mathbb{R}^{d} \rightarrow \mathbb{R}^{d\times d}\) is globally Lipschitz, and consider the backward SDE
where \(F\,:\, \mathbb{R}^{d} \times \mathbb{R}^{k} \times \mathbb{R}^{k\times d} \rightarrow \mathbb{R}^{k}\) is continuous and such that for some K, K ′, μ < 0, p > 0,
We assume moreover that for some λ > 2μ + K 2, and all \(x \in \mathbb{R}^{d}\),
which essentially implies that λ < 0.
Under these assumptions, the BSDE (5.213) has a unique solution, in the sense of Theorem 5.27.
It is not hard to see, using uniqueness for BSDEs, that
Denote by
the infinitesimal generator of the Markov process {X t x ; t ≥ 0}, and consider the following system of semilinear elliptic PDEs in \(\mathbb{R}^{d}\)
As in Sect. 5.4, one easily establishes the following:
Theorem 5.73.
Let \(u \in C^{2}(\mathbb{R}^{d}; \mathbb{R}^{m})\) be a classical solution of (5.216) such that for some M,q > 0,
Then for each \(x \in \mathbb{R}^{d}\) , {(u(X t x ),(∇ug)(X t x )); t ≥ 0} is the solution of the BSDE (5.213) . In particular u(x) = Y 0 x .
We now want to prove that (5.212)–(5.213) provide a viscosity solution to (5.216)
Again, for the notion of a viscosity solution of the system of PDEs we need (5.216) to make sense, therefore we need to make the following restriction: for 0 ≤ i ≤ k,the i -th coordinate of F depends only on the i -th row of the matrix z.
Define the mapping
by
Then the system (5.216) reads
All the assumptions from Theorem 5.37 are assumed to hold below (with of course f, g and F independent of the time variable t). The notion of a viscosity solution of (5.216) is defined by Definition 6.94 in Annex D.
We can now prove the following:
Theorem 5.74.
Under the above assumptions, \(u(x)\mathop{ =}\limits^{ \mathit{def }}Y _{0}^{x}\) is a continuous function which satisfies
for any λ > 2μ + K 2 , and it is a viscosity solution of (5.216) .
Proof.
The continuity follows from the mean-square continuity of \(\{Y _{\cdot }^{x},\,x \in \mathbb{R}^{d}\}\). The inequality (5.217) follows from (5.134) with η = 0 (hence ξ = 0 and ζ = 0).
To prove that u is a viscosity sub-solution, we take any 1 ≤ i ≤ m, \(\varphi \in C^{2}(\mathbb{R}^{d})\) and \(x \in \mathbb{R}^{d}\) such that \(u_{i}-\varphi\) has a local maximum at x. We assume without loss of generality that
We suppose that
and we will find a contradiction.
Let α > 0 be such that whenever | y − x | ≤ α,
and define, for some T > 0,
Let now
\((\overline{Y },\overline{Z})\) solves the one-dimensional BSDE
On the other hand, from Itô’s formula,
solves the BSDE
From \(u_{i} \leq \varphi\), and the choice of α and τ, we deduce with the help of Proposition 5.34 that \(\overline{Y }_{0} <\hat{ Y }_{0}\), i.e. \(u_{i}(x) <\varphi (x)\), which is a contradiction. ■
5.7.2 Elliptic Dirichlet Problem
We now give a similar result for a system of elliptic PDEs in an open bounded subset of \(\mathbb{R}^{d}\), with Dirichlet boundary condition, following [20]. Let \(D \subset \mathbb{R}^{d}\) be a bounded domain (i.e. D is an open bounded subset of \(\mathbb{R}^{d}\)), whose boundary ∂ D is of class C 1. We are given a function \(\chi \in C(\mathbb{R}^{d})\) and we consider the system of elliptic PDEs
The process {X t x; t ≥ 0} is defined as in the preceding subsection. For each \(x \in \overline{D}\), we define the stopping time
Let {(Y t x, Z t x); 0 ≤ t ≤ τ x } be the solution, in the sense of Corollary 5.59, of the BSDE
Using Itô’s formula, it is not hard to establish the following:
Theorem 5.75.
Let \(u \in C^{2}(D; \mathbb{R}^{m}) \cap C^{0}(\overline{D}; \mathbb{R}^{m})\) be a classical solution of (5.218) . Then for each \(x \in \mathbb{R}^{d}\) , {(u(X t x ),(∇ug)(X t x )); t ≥ 0} is the solution of the BSDE (5.219) . In particular u(x) = Y 0 x .
We now assume that P(τ x < ∞) = 1, for all \(x \in \overline{D}\), that the set
and that for some λ > 2μ + K 2, and all \(x \in \overline{D}\),
We again define u(x) = Y 0 x. Besides some arguments which we have already used, the continuity of u also relies on the following:
Proposition 5.76.
Under the condition (5.220) , the mapping x → τ x is a.s. continuous on \(\overline{D}\) .
Proof.
Let \(\{x_{n},\,n \in \mathbb{N}\}\) be a sequence in \(\overline{D}\) such that x n → x, as n → ∞. We first show that
Suppose that (5.221) is false. Then
For each \(\varepsilon > 0\), let
From (5.222), there exists \(\varepsilon\) and T such that
But since \(X_{\cdot }^{x_{n}} \rightarrow X_{\cdot }^{x}\) uniformly on [0, T] a.s., this implies that
which would mean that for some n, \(X^{x_{n}}\) exits the \(\varepsilon /2\)-neighbourhood of D before exiting D, which is impossible.
We next prove that
For this part of the proof, we will need the assumption (5.220) that \(\Lambda \) is closed.
It suffices to prove that (5.223) holds a.s. on \(\Omega _{M} =\{\tau _{x} \leq M\}\), with M arbitrary. From the result of the first step, for almost all \(\omega \in \Omega _{M}\), there exists an n(ω) such that n ≥ n(ω) implies \(\tau _{x_{n}} \leq M + 1\). From the a.s. (on \(\Omega _{M}\)) uniform convergence of \(X_{\cdot }^{x_{n}} \rightarrow X_{\cdot }^{x}\) on the interval [0, M + 1], X ⋅ x hits the set
on the random interval \([0,\liminf _{n}\tau _{x_{n}}]\) a.s. on \(\Omega _{M}\). The result follows, since X ⋅ x exits \(\overline{D}\) when it hits \(\Lambda \). ■
We now prove the following:
Theorem 5.77.
Under the assumptions of Theorem 5.74 , the above conditions on D and the condition (5.220), \(u(x)\mathop{ =}\limits^{ \mathit{def }}Y _{0}^{x}\) is continuous on \(\overline{D}\) and it is a viscosity solution of the system of Eq. (5.218) .
Proof.
We only prove that u is a sub-solution. Let 1 ≤ i ≤ m, \(\varphi \in C^{2}(\mathbb{R}^{d})\ u_{i}-\varphi\) have a local maximum at \(x \in \overline{D}\), such that \(u_{i}(x) =\varphi (x)\). If \(x \in \Lambda \), then τ x = 0, and hence u(x) = χ(x). If however \(x \in D \cup \left (\partial D\setminus \Lambda \right )\), the result follows by the same argument as in the proof of Theorem 5.74.
5.7.3 Elliptic Equations with Neumann Boundary Conditions
The data and assumptions are the same as in Sect. 5.4.3, except that we suppress the dependence of all coefficients upon the time variable. Moreover we also assume that all assumptions of Section 5.4.1 are satisfied.
Consider the following system of semilinear elliptic PDEs with nonlinear Neumann boundary condition
Let X x be the process solution of the reflected stochastic differential equation, for all \(t \geq 0,\ \mathbb{P}\text{ a.s.}\)
To each \(x \in \overline{D}\) we associate the BSDE
Itô’s formula again allows us to establish the following:
Theorem 5.78.
Let \(u \in C^{2}(D; \mathbb{R}^{m}) \cap C^{1}(\overline{D}; \mathbb{R}^{m})\) be a classical solution of (5.224) . Then for each \(x \in \mathbb{R}^{d}\) , {(u(X t x ),(∇ug)(X t x )); t ≥ 0} is the solution of the BSDE (5.226) . In particular u(x) = Y 0 x .
We now have:
Theorem 5.79.
Under the above conditions and those of Theorem 5.43 , u(x):= Y 0 x is a continuous function of x, and it is a viscosity solution of (5.224) .
The proof of this Theorem is easily done by combining the arguments in the proofs of Theorems 5.74 and 5.43.
5.8 Parabolic Variational Inequality
The aim of this section is to prove the existence of a viscosity solution for the following parabolic variational inequality (PVI) with a mixed nonlinear multivalued Neumann–Dirichlet boundary condition:
where the operator \(\mathcal{A}_{t}\) is given by
and D is an open connected bounded subset of \(\mathbb{R}^{d}\) of the form
where \(\phi \in C_{b}^{3}\left (\mathbb{R}^{d}\right )\), \(\left \vert \nabla \phi \left (x\right )\right \vert = 1,\;\) for all \(x \in \mathrm{Bd}\left (D\right )\). The outward normal derivative of v at the point \(x \in \mathrm{Bd}\left (D\right )\) is given by
The functions \(f: \mathbb{R}_{+} \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}\), \(g: \mathbb{R}_{+} \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d\times d}\), \(F: \mathbb{R}_{+} \times \overline{D} \times \mathbb{R} \times \mathbb{R}^{d} \rightarrow \mathbb{R}\), \(G: \mathbb{R}_{+} \times \mathrm{Bd}\left (D\right ) \times \mathbb{R} \rightarrow \mathbb{R}\) and \(\kappa: \overline{D} \rightarrow \mathbb{R}\) are continuous. Assume that for all T > 0, there exist a, L > 0 (which can depend on T) such that \(\forall t \in \left [0,T\right ],\;\forall x,\tilde{x} \in \mathbb{R}^{d}\):
and \(\ \forall t \in \left [0,T\right ]\), \(\forall x \in \overline{D}\), \(x^{{\prime}}\in \mathrm{Bd}\left (D\right )\), \(y,\tilde{y} \in \mathbb{R},z,\tilde{z} \in \mathbb{R}^{d}\):
We also assume that
and the compatibility conditions: for all \(\varepsilon > 0\), t ≥ 0, \(x \in \mathrm{Bd}\left (D\right )\), \(\tilde{x} \in \overline{D}\), \(y \in \mathbb{R}\) and \(z \in \mathbb{R}^{d}\)
where \(a^{+} =\max \left \{0,a\right \}\) and \(\nabla \varphi _{\varepsilon }\left (y\right )\), \(\nabla \psi _{\varepsilon }\left (y\right )\) are the unique solutions U and V, respectively, of equations
(for the Moreau–Yosida approximations \(\nabla \varphi _{\varepsilon }\), \(\nabla \psi _{\varepsilon }\) see section “Convex function” from Annex B and for the compatibility conditions see Example 5.63). We mention that in the one dimensional case (which is our case here)
Since \(\overline{D}\) is bounded and κ is continuous it follows from (5.230-iii) that there exists an M 0 > 0 such that
Let \(\left (t,x\right ) \in \left [0,T\right ] \times \overline{D}\) be arbitrarily fixed. Consider the stochastic basis \(\left (\Omega,\mathcal{F}, \mathbb{P},\left (\mathcal{F}_{s}^{t}\right )_{s\geq 0}\right )\), where the filtration is generated by a d-dimensional Brownian motion as follows: \(\mathcal{F}_{s}^{t} = \mathcal{N}\) if 0 ≤ s ≤ t and
From Theorem 4.54 and Theorem 4.47 we infer that there exists a unique pair \(\left (X^{t,x},A^{t,x}\right ): \Omega \times \left [0,\infty \right [ \rightarrow \mathbb{R}^{d} \times \mathbb{R}^{d}\) of continuous progressively measurable stochastic processes such that, \(\mathbb{P}\text{-a.s.}\):
Then by Proposition 4.55 and Corollary 4.56 we have for all p ≥ 1, λ > 0 and s ≥ t,
and for every T > 0, p ≥ 1 and continuous functions \(h_{1},h_{2}: \left [0,T\right ] \times \overline{D} \rightarrow \mathbb{R}\), the mappings
and
are continuous.
We consider the backward stochastic variational inequality (BSVI):
Note that the backward stochastic variational inequality (5.234) satisfies the assumptions of Theorem 5.69 and Remark 5.70 with τ = T, \(\eta =\kappa \left (X_{T}^{t,x}\right )\) satisfying \(\text{(A}_{7}^{{\prime\prime}}\text{)}\), \(\mu _{s} = \frac{L} {2} \mathbf{1}_{\left [0,T\right ]}\left (s\right )\), \(\ell_{s} = \sqrt{\frac{L} {2}} \mathbf{1}_{\left [0,T\right ]}\left (s\right )\), \(\nu _{s} = L\mathbf{1}_{\left [0,T\right ]}\left (s\right )\), \(V _{s} = LQ_{s\wedge T}^{t,x}\), u 0 = 0, where
Therefore (5.234) has a unique solution \(\left (Y ^{t,x},Z^{t,x},K^{t,x}\right )\) of continuous progressively measurable stochastic processes such that
and \(\mathit{dK}_{s}^{t,x} = U_{s}^{t,x}\mathit{ds} + V _{s}^{t,x}\mathit{dA}_{s}^{t,x}\), where U t, x, V t, x are progressively measurable stochastic processes and \(U_{s}^{t,x} \in \partial \varphi \left (Y _{s}^{t,x}\right )\), \(V _{s}^{t,x} \in \partial \varphi \left (Y _{s}^{t,x}\right )\ d\mathbb{P}\otimes dt - a.e\). on \(\Omega \times \left [t,T\right ]\).
Moreover by (5.162) and (5.164) the solution satisfies for all \(s \in \left [t,T\right ]\):
and
We define
which is a deterministic quantity since Y t t, x is \(\mathcal{F}_{t}^{t} \equiv \mathcal{N}\)-measurable.
From the Markov property, we have
By (5.236) we infer that
In the sequel we shall prove that u defined by (5.237) is a viscosity solution of (5.227). Reversing the time by \(\tilde{u}\left (t,x\right ) = u\left (T - t,x\right )\), the PVI (5.227) becomes (6.137) and the uniqueness of the viscosity solution follows from Theorem 6.112.
We now give the definition of the viscosity solution of the PVI (5.227).
A triple \((p,q,X) \in \mathbb{R} \times \mathbb{R}^{d} \times \mathbb{S}^{d}\) is a parabolic super-jet to u at \(\left (t,x\right ) \in \left (0,T\right ) \times \overline{D}\) if for all \(\left (s,y\right ) \in \left (0,T\right ) \times \overline{D}\)
the set of parabolic super-jets is denoted \(\mathcal{P}^{2,+}u(t,x)\). The set of parabolic sub-jets is defined by \(\mathcal{P}_{\mathcal{O}}^{2,-}u = -\mathcal{P}_{\mathcal{O}}^{2,+}(-u)\).
Let
We clearly have
Definition 5.80.
Let \(u: \left [0,T\right ] \times \overline{D} \rightarrow \mathbb{R}\) be a continuous function, which satisfies \(u(T,x) =\kappa \left (x\right ),\;\forall \ x \in \overline{D}\).
-
(a)
u is a viscosity sub-solution of (5.227) if:
$$\displaystyle{\left \vert \ \begin{array}{l} u(t,x) \in \mathrm{Dom}\left (\varphi \right ),\ \ \forall (t,x) \in (0,T) \times \overline{D},\\ u(t, x) \in \mathrm{Dom } \left (\psi \right ),\ \ \ \forall (t, x) \in (0, T) \times \mathrm{Bd}\left (D\right ), \end{array} \right.}$$and for any \(\left (t,x\right ) \in (0,T) \times \overline{D}\) and any \((p,q,X) \in \mathcal{P}^{2,+}u(t,x)\):
$$\displaystyle{ \left \{\begin{array}{ll} \left (d_{1}\right )&\quad p + \Phi \left (t,x,u(t,x),q,X\right ) +\varphi _{ -}^{{\prime}}\left (u(t,x)\right ) \leq 0\ \;\mbox{ if}\;x \in D, \\ \left (d_{2}\right )&\quad \min \Big\{p + \Phi \left (t,x,u(t,x),q,X\right ) +\varphi _{ -}^{{\prime}}\left (u(t,x)\right ), \\ &\quad \quad \Gamma (t,x,u(t,x),q) +\psi _{ -}^{{\prime}}\left (u(t,x)\right )\Big\} \leq 0\;\ \mbox{ if}\;x \in \mathrm{Bd}\left (D\right ).\end{array} \right. }$$(5.240) -
(b)
The viscosity super-solution of (5.227) is defined in a similar manner as above, with \(\mathcal{P}^{2,+}\) replaced by \(\mathcal{P}^{2,-}\), the left derivative replaced by the right derivative, min by max, and the inequalities ≤ by ≥ .
-
(c)
A continuous function \(u: \left [0,\infty \right ) \times \overline{D}\) is a viscosity solution of (6.137) if it is both a viscosity sub- and super-solution.
Theorem 5.81.
Let the assumptions (5.228), (5.229), (5.230) and (5.231) be satisfied. If u defined by (5.237) is continuous on \(\left [0,T\right ] \times \overline{D}\) , then u is a viscosity solution of PVI (5.227) .
Proof.
We show only that u is a viscosity sub-solution of (5.227) (the proof of the super-solution property is similar).
Let \(\left (t,x\right ) \in \left [0,T\right ] \times \overline{D}\) and \((p,q,X) \in \mathcal{P}^{2,+}u(t,x)\).
Cases.
\(\quad \left (t,x\right ) \in \left [0,T\right ] \times \mathrm{Bd}\left (D\right )\).
Aiming to deduce a contradiction we suppose that
It follows by continuity of F, G, u, f, g, ϕ, \(\Phi \), \(\Gamma \) left continuity and nondecreasing monotonicity of \(\varphi _{-}^{{\prime}}\) and ψ − ′ that there exists \(\varepsilon > 0\), δ > 0 such that for all \(\left (s,x^{{\prime}}\right ) \in \left [0,T\right ] \times \overline{D}\), \(\left \vert s - t\right \vert \leq \delta\), \(\left \vert x^{{\prime}}- x\right \vert \leq \delta\),
and
Now since \((p,q,X) \in \mathcal{P}^{2,+}u(t,x)\) there exists 0 < δ ′ ≤ δ such that for all \(s \in \left [0,T\right ],\ s\neq t,\;x^{{\prime}}\in \overline{D}\), x ′ ≠ x, \(\left \vert s - t\right \vert \leq \delta ^{{\prime}}\), \(\left \vert x^{{\prime}}- x\right \vert \leq \delta ^{{\prime}}\) we have
By (5.239) the condition (5.241) becomes
The condition (5.242) can be written as follows
Let
We note that (Y s t, x, Z s t, x), t ≤ s ≤ θ, solves the BSDE
Moreover, it follows from Itô’s formula that
satisfies
Let \((\tilde{Y }_{s}^{t,x},\tilde{Z}_{s}^{t,x}) = (\hat{Y }_{s}^{t,x} - Y _{s}^{t,x},\hat{Z}_{s}^{t,x} - Z_{s}^{t,x})\). We have
Let
Since \(\vert \hat{\beta }_{s} -\beta _{s}\vert \leq \sqrt{\frac{L} {2}} \ \vert \hat{Z}_{s}^{t,x} - Z_{ s}^{t,x}\vert \), there exists a bounded d-dimensional p.m.s.p. \(\left \{\zeta _{s};t \leq s \leq \theta \right \}\) such that \(\hat{\beta }_{s} -\beta _{s} =\langle \zeta _{s},\tilde{Z}_{s}^{t,x}\rangle\) and therefore
Let
Then by Itô’s formula,
and so
Then
Since \(U_{r}^{t,x} \in \partial \varphi \left (Y _{r}^{t,x}\right )\) and \(V _{r}^{t,x} \in \partial \psi \left (Y _{r}^{t,x}\right )\), we have
and therefore by (5.243) and (5.244)
and
Moreover, the choice of δ ′ and θ implies that \(u(\theta,X_{\theta }^{t,x}) <\hat{ u}(\theta,X_{\theta }^{t,x})\). Hence
which is a contradiction. It follows that (5.240-d 2) holds.
Cases.
\(\left (t,x\right ) \in \left [0,T\right ] \times D\).
The proof follows the same steps from Case 5.8, where we now choose δ and δ ′ such that \(\overline{B\left (x,\delta ^{{\prime}}\right )} \subset \overline{B\left (x,\delta \right )} \subset D\) and, by condition (5.232-iv), A r t, x = 0 for all t ≤ r ≤ θ.
This proves that u is a viscosity sub-solution of PVI (5.227). Symmetric arguments show that u is also a super-solution; hence u is a viscosity solution of PVI (5.227).
Corollary 5.82.
We have
Proof.
Let (t, x) ∈ [0, T] × D be fixed. We have two cases:
-
(1)
\(\mathrm{Dom}\left (\partial \varphi \right ) = \mathrm{Dom}\left (\varphi \right )\), and so, from (5.238), \(u(t,x) \in \mathrm{Dom}\left (\partial \varphi \right )\).
-
(2)
\(\mathrm{Dom}\left (\partial \varphi \right )\neq \mathrm{Dom}\left (\varphi \right )\). Let \(b \in \mathrm{Dom}\,\varphi \setminus \mathrm{Dom}\,(\partial \varphi )\). Then \(b =\sup (\mathrm{Dom}\,\varphi )\) or \(b =\inf \mathrm{Dom}\,\varphi\). If \(b =\sup (\mathrm{Dom}\,\varphi )\) and u(t, x) = b, then \((0,0,0) \in \mathcal{P}^{2,+}u(t,x)\) since
and from (6.143-d 1) it follows that \(\varphi _{-}^{{\prime}}(b) =\varphi _{ -}^{{\prime}}\big(u(t,x)\big) < \infty \) and consequently \(b \in \mathrm{Dom}\,(\partial \varphi )\); a contradiction which shows that u(t, x) < b. Similarly for \(b =\inf (\mathrm{Dom}\,\varphi )\). ■
The problem now is to see when \(\left (t,x\right )\longmapsto u(t,x) = Y _{t}^{tx}: [0,T] \times \overline{D} \rightarrow \mathbb{R}\) is continuous. A recent result of Maticiuc and Răşcanu [46] gives a sufficient condition for u to be continuous. The idea is to show that if \(\left (t_{n},x_{n}\right ) \rightarrow \left (t,x\right )\) then \((Y _{\cdot }^{t_{n},x_{n}})_{n\in \mathbb{N}^{{\ast} }}\) is tight in the Skorohod space \(\mathbb{D}\left (\left [0,T\right ], \mathbb{R}\right )\) of càdlàg functions endowed with the S-topology (introduced by Jakubowski in [41]). This topology makes the mapping \(y\longmapsto \int _{0}^{s}G\left (r,y\left (r\right )\right )\mathit{dA}_{r}\) continuous from \(\mathbb{D}\left (\left [0,T\right ], \mathbb{R}\right )\) into \(\mathbb{R}\). The result is the following:
Proposition 5.83.
Let the assumptions (5.228) , …, (5.231) be satisfied. If moreover there exists an L 0 > 0 such that
then the function
is continuous.
Finally let f, g, F, G be independent of t and (X s x, A s x, Y s x; t, Z s x; t, U s x; t, V s x; t)0 ≤ s ≤ t be defined by
and
Summarizing Theorem 5.81 and Theorem 6.112 we have:
Theorem 5.84.
Let the assumptions (5.228) ,…, (5.231) be satisfied. Assume there exists a continuous function \(\mathbf{m}: [0,\infty ) \rightarrow [0,\infty )\) , \(\mathbf{m}\left (0\right ) = 0\) , such that
If \(\left (t,x\right )\longmapsto u(t,x)\mathop{ =}\limits^{ \mathit{def }}Y _{0}^{x;t}: [0,T] \times \overline{D} \rightarrow \mathbb{R}\) is continuous (this is true in particular under the assumptions of Proposition 5.83), then u is the unique viscosity solution of the parabolic variational inequality
where the operator \(\mathcal{A}\) is given by
5.9 Invariant Sets of BSDEs
Let \(\left \{B_{t}: t \geq 0\right \}\) be a k-dimensional standard Brownian motion defined on some complete probability space \((\Omega,\mathcal{F}, \mathbb{P})\). We denote by \(\left \{\mathcal{F}_{t}: t \geq 0\right \}\) the natural filtration generated by \(\left \{B_{t},t \geq 0\right \}\) and augmented by the \(\mathbb{P}\)-null sets of \(\mathcal{F}\).
Let \(x \in \mathbb{R}^{d}\), \(0 \leq t \leq \tilde{ T} \leq T\). Consider the SDE
and the BSDE
The aim of this section is to state necessary and sufficient conditions which guarantee that the solution of the BSDE (5.249) does not leave a given set
i.e., under which we have that for all \(0 \leq t \leq \tilde{ T} \leq T\), \(x \in \mathbb{R}^{d}\) and \(\kappa (X_{\tilde{T}}^{t,x}) \in E(\tilde{T},X_{\tilde{T}}^{t,x})\;\ a.s.\;\omega \in \Omega \):
As a by-product, we will derive a result on the existence of constrained viscosity solutions to some PDEs. Together with the Eqs. (5.248) and (5.249), we consider the following system of semilinear parabolic PDEs
with the second order differential operator
where \(b: [0,T] \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d},\;\sigma: [0,T] \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d\times k}\) and \(\;f_{i}: [0,T] \times \mathbb{R}^{d} \times \mathbb{R}^{m} \times \mathbb{R}^{k} \rightarrow \mathbb{R},\,1 \leq i \leq n\).
We make the following standard assumptions:
-
\(\left (AV _{1}\right )\) We assume that the functions \(b: [0,T] \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}\), \(\sigma: [0,T] \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d\times k}\), \(F: [0,T] \times \mathbb{R}^{d} \times \mathbb{R}^{m} \times \mathbb{R}^{m\times k} \rightarrow \mathbb{R}^{m}\) and \(f: [0,T] \times \mathbb{R}^{d} \times \mathbb{R}^{m} \times \mathbb{R}^{k} \rightarrow \mathbb{R}^{m}\) are continuous and such that, for some constants L > 0 and q ≥ 2,
$$\displaystyle{\left \vert b\left (t,x\right ) - b\left (t,\tilde{x}\right )\right \vert + \left \Vert \sigma \left (t,x\right ) -\sigma \left (t,\tilde{x}\right )\right \Vert \leq L\left \vert x -\tilde{ x}\right \vert,}$$$$\displaystyle{\begin{array}{rll} i)&&\left \vert F\left (t,x,y,z\right )\right \vert \leq L\left (1 + \left \vert x\right \vert ^{q} + \left \vert y\right \vert + \left \Vert z\right \Vert \right ), \\ \mathit{ii})&&\left \vert F\left (t,x,y,z\right ) - F\left (t,x,y,\tilde{z}\right )\right \vert \leq L\left \Vert z -\tilde{ z}\right \Vert, \\ \mathit{iii})&&\left \langle F\left (t,x,y,z\right ) - F\left (t,x,\tilde{y},z\right ),y -\tilde{ y}\right \rangle \leq L\left \vert y -\tilde{ y}\right \vert ^{2}\end{array} }$$and
$$\displaystyle{\begin{array}{rll} j)&&\left \vert f\left (t,x,y,u\right )\right \vert \leq L\left (1 + \left \vert x\right \vert ^{q} + \left \vert y\right \vert + \left \vert u\right \vert \right ), \\ \mathit{jj})&&\left \vert f\left (t,x,y,u\right ) - f\left (t,x,y,\tilde{u}\right )\right \vert \leq L\left \vert u -\tilde{ u}\right \vert, \\ \mathit{jjj})&&\left \langle f\left (t,x,y,u\right ) - f\left (t,x,\tilde{y},u\right ),y -\tilde{ y}\right \rangle \leq L\left \vert y -\tilde{ y}\right \vert ^{2},\end{array} }$$for all t ∈ [0, T], \(x,\tilde{x} \in \mathbb{R}^{d}\), \(\,y,\tilde{y} \in \mathbb{R}^{m}\), and \(z,\tilde{z} \in \mathbb{R}^{m\times k}\), \(u,\tilde{u} \in \mathbb{R}^{k}\).
-
\(\left (AV _{2}\right )\) We assume that \(\kappa: \mathbb{R}^{d} \rightarrow \mathbb{R}^{m}\) is a Borel measurable function of at most polynomial growth, i.e., there are some a > 0, q ≥ 1 such that, for all \(x \in \mathbb{R}^{d}\),
$$\displaystyle{\vert \kappa (x)\vert \leq a(1 + \vert x\vert ^{q}),\ \forall x \in \mathbb{R}^{d}.}$$
We shall now recall some basic properties of forward and backward stochastic differential equations.
Proposition 5.85.
Under the assumptions \(\left (AV _{1}\right )\) and \(\left (AV _{2}\right )\) :
-
I.
Equations (1.1) and (1.2) have unique solutions \(X^{t,x} \in S_{d}^{2}\left [0,T\right ]\) and
$$\displaystyle{\left (Y ^{t,x},Z^{t,x}\right ) \in S_{ m}^{2}\left [0,T\right ] \times \Lambda _{ m\times k}^{2}\left [0,T\right ]}$$with Z s t,x = 0 for \(s \in [0,t] \cup [\tilde{T},T]\) and the solutions satisfy:
-
II.
For all p ≥ 2, there exist some constants C p > 0, \(q \in \mathbb{N}^{{\ast}}\) , which don’t depend on \(t,t^{{\prime}}\in [0,T]\) and \(x,x^{{\prime}}\in \mathbb{R}^{m}\) , such that
$$\displaystyle{ \begin{array}{ll} a)&\quad \mathbb{E}\left (\sup _{s\in [0,T]}\vert X_{s}^{t,x}\vert ^{p}\right ) \leq C_{p}(1 + \vert x\vert ^{p}), \\ b) &\quad \mathbb{E}\left (\sup _{s\in [0,T]}\vert X_{s}^{t,x} - X_{s}^{t^{{\prime}},x^{{\prime}} }\vert ^{p}\right ) \leq \\ &\leq C_{p}(1 + \vert x\vert ^{p} + \vert x^{{\prime}}\vert ^{pq})(\vert t - t^{{\prime}}\vert ^{p/2} + \vert x - x^{{\prime}}\vert ^{p}),\end{array} }$$(5.251)and
$$\displaystyle{\begin{array}{ll} c) &\quad \mathbb{E}\left (\sup _{s\in [0,T]}\vert Y _{s}^{t,x}\vert ^{p}\right ) \leq C_{p}(1 + \vert x\vert ^{pq}), \\ d)&\quad \mathbb{E}\left (\sup _{s\in [0,T]}\vert Y _{s}^{t,x} - Y _{s}^{t^{{\prime}},x^{{\prime}} }\vert ^{2}\right ) \leq C_{2}[\mathbb{E}\vert \kappa (X_{T}^{t,x}) -\kappa (X_{T}^{t^{{\prime}},x^{{\prime}} })\vert ^{2}, \\ &\begin{array}{l} + \mathbb{E}\int _{0}^{T}\vert 1_{[t,T]}(r)F(r,X_{r}^{t,x},Y _{r}^{t,x},Z_{r}^{t,x}) \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - 1_{[t^{{\prime}},T]}(r)F(r,X_{r}^{t^{{\prime}},x^{{\prime}} },Y _{r}^{t,x},Z_{r}^{t,x})\vert ^{2}\mathit{dr}].\end{array}\end{array} }$$ -
III.
There are some Borel measurable functions \(u: [0,T] \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{m}\) , and \(v: [0,T] \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{m\times d}\) such that for all \(0 \leq t \leq s \leq \tilde{ T} \leq T\)
$$\displaystyle{Y _{s}^{t,x} = u\left (s \wedge \tilde{ T},X_{ s\wedge \tilde{T}}^{t,x}\right ),\quad Z_{ s}^{t,x} = \left (v\sigma \right )\left (s \wedge \tilde{ T},X_{ s\wedge \tilde{T}}^{t,x}\right ),\quad d\mathbb{P} \otimes \mathit{ds} - a.e.}$$(see [30]).
For the convenience of the reader we recall the definition of a viscosity solution corresponding to the PDE (5.250).
Definition 5.86.
a) A lower semicontinuous function \(u: [0,T] \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{m}\) is a viscosity super-solution of (5.250), if, firstly, \(u_{i}\left (T,x\right ) \geq \kappa _{i}\left (x\right )\), for all \(x \in \mathbb{R}^{d}\), 1 ≤ i ≤ n, and secondly, for any 1 ≤ i ≤ n, \(\varphi \in C^{1,2}(\left (0,T\right ) \times \mathbb{R}^{d})\) and \((t,x) \in [0,T] \times \mathbb{R}^{d}\) such that \(u_{i}-\varphi\) achieves a local minimum at (t, x), it holds that
b) An upper semicontinuous function \(u: [0,T] \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{m}\) is a viscosity sub-solution of (5.250), if, firstly, \(u_{i}\left (T,x\right ) \leq \kappa _{i}\left (x\right )\), for all \(x \in \mathbb{R}^{d}\), 1 ≤ i ≤ n, and secondly, for any 1 ≤ i ≤ n, \(\varphi \in C^{1,2}(\left (0,T\right ) \times \mathbb{R}^{d})\) and \((t,x) \in [0,T] \times \mathbb{R}^{d}\) such that \(u_{i}-\varphi\) attains a local maximum at (t, x), we have that
c) Finally, a continuous function \(u: [0,T] \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{m}\) is a viscosity solution of (5.250) if it is both a viscosity super-solution and a viscosity sub-solution of this equation.
From Sect. 5.4.1 of this chapter we have:
Proposition 5.87.
We suppose that the function f satisfies hypothesis (AV 1 ) and that \(\kappa: \mathbb{R}^{d} \rightarrow \mathbb{R}^{m}\) is a continuous function satisfying (AV 2 ). Let X t,x and \(\left (Y ^{t,x},Z^{t,x}\right )\) be the solutions to (5.248) and (5.249) , respectively, where the driver F of BSDE (1.2) is of the form
and e i denotes the unit vector pointing in the i-th coordinate direction of \(\mathbb{R}^{m}\) . Then \(u(t,x) = Y _{t}^{t,x}\), \(\left (t,x\right ) \in [0,T] \times \mathbb{R}^{d}\) , is a deterministic continuous function of at most polynomial growth. This function is a viscosity solution to (5.250) . Moreover if, for each \(R > 0\) , there exists a continuous function \(\alpha _{R}: \mathbb{R}_{+} \rightarrow \mathbb{R}\) , α R (0) = 0, such that, for all t,y,z,x,x ′ with |x|≤ R, |x ′ |≤ R,
then u is the unique viscosity solution in the class \(C_{pol}([0,T] \times \mathbb{R}^{d}, \mathbb{R}^{m})\) .
We now give the notion of the viability property for BSDEs and PDEs. We recall some notations. For any closed set \(S \subset \mathbb{R}^{d}\) we denote by \(x \rightarrow d_{S}(x) =\min \{ \vert x - y: y \in S\}\) the distance function to S, and for \(x \in \mathbb{R}^{d}\), we denote by \(\Pi _{S}(x):=\{ z \in S\): d S (x) = | x − z | } the set of projections of x on S.
For all \(t \in [0,T],\,x \in \mathbb{R}^{d}\), let E(t, x) be a non-empty and closed subset of \(\mathbb{R}^{m}\). We consider the following set of moving constraints
Definition 5.88 (Viability for BSDEs).
The moving set E(t, x), \((t,x) \in [0,T] \times \mathbb{R}^{d}\), is viable (invariant) for the BSDE (5.249) (or Eq. (5.249) is said to be \(\mathcal{E}\)-viable on [0, T]) if, for all \((t,x) \in [0,T] \times \mathbb{R}^{d},\ \tilde{T} \in [t,T]\), and all Borel measurable functions \(\kappa: \mathbb{R}^{d} \rightarrow \mathbb{R}^{m}\) of at most polynomial growth, such that \(\kappa (\tilde{x}) \in E(\tilde{T},\tilde{x}),\, \mathbb{P} \circ \left [X_{\tilde{T}}^{t,s}\right ]^{-1}(d\tilde{x})\)-a.s., it holds that the solution of (1.2) satisfies
Viability for PDEs: Equation (5.250) is said to be \(\mathcal{E}\)-viable (\(\mathcal{E}\)-invariant) on [0, T] if, for all \(\tilde{T} \in [0,T]\) and \(\kappa \in C_{pol}(\mathbb{R}^{d}, \mathbb{R}^{m})\) such that \(\kappa (\tilde{x}) \in E(\tilde{T},\tilde{x}),\) for all \(\tilde{x} \in \mathbb{R}^{d}\), it holds that there exists a viscosity solution \(u \in C_{pol}([0,\tilde{T}] \times \mathbb{R}^{d}, \mathbb{R}^{m})\) of (5.250) with time horizon \(\tilde{T}\) and terminal condition \(u(\tilde{T},x) =\kappa (x)\), x ∈ R m, such that
From Proposition 5.87 we see immediately that:
Remark 5.89.
If BSDE (5.249) is \(\mathcal{E}\)-viable then PDE (5.250) is also \(\mathcal{E}\)-viable.
Therefore the next result also concerns constrained the BSDEs and the PDEs.
Theorem 5.90 (Viability Criterion for BSDEs).
Assume that \(\left (AV _{1}\right )\) and \(\left (AV _{2}\right )\) are satisfied and moreover
-
(i)
the function \((t,x)\mapsto d_{E(t,x)}^{2}(y): [0,T] \times \mathbb{R}^{d} \rightarrow \mathbb{R}\) is jointly upper semicontinuous,
-
(ii)
there exist some constants M > 0, p ≥ 1 such that
$$\displaystyle{d_{E(t,x)}^{2}(0) \leq M(1 + \vert x\vert ^{p}),\ \forall (t,x) \in [0,T] \times \mathbb{R}^{d}.}$$Then the following assertions \(\left (c\right )\) and \(\left (\mathit{cc}\right )\) are equivalent:
-
(c)
Equation (5.249) is \(\mathcal{E}\) -viable on [0,T].
-
(cc)
For any sufficiently large C > 0 and for every \(z \in \mathbb{R}^{m\times d}\) , the function h(t,x,y) = d E(t,x) 2 (y) is an upper semicontinuous viscosity sub-solution of the PDE
$$\displaystyle{ \begin{array}{r} \dfrac{\partial V (t,x,y)} {\partial t} + \mathcal{L}_{z}(t)V (t,x,y) + Cd_{E(t,x)}^{2}(y) = 0, \\ \left (t,x,y\right ) \in [0,T] \in \mathbb{R}^{d} \times \mathbb{R}^{m}.\end{array} }$$(5.253)In the above relation, \(\mathcal{L}_{z}(t)\) denotes the following second order differential operator
$$\displaystyle{ \begin{array}{r} \mathcal{L}_{z}(t)\varphi (x,y) = \dfrac{1} {2}\mathrm{Tr}[\Gamma _{z}\Gamma _{z}^{{\ast}}(t,x,y)D_{\left (x,y\right )}^{2}\varphi (x,y)] \\ + \left \langle B_{z}(t,x,y),\nabla _{\left (x,y\right )}\varphi (x,y)\right \rangle,\end{array} }$$(5.254)where
$$\displaystyle{\Gamma _{z}(t,x,y) ={ \sigma (t,x)\abovewithdelims()0.0pt z\sigma (t,x)},\quad B_{z}(t,x,y) ={ b(t,x)\abovewithdelims()0.0pt -F(t,x,y,z\sigma (t,x))}.}$$
This theorem yields:
Corollary 5.91 (Viability Criterion for BSDEs).
We assume that the moving sets of Theorem 5.90 are independent of the spatial variable, E(t,x) ≡ E(t), \(\left (t,x\right ) \in [0,T] \times \mathbb{R}^{m}\) . Then the following assertions \(\left (j\right )\) and \(\left (\,\mathit{jj}\right )\) are equivalent:
-
(j)
Equation (5.249) is \(\mathcal{E}\) -viable on [0,T].
-
(jj)
The function h(t,y) = d E(t) 2 (y) is an upper semicontinuous viscosity sub-solution of the PDE:
$$\displaystyle{\dfrac{\partial V (t,y)} {\partial t} + \mathcal{A}_{z}(t;x)V (t,y) + Cd_{E(t)}^{2}(y) = 0,\quad (t,y) \in [0,T] \times \mathbb{R}^{m},}$$for all \(x \in \mathbb{R}^{d}\), \(z \in \mathbb{R}^{m\times d}\) , where
$$\displaystyle{\mathcal{A}_{z}\left (t;x\right )\psi (y) = \frac{1} {2}\mathrm{Tr}[z\sigma \sigma ^{{\ast}}(t,x)z^{{\ast}}D_{ y}^{2}\psi (y)] -\left \langle F(t,x,y,z\sigma (t,x)),\nabla _{ y}\psi (y)\right \rangle,}$$and C > 0 is any sufficiently large constant.
Before proving the main results stated above, we shall present some clarifying examples. In the first example we find a criterion such that a family of moving balls has the viability property for a given BSDE.
Example 5.92 (Control Security Tube).
We consider an arbitrary function \(r \in C^{1}([0,T]; \mathbb{R}_{+})\) with \(r\left (t\right ) > 0\) for all t ∈ [0, T], and we put
Then the square-distance function is
and, for \(\left \vert y\right \vert > r(t)\), the operator \(\mathcal{A}_{z}(t)\) applied to h 0 at \(\left (t,y\right )\) takes the form
Hence, the inequality in Corollary 5.91(jj) yields that, for all \((t,x,y,z) \in [0,T] \times \mathbb{R}^{d} \times \mathbb{R}^{m} \times \mathbb{R}^{m\times d}\) with \(\left \vert y\right \vert > r(t)\),
from where we easily deduce the following necessary condition for the \(\mathcal{E}\)-viability of BSDE (5.249):
For all (t, x, y, z) with \(\left \vert y\right \vert = r(t)\) and \(\left (z\sigma \left (t,x\right )\right )^{{\ast}}y = 0\),
If the assumption (AV 1-i) is replaced by
for all \(\left (t,x,y,z\right )\), then this condition is not only necessary but also sufficient as the following argument proves. We fix any \((t,x,y,z) \in [0,T] \times \mathbb{R}^{d} \times \mathbb{R}^{m} \times \mathbb{R}^{m\times d}\) with \(\left \vert y\right \vert > r(t)\), and for simplicity of notation we put \(\overline{y} = \left \vert y\right \vert ^{-1}r\left (t\right )y\) and, for 1 ≤ j ≤ m
-
∘ \(u_{j} = \left (z\sigma \left (t,x\right )\right )_{j} =\sum _{ i=1}^{d}z_{\cdot,i}\sigma _{i,j}\left (t,x\right )\),
-
∘ \(\hat{u}_{j} = \left \vert y\right \vert ^{-2}\left \langle u_{j},y\right \rangle y,\quad u_{j}^{\perp } = u_{j} -\hat{ u}_{j}\),
-
∘ \(\hat{u} = \left (\hat{u}_{1},\ldots,\hat{u}_{m}\right ),\quad u^{\perp } = \left (u_{1}^{\perp },\ldots,u_{m}^{\perp }\right ) = u -\hat{ u}\).
From the assumptions (AV 1) and (AV 1-i ′) we get that, for some generic constant C which can change from line to line but does not depend on (t, x, y, z),
Thus, since \(\left \vert y\right \vert r^{{\prime}}(t) \leq r\left (t\right )r^{{\prime}}(t) + C\left (\left \vert y\right \vert - r\left (t\right )\right )\), for all \(\left (t,y\right ) \in [0,T] \times \mathbb{R}^{m}\), we can deduce from (5.255) that
This proves the sufficiency of (5.255).
The next example shows that, in the general case, there is no possibility of null-controllability of BSDEs; although we don’t consider controlled equations, we can interpret the choice of the coefficients as controls.
Example 5.93.
For any given \(\left (t_{0},y_{0}\right ) \in \left ]0,T\right [ \times \mathbb{R}^{m}\), we introduce the family of moving constraints
The associated square-distance function is of the form:
This function is upper semicontinuous in \((t,y) \in [0,T] \times \mathbb{R}^{m}\), and if t = t 0, y ≠ y 0, then, for every \(a \in \mathbb{R}\), there is some \(\varphi _{a} \in C^{1,2}\left ([0,T] \times \mathbb{R}^{m}\right )\) with
such that \(h -\varphi _{a}\) achieves a local maximum at \(\left (t_{0},y\right )\). Since
does not depend on \(a \in \mathbb{R}\), we can choose a > 0 sufficiently large in order to guarantee that the inequality in Corollary 5.91(jj) is not satisfied. This shows that Eq. (5.249) cannot be \(\mathcal{E}\)-viable.
The proof of Theorem 5.90 reduces to that of the following two lemmas, see [15].
Lemma 5.94.
Under our standard assumptions we have the equivalence between the following statements:
-
i)
Equation (5.249) is \(\mathcal{E}\) -viable on [0,T].
-
ii)
There exists a C > 0 such that, for all \(t,\tilde{T}\) with \(0 \leq t \leq \tilde{ T} \leq T\) , and for all \(x \in \mathbb{R}^{d}\) , the solution of BSDE (5.249) with time horizon \(\tilde{T}\) and arbitrary Borel measurable terminal function \(\kappa: \mathbb{R}^{d} \rightarrow \mathbb{R}^{m}\) of at most polynomial growth satisfies:
$$\displaystyle{d_{E(t,x)}^{2}(Y _{ t}^{t,x}) \leq e^{C(\tilde{T}-t)}\mathbb{E}d_{ E(\tilde{T},X_{\tilde{T}}^{t,x})}^{2}(Y _{\tilde{ T}}^{t,x}).}$$
Lemma 5.95.
Let Y t,x be the solution of BSDE (5.249) with time horizon \(\tilde{T}\) and arbitrary terminal function \(\kappa \in C_{pol}\left ([0,\tilde{T}] \times \mathbb{R}^{d}\right )\) .
Let C be a positive constant and \(h: [0,T] \times \mathbb{R}^{d} \times \mathbb{R}^{m} \rightarrow \mathbb{R}\) be an upper semicontinuous function of at most polynomial growth such that, for some positive constants M,p > 0,
for all \(\left (t,x\right ) \in [0,T] \times \mathbb{R}^{d}\) and all \(y,y^{{\prime}}\in \mathbb{R}^{m}\) . Then the following assertions are equivalent:
-
i)
For all \(x \in \mathbb{R}^{d}\) and \(t,\tilde{T}\) with \(0 \leq t \leq \tilde{ T} \leq T\) , it holds that
$$\displaystyle{h(t,x,Y _{t}^{t,x}) \leq e^{C(\tilde{T}-t)}\mathbb{E}h(\tilde{T},X_{\tilde{ T}}^{t,x},Y _{\tilde{ T}}^{t,x}).}$$ -
ii)
For every \(z \in \mathbb{R}^{m\times d}\) , the function h is a viscosity sub-solution of the equation
$$\displaystyle{ \frac{\partial V \left (t,x,y\right )} {\partial t} + \mathcal{L}_{z}(t)V \left (t,x,y\right ) + Ch(t,x,y) = 0\;\text{on }[0,\tilde{T}] \times \mathbb{R}^{d} \times \mathbb{R}^{m}. }$$(5.257)Recall that \(\mathcal{L}_{z}(t)\) is defined in (5.254).
Proof of Lemma 5.94.
We first remark that (ii) obviously implies (i). Thus, it only remains to show that (ii) can be deduced from (i). Let \(\tilde{T} \in [0,T],\left (t,x\right ) \in [0,\tilde{T}] \times \mathbb{R}^{d}\). For simplicity of notation we put \(u\left (t,x\right ) = Y _{t}^{t,x}\), and we select a Borel measurable mapping \(\hat{u}: [t,T] \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{m}\) such that \(\hat{u}(s,x^{{\prime}}) \in \prod _{E(s,x^{{\prime}})}(u(s,x^{{\prime}}))\), for all \(\left (s,x^{{\prime}}\right ) \in [t,T] \times \mathbb{R}^{d}\). Recall that \(\prod _{E(s,x^{{\prime}})}\left (z\right ) = \left \{y \in E(s,x^{{\prime}}): \left \vert z - y\right \vert = d_{E(s,x^{{\prime}})}\left (z\right )\right \}\). Then, since Eq. (1.2) is \(\mathcal{E}\)-viable, the unique square integrable adapted solution \(\left (\tilde{Y }^{t,x},\tilde{Z}^{t,x}\right )\) of the BSDE
is such that \(\tilde{Y }_{s}^{t,x} \in E(s,X_{s}^{t,x})\), t ≤ s ≤ T, \(\mathbb{P}\text{-}a.s.\)
Consequently, \(\mathbb{E}d_{E(s,X_{s}^{t,x})}^{2}(Y _{s}^{t,x}) \leq \mathbb{E}\vert Y _{s}^{t,x} -\tilde{ Y }_{s}^{t,x}\vert ^{2}\), and a standard estimate of \(\mathbb{E}\vert Y _{s}^{tx} -\tilde{ Y }_{s}^{tx}\vert ^{2}\) involving Itô’s formula and Gronwall’s formula, yields the desired result:
\(0 \leq t \leq s \leq \tilde{ T} \leq T\), \(x \in \mathbb{R}^{d}\). This completes the proof of Lemma 5.94.
We now come to the proof of Lemma 5.95.
Proof of Lemma 5.95.
We first show that, under the assumption (i), we have (ii). To this end we fix an arbitrary function \(\varphi: [0,T] \times \mathbb{R}^{d} \times \mathbb{R}^{m} \rightarrow \mathbb{R}\) of class C pol 1, 2, 2 and a point \((t,x,y) \in \left (0,T\right ) \times \mathbb{R}^{d} \times \mathbb{R}^{m}\) such that the mapping \(h-\varphi\) achieves a global maximum at \(\left (t,x,y\right )\). For an arbitrary but fixed \(z \in \mathbb{R}^{m\times d}\) we denote by \((Y ^{\varepsilon },Z^{\varepsilon }) \in S_{m}^{2}\left [t,t+\varepsilon \right ] \times \Lambda _{m\times k}^{2}\left (t,t+\varepsilon \right )\) the unique solution of the BSDE
where
From the assumption made on h in assertion (i), we obtain
Then, with the help of a Taylor expansion of \(\varphi\), we get
where,
Note that
Hence,
and
Moreover, from the assumptions on h,
Therefore, applying the following auxiliary lemma, the proof of which will be given at the end of this section, we can take the limit as \(\varepsilon \rightarrow 0\) in (5.258) and obtain assertion (ii).
Lemma 5.96.
Under the assumptions of Lemma 5.95 , and with the notations introduced above, we have
for all \(\left (t,x,y\right ) \in [0,T] \times \mathbb{R}^{d} \times \mathbb{R}^{m}\) .
We shall now prove the reverse implication: Under the assumption that (ii) holds we have to show the validity of (i). For this we first remark that, for any continuous function \(\Phi: [0,T] \times \mathbb{R}^{d} \times \mathbb{R}^{m} \times \mathbb{R} \times \mathbb{R}^{m+n} \times \mathbb{S}^{m+n} \times \mathbb{R}^{m\times d} \rightarrow \mathbb{R}\) satisfying the standard assumptions of monotonicity with respect to the \(\mathbb{R}^{m+n}\)-variable and of degenerate ellipticity with respect to the \(\mathbb{S}^{m+n}\)-variable (see Annex D),
if and only if
Indeed, in order to see that (β) implies (α), it suffices to choose \(g \in C_{pol}([0,T] \times \mathbb{R}^{d}; \mathbb{R}^{m\times d})\) with \(g(t,x) = z \in \mathbb{R}^{m\times d}\). On the other hand, to get the necessity of (β) under (α), we remark that, for all test functions \(\varphi \in C^{1,2,2}\) for which \(h-\varphi\) achieves a local maximum at \(\left (t,x,y\right )\), and with the notation
we have that \(\Phi (t,x,y,a,p,S;z) \geq 0,\;\) for all \(z \in \mathbb{R}^{m\times d}\), and hence also for z = g(t, x), where g runs over the set of functions belonging to \(C_{pol}([0,T] \times \mathbb{R}^{d}; \mathbb{R}^{m\times d})\). We now fix any \(g \in C_{pol}([0,T] \times \mathbb{R}^{d}; \mathbb{R}^{m\times d})\) and consider the unique square integrable adapted solution \((X,\overline{Y }^{t,x,y})\) of the (forward) SDE
Of course, here the process X t, x is nothing else than the unique solution of SDE (5.248). Moreover, we denote by \(\left (\tilde{Y }_{k,\cdot }^{t,x,y},\tilde{Z}_{k,\cdot }^{t,x,y}\right ) \in S_{m}^{2}\left [t,\tilde{T}\right ] \times \Lambda _{m\times k}^{2}\left (t,\tilde{T}\right )\) the unique solution of the BSDE
where \(\tilde{T} \in [0,T]\) and \(\left (h_{k}\right )_{k\geq 1} \subset C_{pol}([0,T] \times \mathbb{R}^{d} \times \mathbb{R}^{m})\) is a monotonically decreasing sequence of continuous functions with at most polynomial growth, such that its pointwise limit is equal to h. Then the function
is a continuous viscosity solution of the equation
and it is the unique solution in the class of continuous functions of at most polynomial growth. We also note that, by the Markov property,
Since, due to assumption (ii), h is an upper semicontinuous viscosity sub-solution of at most polynomial growth of the above PDE, we know that h must be smaller than or equal to the viscosity solution V k . Thus,
then, by passing to the limit as k → ∞ and applying Gronwall’s inequality, we obtain the following estimate
Setting s = t and y = u(t, x) = Y t t, x and using the assumption (5.256) we obtain for some positive constant C 1,
for all \(g \in C_{pol}([0,T] \times \mathbb{R}^{d}; \mathbb{R}^{m\times d})\). Since by a result from [30] (Theorem 4.1) there is a Borel measurable function \(v: [0,\tilde{T}] \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{m\times d}\) such that
we deduce that by density (and Lebesgue’s dominated convergence theorem)
Since this result holds true for all \(x \in \mathbb{R}^{d},0 \leq t \leq \tilde{ T} \leq T\), we have proved (i).
Let us now prove Lemma 5.96.
Proof of Lemma 5.96.
We first prove part a) of the lemma. Obviously, we have that
Thus for \(0 <\varepsilon < \dfrac{1} {6L^{2}}\),
which yields
Finally, the proof of part b) of Lemma 5.96 uses the same argument as that of Lemma 4.82. The only difference is that the role of the diffusion process X t, x in the proof of Lemma 4.82 is now replaced by that of the pair \(\left (X^{t,x},Y ^{\varepsilon }\right )\).
5.10 Exercises
Without further mention, \(\left (\Omega,\mathcal{F}, \mathbb{P},\{\mathcal{F}_{t}\}_{t\geq 0}\right )\) will be a stochastic basis, \(\left \{B_{t}: t \geq 0\right \}\) will be a k-dimensional Brownian motion with respect to this basis and \(\mathcal{F}_{t} = \mathcal{F}_{t}^{B}\) for all t ≥ 0.
Exercise 5.1.
Consider the BSDE
under the assumptions (5.41). Let
Show that if p ≥ 2 and for all δ ≥ 0
then the BSDE (5.259) has a unique solution \(\left (Y,Z\right ) \in S_{m}^{0}\left [0,T\right ] \times \Lambda _{m\times k}^{0}\left (0,T\right )\) such that
Remark.
Note that our assumptions hold in particular if both V t has exponential moments of all orders (e.g. the tail of its law behaves like that of a Gaussian random variable) and \(\vert \eta \vert +\mathbb{E}\int _{0}^{T}\vert \Phi (t,0,0)\vert dQ_{t}\) has a finite moment of some order p > 1.
Exercise 5.2 (g-Expectation).
Consider the BSDE: \(\mathbb{P}\text{-a.s.}\), for all \(t \in \left [0,T\right ]\)
where we assume:
-
(i)
\(\eta \in L^{p}\left (\Omega,\mathcal{F}_{T}, \mathbb{P}; \mathbb{R}\right )\), p > 1;
-
(ii)
for every \(\left (y,z\right ) \in \mathbb{R} \times \mathbb{R}^{k}\), the function \(g\left (\cdot,\cdot,y,z\right ): \Omega \times \left [0,T\right ] \rightarrow \mathbb{R}\) is \(\mathcal{P}\)-measurable;
-
(iii)
g satisfies the assumptions of Theorem 5.27 (F replaced by g) and \(g\left (t,y,0\right ) = 0\) for all \(y \in \mathbb{R}\), a.e. \(t \in \left [0,T\right ]\).
Then by Theorem 5.17 the BSDE (5.260) has a unique solution \(\left (Y,Z\right ) \in S_{1}^{p}\left [0,T\right ] \times \Lambda _{k}^{p}\left (0,T\right )\). Moreover if \(\tau: \Omega \rightarrow \left [0,T\right ]\) is a stopping time and \(\eta \in L^{p}\left (\Omega,\mathcal{F}_{\tau }, \mathbb{P}; \mathbb{R}\right )\) then \(\left (Y _{t},Z_{t}\right ) = \left (\eta,0\right )\) for all t ≥ τ.
Define the g-expectation of η by \(\mathbf{E}_{g}\left (\eta \right )\mathop{ =}\limits^{ \mathit{def }}Y _{0}\) and the conditional g-expectation of η with respect to \(\mathcal{F}_{t}\) by \(\mathbf{E}_{g}\left (\eta \vert \mathcal{F}_{t}\right )\mathop{ =}\limits^{ \mathit{def }}Y _{t}\). Clearly \(\mathbf{E}_{0}\left (\eta \right ) = \mathbb{E}\eta\) and \(\mathbf{E}_{0}\left (\eta \vert \mathcal{F}_{t}\right ) = \mathbb{E}\left (\eta \vert \mathcal{F}_{t}\right )\).
Show that:
-
1.
\(\mathbf{E}_{g}\left (a\right ) = a\), for all \(a \in \mathbb{R}.\)
-
2.
η 1 ≤ η 2, \(\mathbb{P}\text{-a.s.}\;\quad \Longrightarrow\quad \mathbf{E}_{g}\left (\eta _{1}\right ) \leq \mathbf{E}_{g}\left (\eta _{2}\right )\).
-
3.
η 1 ≤ η 2, \(\mathbb{P}\text{-a.s.}\) and \(\mathbf{E}_{g}\left (\eta _{1}\right ) = \mathbf{E}_{g}\left (\eta _{2}\right )\quad \Longrightarrow\quad \eta _{1} =\eta _{2}\), \(\mathbb{P}\text{-a.s.}\)
-
4.
If \(g\left (t,\cdot,\cdot \right ): \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) is a convex function, a.e. \(t \in \left [0,T\right ]\), then \(\mathbf{E}_{g}: L^{p}\left (\Omega,\mathcal{F}_{T}, \mathbb{P}\right ) \rightarrow \mathbb{R}\) is convex, too.
-
5.
Let \(U \in L^{p}\left (\Omega,\mathcal{F}_{t}, \mathbb{P}\right )\). Then \(\mathbf{E}_{g}\left (1_{A}\eta \right ) = \mathbf{E}_{g}\left (1_{A}U\right )\), for all \(A \in \mathcal{F}_{t}\), if and only if U = Y t .
-
6.
\(\mathbf{E}_{g}\left (a\vert \mathcal{F}_{t}\right ) = a\), for all \(a \in \mathbb{R}\).
-
7.
\(\mathbf{E}_{g}\left (\eta \vert \mathcal{F}_{t}\right ) =\eta\), for all \(\eta \in L^{p}\left (\Omega,\mathcal{F}_{t}, \mathbb{P}\right )\).
-
8.
η 1 ≤ η 2, \(\mathbb{P}\text{-a.s.}\;\quad \Longrightarrow\quad \mathbf{E}_{g}\left (\eta _{1}\vert \mathcal{F}_{t}\right ) \leq \mathbf{E}_{g}\left (\eta _{2}\vert \mathcal{F}_{t}\right )\), \(\mathbb{P}\text{-a.s.}\)
-
9.
\(\mathbf{E}_{g}\left (1_{A}\eta \vert \mathcal{F}_{t}\right ) = 1_{A}\mathbf{E}_{g}\left (\eta \vert \mathcal{F}_{t}\right )\), for all \(A \in \mathcal{F}_{t}\).
Exercise 5.3 (Peano Type Result).
Consider the BSDE
where \(\eta \in L^{p}\left (\Omega,\mathcal{F}_{T}, \mathbb{P}; \mathbb{R}\right )\), p ≥ 2, and \(G: \left [0,T\right ] \times \mathbb{R} \times \mathbb{R}^{k} \rightarrow \mathbb{R}\) is a function such that
-
\(G\left (\cdot,x,z\right ): \left [0,T\right ] \rightarrow \mathbb{R}\) is measurable for all \(x \in \mathbb{R}\) and \(z \in \mathbb{R}^{k}\),
-
\(G\left (t,\cdot,\cdot \right ): \mathbb{R} \times \mathbb{R}^{k} \rightarrow \mathbb{R}\) is continuous for all \(t \in \left [0,T\right ]\),
-
there exists an L > 0 such that for all \(\left (t,y,z\right ) \in \left [0,T\right ] \times \mathbb{R} \times \mathbb{R}^{k}\),
$$\displaystyle{\left \vert G\left (t,y,z\right )\right \vert \leq L\left (1 + \left \vert y\right \vert + \left \vert z\right \vert \right ).}$$Under these conditions we shall prove that the BSDE (5.260) has at least one solution \(\left (Y,Z\right ) \in S^{p}\left [0,T\right ] \times \Lambda _{k}^{p}\left (0,T\right )\).
Let \(0 <\varepsilon \leq \varepsilon _{0} = 1 \wedge \left (1/L\right )\) and \(G_{\varepsilon }: \left [0,T\right ] \times \mathbb{R} \times \mathbb{R}^{k} \rightarrow \mathbb{R}\),
Prove that:
-
1.
For all \(t \in \left [0,T\right ]\), \(y,y^{{\prime}}\in \mathbb{R}\) and \(z,z^{{\prime}}\in \mathbb{R}^{k}\):
-
(a)
\(\left \vert G_{\varepsilon }\left (t,y,z\right )\right \vert \leq L\left (1 + \left \vert y\right \vert + \left \vert z\right \vert \right );\)
-
(b)
\(\left \vert G_{\varepsilon }\left (t,y,z\right ) - G_{\varepsilon }\left (t,y^{{\prime}},z^{{\prime}}\right )\right \vert \leq \dfrac{1} {\varepsilon } \left (\left \vert y - y^{{\prime}}\right \vert + \left \vert z - z^{{\prime}}\right \vert \right );\)
-
(c)
\(yG_{\varepsilon }\left (t,y,z\right ) \leq L\left \vert y\right \vert + \left (L + L^{2}\right )\left \vert y\right \vert ^{2} + \dfrac{1} {4}\left \vert z\right \vert ^{2};\)
-
(d)
\(0 <\delta <\varepsilon \quad \Longrightarrow\quad G_{\delta }\left (t,y,z\right ) \geq G_{\varepsilon }\left (t,y,z\right );\)
-
(e)
if \(\lim \limits _{\varepsilon \rightarrow 0}\left (y_{\varepsilon },z_{\varepsilon }\right ) = \left (y,z\right )\), then \(\lim \limits _{\varepsilon \rightarrow 0}G_{\varepsilon }\left (t,y_{\varepsilon },z_{\varepsilon }\right ) = G\left (t,y,z\right )\).
-
(a)
-
2.
The BSDEs
$$\displaystyle\begin{array}{rcl} Y _{t}^{\varepsilon }& =& \eta +\int _{ t}^{T}G_{\varepsilon }\left (s,Y _{ s}^{\varepsilon },Z_{ s}^{\varepsilon }\right )\mathit{ds} -\int _{ t}^{T}Z_{ s}^{\varepsilon }\mathit{dB}_{ s}, {}\\ U_{t}& =& \eta +\int _{t}^{T}L\left (1 + \left \vert U_{ s}\right \vert + \left \vert V _{s}\right \vert \right )\mathit{ds} -\int _{t}^{T}Z_{ s}\mathit{dB}_{s} {}\\ \end{array}$$have unique solutions \(\left (Y ^{\varepsilon },Z^{\varepsilon }\right )\), \(\left (U,V \right ) \in S^{p}\left [0,T\right ] \times \Lambda _{m\times k}^{p}\left (0,T\right )\) and:
-
(a)
$$\displaystyle\begin{array}{rcl} & & \mathbb{E}^{\mathcal{F}_{t} }\ \left (\sup \limits _{s\in \left [t,T\right ]}\left \vert Y _{s}^{\varepsilon }\right \vert ^{p}\right ) + \mathbb{E}^{\mathcal{F}_{t} }\ \left (\int _{t}^{T}\left \vert Z_{ s}^{\varepsilon }\right \vert ^{2}\mathit{ds}\right )^{p/2} {}\\ & & \leq C_{p}\exp \left [\left (L + L^{2}\right )\left (T - t\right )\right ]\bigg[\mathbb{E}^{\mathcal{F}_{t} }\left \vert \eta \right \vert ^{p} + L^{p}\left (T - t\right )^{p}\bigg] {}\\ \end{array}$$
where C p is a constant depending only on p.
-
(b)
For all \(0 <\delta <\varepsilon \leq \varepsilon _{0} = 1 \wedge \left (1/L\right )\), \(\mathbb{P}\text{-a.s.}\),
$$\displaystyle{Y _{t}^{\varepsilon _{0} } \leq Y _{t}^{\varepsilon } \leq Y _{ t}^{\delta } \leq U_{ t},\quad \text{for all }t \in \left [0,T\right ],}$$and there exists a \(Y \in S^{p}\left [0,T\right ]\) such that
$$\displaystyle{\lim \limits _{\varepsilon \rightarrow 0}\mathbb{E}\left (\sup \limits _{s\in \left [0,T\right ]}\left \vert Y _{s}^{\varepsilon } - Y _{ s}\right \vert ^{p}\right ) = 0.}$$ -
(c)
There exists a \(Z \in \Lambda _{m\times k}^{p}\left (0,T\right )\) such that
$$\displaystyle{\lim _{\varepsilon \rightarrow 0}\mathbb{E}\ \left (\int _{0}^{T}\left \vert Z_{ s}^{\varepsilon } - Z_{ s}\right \vert ^{2}\mathit{ds}\right )^{p/2} = 0.}$$
-
(a)
Exercise 5.4 (BSDE Reflected Above 0).
Let \(\xi \in L^{2}\left (\Omega,\mathcal{F}_{T}^{B}, \mathbb{P}; \mathbb{R}\right )\), where {B t , 0 ≤ t ≤ T} is a k-dimensional BM, and \(F: \mathbb{R} \times \mathbb{R}^{k} \rightarrow \mathbb{R}\) be a Lipschitz continuous mapping. Consider for each \(n \in \mathbb{N}\) the solution \(\{(Y _{t}^{n},Z_{t}^{n}),\,0 \leq t \leq T\}\) of the BSDE
and let \(K_{t}^{n} = n\int _{0}^{t}(Y _{s}^{n})^{-}\mathit{ds}\).
-
1.
Show that Y t n+1 ≥ Y t n, 0 ≤ t ≤ T.
-
2.
Show that
$$\displaystyle{\sup _{n}\mathbb{E}\left (\sup _{0\leq t\leq T}\vert Y _{t}^{n}\vert ^{2}\right ) < \infty.}$$ -
3.
Deduce that there exists a progressively measurable process {Y t , 0 ≤ t ≤ T} such that Y t n → Y t a.s. for all t ∈ [0, T], and
$$\displaystyle{\mathbb{E}\left (\sup _{0\leq t\leq T}\vert Y _{t}\vert ^{2}\right ) < \infty.}$$ -
4.
Show that \(Y _{t}^{n} \geq \tilde{ Y }_{t}^{n}\), where \(\{\tilde{Y }_{t}^{n},\ 0 \leq t \leq T\}\) solves the BSDE
$$\displaystyle{\tilde{Y }_{t}^{n} =\xi +\int _{ t}^{T}F(\tilde{Y }_{ s}^{n},\tilde{Z}_{ s}^{n})\mathit{ds} - n\int _{ t}^{T}\tilde{Y }_{ s}^{n}\mathit{ds} -\int _{ 0}^{t}\left \langle \tilde{Z}_{ s}^{n},\mathit{dB}_{ s}\right \rangle.}$$ -
5.
Identify \(\lim _{n\rightarrow \infty }\tilde{Y }_{t}^{n}\) and deduce that Y t ≥ 0, 0 ≤ t ≤ T, a.s., and (with the help of Dini’s theorem) that \(\sup _{0\leq t\leq T}(Y _{t}^{n})^{-}\rightarrow 0\) in mean square.
-
6.
Show that {Z t n, 0 ≤ t ≤ T} is a Cauchy sequence in \(\Lambda _{k}^{2}\left (0,T\right )\). Hint: check that
$$\displaystyle{\int _{t}^{T}(Y _{ s}^{n} - Y _{ s}^{p})(\mathit{dK}_{ s}^{n} -\mathit{dK}_{ s}^{p}) \leq \int _{ t}^{T}\left [(Y _{ s}^{p})^{-}\mathit{dK}_{ s}^{n} + (Y _{ s}^{n})^{-}\mathit{dK}_{ s}^{p}\right ] \rightarrow 0.}$$ -
7.
Deduce that K t n converges in probability to a progressively measurable increasing continuous stochastic process K t .
-
8.
Show that the just constructed triple {(X t , Z t , K t ), 0 ≤ t ≤ T} is a unique progressively measurable solution of the reflected BSDE: for all \(t \in \left [0,T\right ]\), \(\mathbb{P}\text{-a.s.}\)
$$\displaystyle{\left \{\begin{array}{ll} \left (i\right ) &\quad Y \text{ is a continuous stochastic process, }Y _{t} \geq 0, \\ \left (\mathit{ii}\right ) &\quad K\text{ is c.i.s.p.},\ \int _{0}^{T}Y _{ s}\mathit{dK}_{s} = 0, \\ \left (\mathit{iii}\right )&\quad \mathbb{E}\int _{0}^{T}\vert Z_{ s}\vert ^{2}dt < \infty, \\ \left (\mathit{iv}\right ) &\quad Y _{t} =\xi +\int _{t}^{T}F(Y _{ s},Z_{s})\mathit{ds} + K_{T} - K_{t} -\int _{t}^{T}\left \langle Z_{ s},\mathit{dB}_{s}\right \rangle.\end{array} \right.}$$ -
9.
With the help of Tanaka’s formula applied to (Y t )+ = Y t , show that in the sense of inequality between measures,
$$\displaystyle{0 \leq \mathit{dK}_{t} \leq \mathbf{1}_{\{Y _{t}=0\}}\left [F(Y _{t},Z_{t})\right ]^{-}dt.}$$Deduce that K is absolutely continuous.
-
10.
Show that the points 2–9 constitute a particular case of Theorem 5.52.
Exercise 5.5.
Let \(\eta \in L^{0}\left (\Omega,\mathcal{F}_{T}, \mathbb{P}; \mathbb{R}\right )\) be such that 0 ≤ η ≤ 1, \(\mathbb{P}\text{-a.s.}\) Prove that the BSDE
has a unique solution \(\left (Y,Z\right ) \in S_{1}^{2}\left [0,T\right ] \times \Lambda _{k}^{2}\left (0,T\right )\). Moreover
\(0 \leq Y _{t} \leq 1,\quad \mathbb{P} - a.s\).
Exercise 5.6.
Let \(\varepsilon > 0\), \(\kappa: \mathbb{R} \rightarrow \mathbb{R}\) be a continuous bounded function and \(g: \mathbb{R} \rightarrow \mathbb{R}\) be a bounded Lipschitz continuous function. Consider the PDEs
and
-
1.
Write the BSDEs in \(\left (Y ^{t,x},Z^{t,x}\right )\) and respectively in \(\left (Y ^{\varepsilon;t,x},Z^{\varepsilon;t,x}\right )\) such that \(u\left (t,x\right ) = Y _{t}^{t,x}\) and \(u^{\varepsilon }\left (t,x\right ) = Y _{t}^{\varepsilon;t,x}\) are viscosity solutions of the PDEs (5.261) and, respectively, (5.262). Are the corresponding viscosity solutions unique?
-
2.
Prove that
$$\displaystyle{\lim _{\varepsilon \rightarrow 0}u^{\varepsilon }\left (0,x\right ) = u\left (0,x\right ),\quad \text{for all }x \in \mathbb{R}.}$$
Exercise 5.7.
Let E be a non-empty closed subset of \(\mathbb{R}^{m}\), \(g: \mathbb{R}^{k} \rightarrow E\) be a bounded Borel measurable function and \(F: \Omega \times \left [0,T\right ] \rightarrow \mathbb{R}^{m}\) be a bounded progressively measurable stochastic process. Let \(\left (Y,Z\right ) \in S_{m}^{1}\left [0,T\right ] \times \Lambda _{m\times k}^{1}\left (0,T\right )\) be such that
Show that \(\left (i\right ) \Rightarrow \left (\mathit{ii}\right )\), where:
-
(i)
\(\mathbb{P}\text{-a.s.}\), \(\left \{Y _{t}: t \in \left [0,T\right ]\right \} \subset E\), for all bounded Borel measurable function \(g: \mathbb{R}^{k} \rightarrow E\);
-
(ii)
E is a convex set.
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Pardoux, E., Răşcanu, A. (2014). Backward Stochastic Differential Equations. In: Stochastic Differential Equations, Backward SDEs, Partial Differential Equations. Stochastic Modelling and Applied Probability, vol 69. Springer, Cham. https://doi.org/10.1007/978-3-319-05714-9_5
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