Some Further Results of Itô’s Calculus

Abstract

In this chapter, we use the quasi-surely analysis theory to develop Itô’s integrals without the quasi-continuity condition. This allows us to define Itô’s integral on stopping time interval. In particular, this new formulation can be applied to obtain Itô’s formula for a general $C^{1,2}$-function, thus extending previously available results.

In this chapter, we use the quasi-surely analysis theory to develop Itô’s integrals without the quasi-continuity condition. This allows us to define Itô’s integral on stopping time interval. In particular, this new formulation can be applied to obtain Itô’s formula for a general $C^{1,2}$-function, thus extending previously available results.

1 A Generalized Itô’s Integral

Recall that $B_b(\Omega )$ is the space of all bounded and Borel measurable real functions defined on $\Omega =C^d_0(\mathbb {R}^+)$. We denote by $L^p_*(\Omega )$ the completion of $B_b(\Omega )$ under the natural norm $\Vert X\Vert _p := \hat{\mathbb {E}} [|X|^p]^{{1}/{p}}$. Similarly, we can define $L^p_*(\Omega _T)$ for any fixed $T \ge 0$. For any fixed $\mathbf {a}\in \mathbb {R}^d$, we still use the notation $B^\mathbf {a}_t:= \langle \mathbf {a}, B_t \rangle$. Then we introduce the following properties, which are important in our stochastic calculus.

Proposition 8.1.1

For any $0\le t<T$, $\xi \in L^2_*(\Omega _{t})$, we have

$$ \mathbb {\hat{E}}[\xi (B^\mathbf {a}_{T}-B^\mathbf {a}_{t})]=0. $$

Proof

For a fixed $P\in \mathcal {P}$, $B^\mathbf {a}$ is a martingale on $(\Omega , \mathcal {F}_t, P)$. Then we have

$$ E_{P}[\xi (B^\mathbf {a}_{T}-B^\mathbf {a}_{t})]=0, $$

which completes the proof.   $\square$

Proposition 8.1.2

For any $0\le t\le T$ and $\xi \in B_{b}({\Omega _{t}})$, we have

$$\begin{aligned} \hat{\mathbb {E}}[\xi ^{2}(B^\mathbf {a}_{T}-B^\mathbf {a}_{t})^{2}-{\sigma }^{2}_{\mathbf {a}\mathbf {a}^T}\xi ^{2}(T-t)]\le 0. \end{aligned}$$

(8.1.1)

Proof

If $\xi \in C_{b}(\Omega _{t})$, then we get that $\mathbb {\hat{E}[}\xi ^{2}(B^\mathbf {a}_{T}-B^\mathbf {a}_{t})^{2}-\xi ^{2}{\sigma }^{2}_{\mathbf {a}\mathbf {a}^T}(T-t)]=0$. Thus (8.1.1) holds for $\xi \in C_{b}(\Omega _{t})$. This implies that, for a fixed $P\in \mathcal {P}$,

$$\begin{aligned} E_{P}\mathbb {[}\xi ^{2}(B^\mathbf {a}_{T}-B^\mathbf {a}_{t})^{2}-\xi ^2{\sigma }^{2}_{\mathbf {a}\mathbf {a}^T}(T-t)]\le 0. \end{aligned}$$

(8.1.2)

If we take $\xi \in B_{b}(\Omega _{t})$, we can find a sequence $\{ \xi _{n}\}_{n=1}^{\infty }$ in $C_{b}(\Omega _{t})$, such that $\xi _{n} \rightarrow \xi$ in $L^{p}(\Omega ,\mathcal {F}_{t}, P)$, for some $p>2$. Thus we conclude that

$$ E_{P}\mathbb {[}\xi _{n}^{2}(B^\mathbf {a}_{T}-B^\mathbf {a}_{t})^{2}-\xi _{n}^{2}{\sigma }^{2}_{\mathbf {a}\mathbf {a}^T}(T-t)]\le 0. $$

Then letting $n\rightarrow \infty$, we obtain (8.1.2) for $\xi \in B_{b}(\Omega _{t})$.    $\square$

In what follows, we use the notation ${L}^p_*(\Omega )$, instead of $L^p_G(\Omega )$, to generalize Itô’s integral on a larger space of stochastic processes $M^2_*(0, T )$ defined as follows. For fixed $p\ge 1$ and $T\in \mathbb {R}_{+}$, we first consider the following simple type of processes:

$$\begin{aligned} M_{b, 0}(0,T)&=\Big \{ \eta :\eta _{t}(\omega )=\sum _{j=0}^{N-1}\xi _{j} (\omega )\mathbf {1}_{[t_{j}, t_{j+1})}(t),\\&\forall N>0,\,\,\, 0=t_{0}<\cdots <t_{N}=T,\,\,\,\xi _{j}(\omega )\in B_{b}(\Omega _{t_{j} }), j=0,\cdots , N-1\Big \}. \end{aligned}$$

Definition 8.1.3

For an element $\eta \in M_{b, 0}(0,T)$ with $\eta _{t}=\sum _{j=0}^{N-1}\xi _{j} (\omega )\mathbf {1}_{[t_{j}, t_{j}+1)}(t)$, the related Bochner integral is

$$ \int _{0}^{T}\eta _{t}(\omega )dt=\sum _{j=0}^{N-1}\xi _{j}(\omega )(t_{j+1} -t_{j}). $$

For any $\eta \in M_{b, 0}(0,T)$ we set

$$ \tilde{\mathbb {E}}_{T}[\eta ]:=\tfrac{1}{T}\hat{\mathbb {E}}\left[ \int _{0}^{T}\eta _{t}dt\right] =\frac{1}{T}\hat{\mathbb {E}}\left[ \sum _{j=0}^{N-1}\xi _{j}(\omega )(t_{j+1}-t_{j})\right] . $$

Then $\mathbb {\tilde{E}}: M_{b, 0}(0, T )\mapsto \mathbb {R}$ forms a sublinear expectation. We can introduce a natural norm $\Vert \eta \Vert _{M^{p}(0,T)}=\left\{ \hat{\mathbb {E} }\left[ \int _{0}^{T}|\eta _{t}|^{p}dt\right] \right\} ^{1/p}$.

Definition 8.1.4

For any $p\ge 1$, we denote by $M_{*}^{p}(0,T)$ the completion of $M_{b, 0}(0,T)$ under the norm

$$ ||\eta ||_{M^{p}(0,T)}=\left\{ \hat{\mathbb {E}}\left[ \int _{0}^{T}|\eta _{t}|^{p} dt\right] \right\} ^{1/p}. $$

We have $M_{*}^{p}(0,T)\supset M_{*}^{q}(0,T)$, for $p\le q$. The following process

$$ \eta _{t}(\omega )=\sum _{j=0}^{N-1}\xi _{j}(\omega )\mathbf {1}_{[t_{j}, t_{j+1} )}(t),\ \xi _{j}\in {L}_{*}^{p}(\Omega _{t_{j}}),\ j=1,\cdots , N $$

is also in $M_{*}^{p}(0,T)$.

Definition 8.1.5

For any $\eta \in M_{b, 0}(0,T)$ of the form

$$ \eta _{t}(\omega )=\sum _{j=0}^{N-1}\xi _{j}(\omega )\mathbf {1}_{[t_{j}, t_{j+1} )}(t), $$

we define Itô’s integral

$$ I(\eta )=\int _{0}^{T}\eta _{s}dB^{\mathbf {a}}_{s}:=\sum _{j=0}^{N-1}\xi _{j}(B^\mathbf {a}_{t_{j+1} }-B^\mathbf {a}_{t_{j}})\mathbf {.} $$

Lemma 8.1.6

The mapping $I:M_{b, 0}(0,T)\mapsto {L}_{*} ^{2}(\Omega _{T})$ is a linear continuous mapping and thus can be continuously extended to $I:M_{*}^{2}(0,T)\mapsto {L}^{2}_*(\Omega _{T})$. Moreover, we have

$$\begin{aligned} \mathbb {\hat{E}}\left[ \int _{0}^{T}\eta _{s}dB^{\mathbf {a}}_{s}\right]&=0,\ \ \end{aligned}$$

(8.1.3)

$$\begin{aligned} \mathbb {\hat{E}}\left[ (\int _{0}^{T}\eta _{s}dB^{\mathbf {a}}_{s})^{2}\right]&\le {\sigma }^{2}_{{\mathbf {a}}{\mathbf {a}}^T}\hat{\mathbb {E}}\left[ \int _{0}^{T}|\eta _{t}|^{2}dt\right] . \end{aligned}$$

(8.1.4)

Proof

It suffices to prove (8.1.3) and (8.1.4) for any $\eta \in M_{b, 0}(0,T)$. From Proposition 8.1.1, for any j,

$$ \mathbb {\hat{E}}\mathbf {[}\xi _{j}(B^{\mathbf {a}}_{t_{j+1}}-B^{\mathbf {a}}_{t_{j}})]=\mathbb {\hat{E} }\mathbf {[-}\xi _{j}(B^{\mathbf {a}}_{t_{j+1}}-B^{\mathbf {a}}_{t_{j}})]=0. $$

Thus we obtain (8.1.3):

$$\begin{aligned} \mathbb {\hat{E}}\left[ \int _{0}^{T}\eta _{s}dB^{\mathbf {a}}_{s}\right]&=\mathbb {\hat{E}}\left[ \int _{0}^{t_{N-1}}\eta _{s}dB^{\mathbf {a}}_{s}+\xi _{N-1}(B^{\mathbf {a}}_{t_{N}}-B^{\mathbf {a}}_{t_{N-1}})\right] \\&=\mathbb {\hat{E}}\left[ \int _{0}^{t_{N-1}}\eta _{s}dB^{\mathbf {a}}_{s}\right] =\cdots =\hat{\mathbb {E} }[\xi _{0}(B^{\mathbf {a}}_{t_{1}}-B^{\mathbf {a}}_{t_{0}})]=0. \end{aligned}$$

We now prove (8.1.4). By a similar analysis as in Lemma 3.3.4 of Chap. 3, we derive that

$$\begin{aligned} \hat{\mathbb {E}}\left[ \left( \int _{0}^{T}\eta _{t}dB^{\mathbf {a}}_{t}\right) ^{2}\right] =\hat{\mathbb {E}}\left[ \sum _{i=0}^{N-1}\xi _{i}^{2}(B^{\mathbf {a}}_{t_{i+1}}-B^{\mathbf {a}}_{t_{i} })^{2}\right] . \end{aligned}$$

Then from Proposition 8.1.2, we obtain that

$$ \hat{\mathbb {E}}\left[ \xi _{j}^{2}(B^{\mathbf {a}}_{t_{j+1}}-B^{\mathbf {a}}_{t_{j}})^{2}-{\sigma }^2_{{\mathbf {a}}{\mathbf {a}}^T} \xi _{j}^{2}(t_{j+1}-t_{j})\right] \le 0. $$

Thus

$$\begin{aligned}&\hat{\mathbb {E}}\left[ \left( \int _{0}^{T}\eta _{t}dB^{\mathbf {a}}_{t}\right) ^{2}\right] =\hat{\mathbb {E}} [\sum _{i=0}^{N-1}\xi _{i}^{2}(B^{\mathbf {a}}_{t_{N}}-B^{\mathbf {a}}_{t_{N-1}})^{2}]\\&\ \ \le \hat{\mathbb {E}}\left[ \sum _{i=0}^{N-1}\xi _{i}^{2}[(B^{\mathbf {a}}_{t_{N}}-B^{\mathbf {a}}_{t_{N-1} })^{2}-{\sigma }^{2}_{{\mathbf {a}}{\mathbf {a}}^T}(t_{i+1}-t_{i})]\right] +\hat{\mathbb {E}}\left[ \sum _{i=0}^{N-1}{\sigma }^{2}_{{\mathbf {a}}{\mathbf {a}}^T}\xi _{i}^{2}(t_{i+1}-t_{i})\right] \\&\ \ \le \hat{\mathbb {E}}\left[ \sum _{i=0}^{N-1}{\sigma }^{2}_{{\mathbf {a}}{\mathbf {a}}^T}\xi _{i} ^{2}(t_{i+1}-t_{i})\right] ={\sigma }^{2}_{{\mathbf {a}}{\mathbf {a}}^T}\hat{\mathbb {E}}\left[ \int _{0}^{T}|\eta _{t}| ^{2}dt\right] , \end{aligned}$$

which is the desired result.    $\square$

The following proposition can be verified directly by the definition of Itô’s integral with respect to G-Brownian motion.

Proposition 8.1.7

Let $\eta ,\theta \in M_{*}^{2}(0,T)$. Then for any $0\le s\le r\le t\le T$, we have:

(i):

$\int _{s}^{t}\eta _{u}dB^{\mathbf {a}}_{u}=\int _{s}^{r} \eta _{u}d B^{\mathbf {a}}_{u}+\int _{r}^{t} \eta _{u}d B^{\mathbf {a}}_{u};$

(ii):

$\int _{s}^{t}(\alpha \eta _{u}+\theta _{u})dB^{\mathbf {a}}_{u}=\alpha \int _{s}^{t}\eta _{u}dB^{\mathbf {a}}_{u}+\int _{s}^{t}\theta _{u}dB^{\mathbf {a}}_{u}$, where $\alpha \in B_{b}(\Omega _{s})$.

Proposition 8.1.8

For any $\eta \in M_{*}^{2}(0,T)$, we have

$$\begin{aligned} \hat{\mathbb {E}}\left[ \sup _{0\le t\le T}\left| \int _{0}^{t}\eta _{s}dB^{\mathbf {a}}_{s}\right| ^{2} \right]&\le 4{\sigma }^{2}_{{\mathbf {a}}{\mathbf {a}}^T}\hat{\mathbb {E}}\left[ \int _{0}^{T}\eta _{s} ^{2}ds\right] . \end{aligned}$$

(8.1.5)

Proof

Since for any $\alpha \in B_{b}(\Omega _{t})$, we have

$$ \mathbb {\hat{E}}\left[ \alpha \int _{t}^{T}\eta _{s}dB^{\mathbf {a}}_{s}\right] =0. $$

Then, for a fixed $P\in \mathcal {P}$, the process $\int _{0}^{\cdot }\eta _{s}dB^{\mathbf {a}}_{s}$ is a martingale on $(\Omega ,\mathcal {F}_t, P)$. It follows from the classical Doob’s maximal inequality (see Appendix B) that

$$\begin{aligned} E_{P}\left[ \sup _{0\le t\le T}\left| \int _{0}^{t}\eta _{s}dB^\mathbf {a}_{s}\right| ^{2}\right] \le 4E_{P} \left[ \left| \int _{0}^{T}\eta _{s}dB^\mathbf {a}_{s}\right| ^{2}\right]&\le 4\hat{\mathbb {E}}\left[ \left| \int _{0}^{T}\eta _{s}dB^\mathbf {a}_{s}\right| ^{2}\right] \\ {}&\le 4{\sigma }^{2}_{\mathbf {a}\mathbf {a}^T}\hat{\mathbb {E}}\left[ \int _{0}^{T} \eta _{s}^{2}ds\right] . \end{aligned}$$

Thus (8.1.5) holds.    $\square$

Proposition 8.1.9

For any $\eta \in M_{*}^{2}(0,T)$ and $0\le t\le T$, the integral $\int _{0}^{t}\eta _{s}dB^\mathbf {a}_{s}$ is continuous q.s., i.e., $\int _{0}^{t}\eta _{s}dB^\mathbf {a}_{s}$ has a modification whose paths are continuous in t.

Proof

The claim is true for $\eta \in M_{b, 0}(0,T)$ since $(B^\mathbf {a}_{t})_{t\ge 0}$ is a continuous process. In the case of $\eta \in M_{*}^{2}(0,T)$, there exists $\eta ^{(n)}\in M_{b, 0}(0,T)$, such that $\hat{\mathbb {E}}[\int _{0} ^{T}(\eta _{s}-\eta _{s}^{(n)})^{2}ds]\rightarrow 0$, as $n\rightarrow \infty$. By Proposition 8.1.8, we have

$$ \hat{\mathbb {E}}\left[ \sup _{0\le t\le T}\left| \int _{0}^{t}(\eta _{s}-\eta _{s} ^{(n)})dB^\mathbf {a}_{s}\right| ^{2}\right] \le 4{\sigma }^{2}_{\mathbf {a}\mathbf {a}^T}\hat{\mathbb {E}}\left[ \int _{0}^{T} (\eta _{s}-\eta _{s}^{(n)})^{2}ds\right] \rightarrow 0,\,\,\, \text { as }\,\, n\rightarrow \infty . $$

Then choosing a subsequence if necessary, we can find a set $\hat{\Omega }\subset \Omega$ with $\hat{c}(\hat{\Omega }^c)=0$ so that, for any $\omega \in \hat{\Omega }$ the sequence of processes $\int _{0}^{\cdot }\eta _{s}^{(n)}dB^{\mathbf {a}}_{s}(\omega )$ uniformly converges to $\int _{0}^{\cdot }\eta _{s}dB^\mathbf {a}_{s}(\omega )$ on [0, T]. Thus for any $\omega \in \hat{\Omega }$, we get that $\int _{0}^{\cdot }\eta _{s}dB^\mathbf {a}_{s}(\omega )$ is continuous in t. For any $(\omega , t)\in [0,T]\times \Omega$, we take the process

$$J_t(\omega )=\left\{ \begin{aligned}&\int _{0}^{t}\eta _{s}dB^\mathbf {a}_{s}(\omega ),\ \omega \in \hat{\Omega };\\&\,\,\, 0,\ \,\,\,\,\,\,\, \,\,\,\,\,\,\,\, \mathrm{otherwise}, \end{aligned}\right. $$

as the desired t-continuous modification. This completes the proof.    $\square$

We now define the integral of a process $\eta \in M_{*}^{1}(0,T)$ with respect to $\left\langle B^{\mathbf {a}}\right\rangle$. We also define a mapping:

$$ Q_{0,T}(\eta )=\int _{0}^{T}\eta _td\left\langle B^{\mathbf {a}}\right\rangle _{t}:=\sum _{j=0}^{N-1}\xi _{j}(\left\langle B^{\mathbf {a}}\right\rangle _{t_{j+1}}-\left\langle B^{\mathbf {a}}\right\rangle _{t_{j}}):M_{b}^{1,0}(0,T)\rightarrow L_{*}^{1}(\Omega _{T}). $$

Proposition 8.1.10

The mapping $Q_{0,T}:M_{b}^{1,0}(0,T)\mapsto L_{*}^{1}(\Omega _{T})$ is a continuous linear mapping and $Q_{0,T}$ can be uniquely extended to $M_{*}^{1}(0,T)$. Moreover, we have

$$\begin{aligned} \hat{\mathbb {E}}\left[ \left| \int _{0}^{T}\eta _td\left\langle B^{\mathbf {a}}\right\rangle _{t}\right| \right] \le {\sigma }^{2}_{{\mathbf {a}}{\mathbf {a}}^T}\hat{\mathbb {E}}\left[ \int _{0}^{T}|\eta _{t}|dt\right] \ \ \text { for any }\ \eta \in M_{*}^{1}(0,T)\text {.} \end{aligned}$$

(8.1.6)

Proof

From the relation

$$ {\sigma }^{2}_{-{\mathbf {a}}{\mathbf {a}}^T}(t-s)\le \left\langle B^{\mathbf {a}}\right\rangle _t-\left\langle B^{\mathbf {a}}\right\rangle _s\le {\sigma }^{2}_{{\mathbf {a}}{\mathbf {a}}^T}(t-s) $$

it follows that

$$\begin{aligned}&\hat{\mathbb {E}}\left[ |\xi _j|(\langle B^{\mathbf {a}}\rangle _{t_{j+1}}-\langle B^{\mathbf {a}}\rangle _{t_j})-{\sigma }^{2}_{{\mathbf {a}}{\mathbf {a}}^T}|\xi _j|(t_{j+1}-t_j)\right] \le 0,\,\,\,\, \text { for any } \,\,\, j=1,\cdots , N-1. \end{aligned}$$

Therefore, we deduce the following chain of inequalities:

$$\begin{aligned}&\hat{\mathbb {E}}\left[ \left| \sum _{j=0}^{N-1}\xi _{j}(\left\langle B^{\mathbf {a}}\right\rangle _{t_{j+1}}-\left\langle B^{\mathbf {a}}\right\rangle _{t_{j}})\right| \right] \le \hat{\mathbb {E}}\left[ \sum _{j=0} ^{N-1}|\xi _{j}|\left\langle B^{\mathbf {a}}\right\rangle _{t_{j+1}}-\left\langle B^{\mathbf {a}}\right\rangle _{t_{j}}\right] \\ \le&\hat{\mathbb {E}}\left[ \sum _{j=0} ^{N-1}|\xi _{j}|[(\langle B^{\mathbf {a}}\rangle _{t_{j+1}}-\langle B^{\mathbf {a}}\rangle _{t_{j}})-{\sigma }^2_{{\mathbf {a}}{\mathbf {a}}^T}(t_{j+1}-t_j)]\right] +\hat{\mathbb {E}}\left[ {\sigma }^2_{{\mathbf {a}}{\mathbf {a}}^T}\sum _{j=0}^{N-1}|\xi _j|(t_{j+1}-t_j)\right] \\ \le&\sum _{j=0}^{N-1}\hat{\mathbb {E}}[|\xi _j|[(\langle B^{\mathbf {a}}\rangle _{t_{j+1}}-\langle B^{\mathbf {a}}\rangle _{t_j})-{\sigma }^2_{{\mathbf {a}}{\mathbf {a}}^T}(t_{j+1}-t_j)]]+\hat{\mathbb {E}}\left[ {\sigma }^2_{{\mathbf {a}}{\mathbf {a}}^T}\sum _{j=0}^{N-1}|\xi _j|(t_{j+1}-t_j)\right] \\ \le&\hat{\mathbb {E}}\left[ {\sigma }^2_{{\mathbf {a}}{\mathbf {a}}^T}\sum _{j=0}^{N-1}|\xi _j|(t_{j+1}-t_j)\right] = {\sigma }^2_{{\mathbf {a}}{\mathbf {a}}^T}\hat{\mathbb {E}}\left[ \int _{0} ^{T}|\eta _{t}|dt\right] . \end{aligned}$$

This completes the proof.    $\square$

From the above Proposition 8.1.9, we obtain that $\left\langle B^{\mathbf {a}}\right\rangle _t$ is continuous in t q.s.. Then for any $\eta \in M_{*}^{1}(0,T)$ and $0\le t\le T$, the integral $\int _{0}^{t}\eta _{s}d\langle B^\mathbf {a}\rangle _{s}$ also has a t-continuous modification. In the sequel, we always consider the t-continuous modification of Itô’s integral. Moreover, Itô’s integral with respect to $\langle B^i, B^j\rangle =\langle B\rangle ^{ij}$ can be similarly defined. This is left as an exercise for the readers.

Lemma 8.1.11

Let $\eta \in M_{b}^{2}(0,T)$. Then $\eta$ is Itô-integrable for every $P \in P$. Moreover,

$$ \int ^T_0\eta _sdB^{\mathbf {a}}_s=\int ^T_0\eta _sd_PB^{\mathbf {a}}_s, \ \ P\text {-a.s.}, $$

where the right hand side is the usual Itô integral.

We leave the proof of this lemma to readers as an exercise.

Lemma 8.1.12

(Generalized Burkholder-Davis-Gundy (BDG) inequality) For any $\eta \in M^2_*(0,T)$ and $p>0$, there exist constants $c_p$ and $C_p$ with $0<c_{p}<C_{p}<\infty$, depending only on p, such that

$$\begin{aligned} \sigma _{-\mathbf {aa}^{T}}^{p}c_{p}\mathbb {\hat{E}}\left[ \left( \int _{0}^{T}| \eta _{s}|^{2}ds\right) ^{p/2}\right] \le \mathbb {\hat{E}}\left[ \sup _{t\in [0,T]}\left| \int _{0}^{t}\eta _{s}dB^{\mathbf {a}}_{s}\right| ^{p}\right] \le \sigma _{\mathbf {aa}^{T}}^{p}C_{p} \mathbb {\hat{E}}\left[ \left( \int _{0}^{T}|\eta _{s}|^{2}ds\right) ^{p/2}\right] . \end{aligned}$$

Proof

Observe that, under any $P\in \mathcal {P}$, $B^{\mathbf {a}}$ is a P-martingale with

$${\sigma }^2_{-{\mathbf {a}}{\mathbf {a}}^T} dt \le d\langle B^{\mathbf {a}}\rangle _t \le {\sigma }^2_{{\mathbf {a}}{\mathbf {a}}^T} dt. $$

The proof is then a simple application of the classical BDG inequality.    $\square$

2 Itô’s Integral for Locally Integrable Processes

So far we have considered Itô’s integral $\int ^T_0\eta _tdB^{\mathbf {a}}_t$ where $\eta$ in $M^2_*(0,T)$. In this section we continue our study of Itô’s integrals for a type of locally integrable processes.

We first give some properties of $M^p_*(0,T)$.

Lemma 8.2.1

For any $p\ge 1$ and $X\in M_{*}^{p}(0,T)$, the following relation holds:

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathbb {\hat{E}}\left[ \int _{0}^{T}|X_{t}|^{p} \mathbf {1}_{\{|X_{t}|>n\}}dt\right] =0. \end{aligned}$$

(8.2.1)

Proof

The proof is similar to that of Proposition 6.1.22 in Chap. 6.    $\square$

Corollary 8.2.2

For any $\eta \in M_{*}^{2}(0,T)$, let $\eta _{s}^{(n)} =(-n)\vee (\eta _{s}\wedge n)$, then, as $n\rightarrow \infty$, we have $\int _{0}^{t}\eta _{s}^{(n)} dB^\mathbf {a}_{s}\rightarrow \int _{0}^{t}\eta _{s}dB^\mathbf {a}_{s}$ in $L_{*}^{2}(0,T)$ for any $t\le T$.

Proposition 8.2.3

Let $X\in M_{*}^{p}(0,T)$. Then for any $\varepsilon >0$, there exists a constant $\delta >0$ such that for all $\eta \in M_{*}^p(0,T)$ satisfying $\mathbb {\hat{E}}\left[ \int _{0}^{T}|\eta _{t}|dt\right] \le \delta$ and $|\eta _{t} (\omega )|\le 1$, we have $\mathbb {\hat{E}}\left[ \int _{0}^{T}|X_{t}|^{p}|\eta _{t}|dt\right] \le \varepsilon$.

Proof

For any $\varepsilon >0$, according to Lemma 8.2.1, there exists a number $N>0$ such that $\mathbb {\hat{E}}\left[ \int _{0}^{T}|X|^{p}\mathbf {1}_{\{|X|>N\}}\right] \le {\varepsilon }/{2}$. Take $\delta ={\varepsilon }/{2N^{p}}$. Then we derive that

$$\begin{aligned} \mathbb {\hat{E}}\left[ \int _{0}^{T}|X_{t}|^{p}|\eta _{t}|dt \right]&\le \mathbb {\hat{E} }\left[ \int _{0}^{T}|X_{t}|^{p}|\eta _{t}|\mathbf {1}_{\{|X_{t}|>N\}}dt\right] +\mathbb {\hat{E}}\left[ \int _{0}^{T}|X_{t}|^{p}|\eta _{t}|\mathbf {1}_{\{|X_{t}|\le N\}}dt\right] \\&\le \mathbb {\hat{E}}\left[ \int _{0}^{T}|X_{t}|^{p}\mathbf {1}_{\{|X_{t}|>N\}}dt\right] +N^{p} \mathbb {\hat{E}}\left[ \int _{0}^{T}|\eta _{t}|dt\right] \le \varepsilon \text {,} \end{aligned}$$

which is the desired result.    $\square$

Lemma 8.2.4

If $p\ge 1$ and $X,\eta \in M_{*}^{p}(0,T)$ are such that $\eta$ is bounded, then the product $X\eta \in M_{*}^{p}(0,T)$.

Proof

We can find $X^{(n)}, \eta ^{(n)} \in M_{b, 0}(0, T )$ for $n= 1, 2,\ldots$, such that $\eta ^{(n)}$ is uniformly bounded and

$$ \Vert X-X^{(n)}\Vert _{M^p(0,T)}\rightarrow 0, \ \ \Vert \eta -\eta ^{(n)}\Vert _{M^p(0,T)}\rightarrow 0,\,\,\, \text { as } \,\, \, n\rightarrow \infty . $$

Then we obtain that

$$\begin{aligned} \mathbb {\hat{E}}\left[ \int ^T_0|X_t\eta _t-X_t^{(n)}\eta _t^{(n)}|^pdt\right] \le 2^{p-1}\left( \mathbb {\hat{E}}\left[ \int ^T_0|X_t|^p|\eta _t-\eta _t^{(n)}|^pdt\right] +\mathbb {\hat{E}}\left[ \int ^T_0|X_t-X_t^{(n)}|^p|\eta _t^{(n)}|^pdt\right] \right) . \end{aligned}$$

By Proposition 8.2.3, the first term on the right-hand side tends to 0 as $n\rightarrow \infty$. Since $\eta ^{(n)}$ is uniformly bounded, the second term also tends to 0.     $\square$

Now we are going to study Itô’s integrals on an interval $[0, \tau ]$, where $\tau$ is a stopping time relative to the G-Brownian pathes.

Definition 8.2.5

A stopping time $\tau$ relative to the filtration $(\mathcal {F}_{t})$ is a map on $\Omega$ with values in [0, T] such that $\{ \tau \le t\} \in \mathcal {F}_{t}$, for every $t\in [0,T]$.

Lemma 8.2.6

For any stopping time $\tau$ and any $X\in M_{*}^{p}(0,T),$ we have $\mathbf {1}_{[0,\tau ]} (\cdot )X\in M_{*}^{p}(0,T)$.

Proof

Related to the given stopping time $\tau$, we consider the following sequence:

$$ \tau _{n}=\sum _{k=0}^{2^{n}-1}\frac{(k+1)T}{2^{n}}\mathbf {1}_{[\frac{kT}{2^{n}} \le \tau <\frac{(k+1)T}{2^{n}})}+T\mathbf {1}_{[\tau \ge T]}. $$

It is clear that $2^{-n}\ge \tau _{n}-\tau \ge 0$. It follows from Lemma 8.2.4 that any element of the sequence $\{1_{[0,\tau _{n}]}X\}_{n=1}^{\infty }$ is in $M^p_*(0,T)$. Note that, for $m\ge n$, we have

$$\begin{aligned} \mathbb {\hat{E}}\left[ \int _{0}^{T}|\mathbf {1}_{[0,\tau _{n}]}(t)-\mathbf {1}_{[0,\tau _{m}]}(t)|dt\right]&\le \mathbb {\hat{E}}\left[ \int _{0}^{T}|\mathbf {1}_{[0,\tau _{n}]}(t)-\mathbf {1}_{[0,\tau ]}(t)|dt\right] \\&=\mathbb {\hat{E}}[\tau _{n}-\tau ]\le 2^{-n}T\text {.} \end{aligned}$$

Then applying Proposition 8.2.3, we derive that $\mathbf {1}_{[0,\tau ]}X\in M_{*}^{p}(0,T)$ and the proof is complete.    $\square$

Lemma 8.2.7

For any stopping time $\tau$ and any $\eta \in M_{*}^{2}(0,T)$, we have

$$\begin{aligned} \int _{0}^{t\wedge \tau }\eta _{s}dB^{\mathbf {a}}_{s}(\omega )=\int _{0}^{t}\mathbf {1}_{[0,\tau ]} (s)\eta _{s}dB^\mathbf {a}_{s}(\omega ), \ \text {for all t}\in [0,T] \text { q.s.} \end{aligned}$$

(8.2.2)

Proof

For any $n\in \mathbb {N}$, let

$$ \tau _{n}:=\sum _{k=1}^{[t\cdot 2^{n}]}\frac{k}{2^{n}}\mathbf {1}_{[ \frac{(k-1)t}{2^{n}}\le \tau <\frac{kt}{2^{n}})}+t\mathbf {1}_{[\tau \ge t]} =\sum _{k=1}^{2^{n}}\mathbf {1}_{A_{n}^{k}}t_{n}^{k}. $$

Here $t_{n}^{k}=k2^{-n}t$, $A_{n}^{k}=[t_{n}^{k-1}<t\wedge \tau \le t_{n} ^{k}]$, for $k<2^{n}$, and $A_{n}^{2^{n}}=[\tau \ge t]$. We see that $\{ \tau _{n} \}_{n=1}^{\infty }$ is a decreasing sequence of stopping times which converges to $t\wedge \tau$.

We first show that

$$\begin{aligned} \int _{\tau _{n}}^{t}\eta _{s}dB^{\mathbf {a}}_{s}=\int _{0}^{t}\mathbf {1}_{[\tau _{n}, t]} (s)\eta _{s}dB^\mathbf {a}_{s}, \ \ \text {q.s.} \end{aligned}$$

(8.2.3)

By Proposition 8.1.7 we have

$$\begin{aligned} \int _{\tau _{n}}^{t}\eta _{s}dB^\mathbf {a}_{s}&=\int _{\sum _{k=1}^{2^{n}}\mathbf {1} _{A_{n}^{k}}t_{n}^{k}}^{t}\eta _{s}dB^\mathbf {a}_{s}=\sum _{k=1}^{2^{n}}\mathbf {1} _{A_{n}^{k}}\int _{t_{n}^{k}}^{t}\eta _{s}dB^\mathbf {a}_{s}\\&=\sum _{k=1}^{2^{n}}\int _{t_{n}^{k}}^{t}\mathbf {1}_{A_{n}^{k}}\eta _{s} dB^\mathbf {a}_{s} =\int _{0}^{t}\sum _{k=1}^{2^{n}}\mathbf {1}_{[t_{n}^{k}, t]}(s)\mathbf {1} _{A_{n}^{k}}\eta _{s}dB^\mathbf {a}_{s}, \end{aligned}$$

from which (8.2.3) follows. Hence we obtain that

$$ \int _{0}^{\tau _{n}}\eta _{s}dB^{\mathbf {a}}_{s}=\int _{0}^{t}\mathbf {1}_{[0,\tau _{n}]} (s)\eta _{s}dB^{\mathbf {a}}_{s}, \ \text {q.s.} $$

Observe now that $0\le \tau _{n}-\tau _{m}\le \tau _{n}-t\wedge \tau \le 2^{-n}t$, for $n\le m$. Then Proposition 8.2.3 yields that $\mathbf {1} _{[0,\tau _{n}]}\eta$ converges in $M_{*}^{2}(0,T)$ to $\mathbf {1} _{[0,\tau \wedge t]}\eta$ as $n\rightarrow \infty$, which implies that $\mathbf {1}_{[0,\tau \wedge t]}\eta \in M_{*}^{2}(0,T)$. Consequently,

$$ \lim _{n\rightarrow \infty }\int _{0}^{\tau _{n}} \eta _{s}dB^{\mathbf {a}}_{s}=\int _{0} ^{t}\mathbf {1}_{[0,\tau ]} (s)\eta _{s}dB^{\mathbf {a}}_{s},\ \ \text {q.s.} $$

Note that $\int _{0}^{t}\eta _{s}dB^{\mathbf {a}}_{s}$ is continuous in t, hence (8.2.2) is proved.    $\square$

The space of processes $M_*^p(0,T)$ can be further enlarged as follows.

Definition 8.2.8

For fixed $p\ge 1$, a stochastic process $\eta$ is said to be in $M^p_w(0, T)$, if it is associated with a sequence of increasing stopping times $\{\sigma _m\}_{m\in \mathbb {N}}$, such that:

• (i) For any $m\in \mathbb {N}$, the process $\left( \eta _t\mathbf {1}_{[0, \sigma _m]}(t)\right) _{t\in [0,T]} \in M^p_*(0, T)$;

• (ii) If $\Omega ^{(m)}:=\{\omega \in \Omega : \sigma _m(\omega )\wedge T=T\}$ and $\hat{\Omega }:= \lim _{m\rightarrow \infty }\Omega ^{(m)}$, then $\hat{c}({\hat{\Omega }}^c)=0.$

Remark 8.2.9

Suppose there is another sequence of stopping times $\{\tau _m\}_{m=1}^\infty$ that satisfies the second condition in Definition 8.2.8. Then the sequence $\{\tau _m\wedge \sigma _m\}_{m\in \mathbb {N}}$ also satisfies this condition. Moreover, by Lemma 8.2.6, we know that for any $m\in \mathbb {N}$, $\eta \mathbf {1}_{[0, \tau _m\wedge \sigma _m]} \in M^p_*(0, T)$. This property allows to associate the same sequence of stopping times with several different processes in $M_w^p(0,T)$.

For given $\eta \in M^2_w(0, T)$ associated with $\{\sigma _m\}_{m\in \mathbb {N}}$, we consider, for any $m\in \mathbb {N}$, the t-continuous modification of the process $\left( \int ^t_0 \eta _s\mathbf{1}_{[0, \sigma _m]}(s)dB^\mathbf{a}_s\right) _{0\le t\le T}$. For any m, $n\in \mathbb {N}$ with $n >m$, by Lemma 8.2.7 we can find a polar set $\hat{A}_{m, n}$, such that for all $\omega \in (\hat{A}_{m, n})^c$, the following equalities hold:

$$\begin{aligned} \int ^{t\wedge \sigma _m}_0\eta _s{\mathbf {1}}_{[0, \sigma _m]}(s)dB^\mathbf{a}_s(\omega )&=\int ^{t}_0\eta _s{\mathbf {1}}_{[0, \sigma _m]}(s)dB^\mathbf{a}_s(\omega )\nonumber \\&=\int ^{t}_0\eta _s{\mathbf {1}}_{[0, \sigma _m]}(s)\mathbf{1}_{[0, \sigma _n]}(s)dB^\mathbf{a}_s(\omega )\\&=\int ^{t\wedge \sigma _m}_0\eta _s{\mathbf {1}}_{[0, \sigma _n]}(s)dB_s^\mathbf{a}(\omega ),\ 0\le t\le T.\nonumber \end{aligned}$$

(8.2.4)

Define the polar set

$$\begin{aligned} \hat{A}:=\bigcup _{m=1}^{\infty }\bigcup _{n=m+1}^{\infty }\hat{A}_{m, n}. \end{aligned}$$

For any $m\in \mathbb {N}$ and any $(\omega , t)\in \Omega \times [0, T]$, we set

$$ X^{(m)}_t(\omega ):= \left\{ \begin{aligned}&\int ^{t}_0\eta _s\mathbf{1}_{[0, \sigma _m]}(s)dB^\mathbf{a}_s(\omega ), \,\,\,\,\,\,\, \omega \in \hat{A}^c\cap \hat{\Omega };\\&\,\,\,\, 0,\ \ \,\,\,\,\,\,\,\,\,\,\,\, \text { otherwise. } \end{aligned}\right. $$

From (8.2.4), for any m, $n\in \mathbb {N}$ with $n>m$, $X^{(n)}(\omega )\equiv X^{(m)}(\omega )$ on $[0, \sigma _m(\omega )\wedge T]$ for any $\omega \in \hat{A}^c\cap \hat{\Omega }$ and $X^{(n)}(\omega )\equiv X^{(m)}(\omega )$ on [0, T] for all other $\omega$. Note that for $\omega \in \hat{A}^c\cap \hat{\Omega }$, we can find $m\in \mathbb {N}$, such that $\sigma _m(\omega )\wedge T=T$. Consequently, for any $\omega \in \Omega$, $\lim _{m\rightarrow \infty }X^{(m)}_t(\omega )$ exists for any t. From Lemma  8.2.7, it is not difficult to verify that choosing a different sequence of stopping times will only alter this limitation on the polar set. The details are left to the reader. Thus, the following definition is well posed.

Definition 8.2.10

Giving $\eta \in M^2_w([0, T])$, for any $(\omega , t)\in \Omega \times [0, T]$, we define

$$\begin{aligned} \int ^{t}_0\eta _sdB^\mathbf{a}_s(\omega ):=\lim _{m\rightarrow \infty }X^{(m)}_t(\omega ). \end{aligned}$$

(8.2.5)

For any $\omega \in \Omega$ and $t\in [0,\sigma _m]$, $\int ^{t}_0\eta _sdB^\mathbf{a}_s(\omega )=X^{(m)}_t(\omega )$, $0\le t\le T$. Since each of the processes $\{X^{(m)}_t\}_{0\le t\le T}$ has t-continuous paths, we conclude that the paths of $\left( \int ^{t}_0\eta _sdB^\mathbf{a}_s\right) _{0\le t\le T}$ are also t-continuous. The following theorem is an direct consequence of the above discussion.

Theorem 8.2.11

Assume that $\eta \in M^2_w([0, T])$. Then the stochastic process $\int ^{\cdot }_0\eta _sdB^\mathbf{a}_s$ is a well-defined continuous process on [0, T].

For any $\eta \in M_{w}^{1}(0,T)$, the integrals $\int _{0}^{t}\eta _{s}d\langle B^\mathbf {a}\rangle _{s}$ and $\int _{0}^{t}\eta _{s}d\langle B\rangle _s^{ij}$ are both well-defined continuous stochastic processes on [0, T] by a similar analysis.

3 Itô’s Formula for General $C^2$ Functions

The objective of this section is to give a very general form of Itô’s formula with respect to G-Brownian motion, which is comparable with that from the classical Itô’s calculus.

Consider the following G-Itô diffusion process:

$$ X_{t}^{\nu }=X_{0}^{\nu }+\int _{0}^{t}\alpha _{s}^{\nu }ds+\int _{0}^{t}\eta _{s}^{\nu ij}d\left\langle B\right\rangle ^{ij} _{s}+\int _{0}^{t}\beta _{s}^{\nu j}dB_{s}^{j}. $$

Lemma 8.3.1

Suppose that $\Phi \in C^{2}(\mathbb {R}^{n})$ and that all first and second order derivatives of $\Phi$ are in $C_{b, Lip}(\mathbb {R}^n)$. Let $\alpha ^{\nu }$, $\beta ^{\nu j}$ and $\eta ^{\nu ij}$, $\nu =1,\cdots , n$, $i, j=1,\cdots , d$, be bounded processes in $M_{*}^{2}(0,T)$. Then for any $t\ge 0$, we have in $L^2_*(\Omega _t)$,

$$\begin{aligned} \Phi (X_{t})-\Phi (X_{0})&=\int _{0}^{t}\partial _{x^{\nu }}\Phi (X_{u})\beta _{u}^{\nu j}dB_{u}^{j}+\int _{0}^{t}\partial _{x^{\nu }}\Phi (X_{u})\alpha _{u}^{\nu }du\\&\ \ \ +\int _{0}^{t}[\partial _{x^{\nu }}\Phi (X_{u})\eta _{u}^{\nu ij}+\frac{1}{2}\partial _{x^{\mu }x^{\nu }}^{2}\Phi (X_{u})\beta _{u}^{\mu i}\beta _{u}^{\nu j}]d\left\langle B\right\rangle ^{ij} _{u}.\nonumber \end{aligned}$$

(8.3.1)

The proof is parellel to that of Proposition 6.3, in Chap. 3. The details are left as an exercise for the readers.

Lemma 8.3.2

Suppose that $\Phi \in C^{2}(\mathbb {R}^{n})$ and all first and second order derivatives of $\Phi$ are in $C_{b, Lip}(\mathbb {R}^n)$. Let $\alpha ^{\nu }$, $\beta ^{\nu j}$ be in $M_{*}^{1}(0,T)$ and $\eta ^{\nu ij}$ belong to $M_{*}^{2}(0,T)$ for $\nu =1,\cdots , n$, $i, j=1,\cdots , d$. Then for any $t\ge 0$, relation (8.3.1) holds in $L^1_*(\Omega _t)$.

Proof

For simplicity, we only deal with the case $n=d=1$. Let $\alpha ^{(k)}$, $\beta ^{(k)}$ and $\eta ^{(k)}$ be bounded processes such that, as $k\rightarrow \infty$,

$$ \alpha ^{(k)}\rightarrow \alpha ,\,\, \eta ^{(k)}\rightarrow \eta \text { in }M_{*} ^{1}(0,T)\,\, \text { and }\,\, \beta ^{(k)}\rightarrow \beta \text { in }M_{*}^{2}(0,T) $$

and let

$$ X_{t}^{(k)}=X_{0}+\int _{0}^{t}\alpha _{s}^{(k)}ds+\int _{0}^{t}\eta _{s} ^{(k)}d\langle B\rangle _{s}+\int _{0}^{t}\beta _{s}^{(k)}dB_{s}. $$

Then applying Hölder’s inequality and BDG inequality yields that

$$ \lim _{k\rightarrow \infty }\mathbb {\hat{E}}[\sup _{0\le t\le T}|X_{t}^{(k)} -X_{t}|]=0 \ \ \text {and} \ \ \lim _{k\rightarrow \infty }\mathbb {\hat{E}}[\sup _{0\le t\le T}|\Phi (X_{t}^{(k)})-\Phi (X_{t}) |]=0. $$

Note that

$$\begin{aligned}&\mathbb {\hat{E}}\left[ \int _{0}^{T}|\partial _{x}\Phi (X_{t}^{(k)})\beta _{t} ^{(k)}-\partial _{x}\Phi (X_{t})\beta _{t}|^{2}dt \right] \\&\ \ \ \le 2\mathbb {\hat{E}} \left[ \int _{0}^{T}|\partial _{x}\Phi (X_{t}^{(k)})\beta _{t}^{(k)}-\partial _{x}\Phi (X_{t}^{(k)})\beta _{t}|^{2}dt\right] \\&\ \ \ \ \ \ \ +2\mathbb {\hat{E}}\left[ \int _{0}^{T}|\partial _{x}\Phi (X_{t}^{(k)})\beta _{t}-\partial _{x}\Phi (X_{t})\beta _{t}|^{2}dt\right] \\&\ \ \ \le 2C^2\mathbb {\hat{E}}\left[ \int _{0}^{T}|\beta _{t}^{(k)}-\beta _{t}|^{2}dt\right] +2\mathbb {\hat{E}}\left[ \int _{0}^{T}|\beta _{t}|^{2}|\partial _{x}\Phi (X_{t}^{(k)})-\partial _{x}\Phi (X_{t})|^{2}dt\right] , \end{aligned}$$

where C is the upper bound of $\partial _x\Phi$. Since $\sup _{0\le t\le T}|\partial _{x}\Phi (X_{t}^{(k)} )-\partial _{x}\Phi (X_{t})|^{2}\le 4C^2$, we conclude that

$$ \mathbb {\hat{E}}\left[ \int _{0}^{T}|\partial _{x}\Phi (X_{t}^{(k)})-\partial _{x}\Phi (X_{t})|^2dt\right] \rightarrow 0\text {, as }k\rightarrow \infty . $$

Thus we can apply Proposition 8.2.3 to prove that, in $M_{*}^{2}(0,T)$, as $k\rightarrow \infty$,

$$\begin{aligned} \partial _{x}&\Phi (X^{(k)})\beta ^{(k)}\rightarrow \partial _{x}\Phi (X)\beta ,\,\,\,\, \partial _{x}\Phi (X^{(k)})\alpha ^{(k)}\rightarrow \partial _{x}\Phi (X)\alpha , \\&\partial _{x}\Phi (X^{(k)})\eta ^{(k)}\rightarrow \partial _{x}\Phi (X)\eta , \,\,\,\, \partial _{xx}^{2}\Phi (X^{(k)})(\beta ^{(k)})^{2}\rightarrow \partial _{xx}^{2} \Phi (X)\beta ^{2}. \end{aligned}$$

However, from the above lemma we have

$$\begin{aligned} \Phi (X_{t}^{(k)})-\Phi (X_{0}^{(k)})&=\int _{0}^{t}\partial _{x} \Phi (X_{u}^{(k)})\beta _{u}^{(k)}dB_{u}+\int _{0}^{t}\partial _{x} \Phi (X_{u}^{(k)})\alpha _{u}^{(k)}du\\&+\int _{0}^{t}[\partial _{x}\Phi (X_{u}^{(k)})\eta _{u}^{(k)}+\tfrac{1}{2}\partial _{xx}^{2}\Phi (X_{u}^{(k)})(\beta _{u}^{(k)})^{2}]d\langle B\rangle _{u}. \end{aligned}$$

Therefore passing to the limit on both sides of this equality, we obtain the desired result.    $\square$

Lemma 8.3.3

Let X be given as in Lemma 8.3.2 and let $\Phi \in C^{1,2}([0,T]\times \mathbb {R}^{n})$ be such that $\Phi$, $\partial _{t}\Phi$, $\partial _{x}\Phi$ and $\partial _{xx}^{2}\Phi$ are bounded and uniformly continuous on $[0,T]\times \mathbb {R}^{n}$. Then we have the following relation in $L^1_*(\Omega _t)$:

$$\begin{aligned} \Phi (t, X_{t})-\Phi (0,X_{0})&=\int _{0}^{t}\partial _{x^{\nu }}\Phi (u, X_{u})\beta _{u}^{\nu j}dB_{u}^{j}+\int _{0}^{t}[\partial _t\Phi (u,X_{u})+\partial _{x^{\nu }}\Phi (u, X_{u})\alpha _{u}^{\nu }]du\\&\ \ \ +\int _{0}^{t}[\partial _{x^{\nu }}\Phi (u, X_{u})\eta _{u}^{\nu ij}+\tfrac{1}{2}\partial _{x^{\mu }x^{\nu }}^{2}\Phi (u, X_{u})\beta _{u}^{\mu i}\beta _{u}^{\nu j}]d\left\langle B\right\rangle ^{ij} _{u}. \end{aligned}$$

Proof

Choose a sequence of functions $\{ \Phi _{k}\}_{k=1}^{\infty }$ such that, $\Phi _{k}$ and all its first order and second order derivatives are in $C_{b, Lip}([0,T]\times \mathbb {R}^{n})$. Moreover, as $n\rightarrow \infty$, $\Phi _{n}$, $\partial _{t}\Phi _{n}$, $\partial _{x}\Phi _{n}$ and $\partial _{xx}^{2}\Phi _{n}$ converge respectively to $\Phi$, $\partial _{t}\Phi$, $\partial _{x}\Phi$ and $\partial _{xx} ^{2}\Phi$ uniformly on $[0,T]\times \mathbb {R}$. Then we use the above Itô’s formula to $\Phi _{k}(X_{t}^{0}, X_{t})$, with $Y_{t}=(X_{t} ^{0}, X_{t})$, where $X_{t}^{0}\equiv t$:

$$\begin{aligned} \Phi _k(t, X_{t})-\Phi _k(0,X_{0})&=\int _{0}^{t}\partial _{x^{\nu }}\Phi _k(u, X_{u})\beta _{u}^{\nu j}dB_{u}^{j}+\int _{0}^{t}[\partial _t\Phi _k(u,X_{u})+\partial _{x^{\nu }}\Phi _k(u, X_{u})\alpha _{u}^{\nu }]du\\&\ \ \ +\int _{0}^{t}[\partial _{x^{\nu }}\Phi _k(u, X_{u})\eta _{u}^{\nu ij}+\tfrac{1}{2}\partial _{x^{\mu }x^{\nu }}^{2}\Phi _k(u, X_{u})\beta _{u}^{\mu i}\beta _{u}^{\nu j}]d\left\langle B\right\rangle ^{ij} _{u}. \end{aligned}$$

It follows that, as $k\rightarrow \infty$, the following uniform convergences:

$$\begin{aligned} |\partial _{x_{\nu }}\Phi _{k}(u,X_{u})-\partial _{x_{\nu }}\Phi (u, X_{u})|&\rightarrow 0\text {, }|\partial _{x_{\mu }x_{\nu }}^{2}\Phi _{k}(u, X_{u} )-\partial _{x_{\mu }x_{\nu }}^{2}\Phi _{k}(u, X_{u})|\rightarrow 0\text {,}\\ |\partial _{t}\Phi _{k}(u,X_{u})-\partial _{t}\Phi (u, X_{u})|&\rightarrow 0\text {. } \end{aligned}$$

Sending $k\rightarrow \infty$, we arrive at the desired result.    $\square$

Theorem 8.3.4

Suppose $\Phi \in C^{1,2}([0,T]\times \mathbb {R}^n)$. Let $\alpha ^\nu ,\eta ^{\nu i j}$ be in $M_w^{1}(0,T)$ and $\beta ^{\nu j}$ be in $M_w^{2}(0,T)$ associated with a common stopping time sequence $\{\sigma _m\}_{m=1}^\infty$. Then for any $t\ge 0$, we have q.s.

$$\begin{aligned} \Phi (t, X_{t})-\Phi (0,X_{0})&=\int _{0}^{t}\partial _{x^{\nu }}\Phi (u, X_{u})\beta _{u}^{\nu j}dB_{u}^{j}+\int _{0}^{t}[\partial _t\Phi (u,X_{u})+\partial _{x^{\nu }}\Phi (u, X_{u})\alpha _{u}^{\nu }]du\\&\ \ \ +\int _{0}^{t}[\partial _{x^{\nu }}\Phi (u, X_{u})\eta _{u}^{\nu ij}+\tfrac{1}{2}\partial _{x^{\mu }x^{\nu }}^{2}\Phi (u, X_{u})\beta _{u}^{\mu i}\beta _{u}^{\nu j}]d\left\langle B\right\rangle ^{ij} _{u}. \end{aligned}$$

Proof

For simplicity, we only deal with the case $n = d = 1$. We set, for $k=1,2,\cdots ,$

$$ \tau _{k}:=\inf \{t\ge 0|\,\,\,\, |X_{t}-X_{0}|>k\} \wedge \sigma _{k}. $$

Let $\Phi _{k}$ be a $C^{1,2}$-function on $[0,T]\times \mathbb {R}^{n}$ such that $\Phi _k$, $\partial _{t}\Phi _k$, $\partial _{x_{i}}\Phi _k$ and $\partial _{x_{i}x_{j}}^{2}\Phi _k$ are uniformly bounded continuous functions satisfying $\Phi _{k}=\Phi$, for $|x|\le 2k$, $t\in [0,T]$. It is clear that the process $\mathbf{1}_{[0,\tau _{k}]}\beta$ is in $M_{w}^{2}(0,T)$, while $\mathbf{1}_{[0,\tau _{k}]}\alpha$ and $\mathbf{1}_{[0,\tau _{k}]}\eta$ are in $M_{w} ^{1}(0,T)$ and they are all associated to the same sequence of stopping times $\{\tau _k\}_{k=1}^\infty$. We also have

$$ X_{t\wedge \tau _{k}}=X_{0}+\int _{0}^{t}\alpha _{s}{} \mathbf{1} _{[0,\tau _{k}]}ds+\int _{0}^{t}\eta _{s}{} \mathbf{1}_{[0,\tau _{k}]}d\langle B\rangle _{s}+\int _{0}^{t}\beta _{s}{} \mathbf{1}_{[0,\tau _{k}]}dB_{s} $$

Then we can apply Lemma 8.3.3 to $\Phi _k(s, X_{s\wedge \tau _{k}})$, $s\in [0,t]$, to obtain

$$\begin{aligned}&\Phi (t, X_{t\wedge \tau _{k}})-\Phi (0,X_{0})\\&\ \ =\int _{0}^{t} \partial _{x}\Phi (u, X_{u})\beta _{u}{} \mathbf{1}_{[0,\tau _{k}]} dB_{u}+\int _{0}^{t}[\partial _{t}\Phi (u,X_{u})+\partial _{x} \Phi (u, X_{u})\alpha _{u}]\mathbf{1}_{[0,\tau _{k}]}du\\&\ \ \ \ \ +\int _{0}^{t}[\partial _{x}\Phi (u, X_{u})\eta _{u}{} \mathbf{1} _{[0,\tau _{k}]}+\tfrac{1}{2}\partial _{xx}^{2}\Phi (u, X_{u})|\beta _{u}|^2\mathbf{1}_{[0,\tau _{k}]}]d\langle B\rangle _{u}. \end{aligned}$$

Letting $k\rightarrow \infty$ and noticing that $X_t$ is continuous in t, we get the desired result.

   $\square$

Example 8.3.5

For given $\varphi \in C^{2}(\mathbb {R})$, we have

$$ \Phi (B_{t})-\Phi (B_{{0}})=\int _{{0}}^{t}\Phi _{x}(B_{s} )dB_{s}+\tfrac{1}{2}\int _{0}^{t}\Phi _{xx}(B_{s})d\left\langle B\right\rangle _{s}. $$

This generalizes the previous results to more general situations.

Notes and Comments

The results in this chapter were mainly obtained by Li and Peng [110, 2011] .  Li and Lin [109, 2013] found a point of incompleteness and proposed to use a more essential condition (namely, Condition (ii) in Definition 8.2.8) to replace the original one which was $\int _0^T|\eta _t|^pdt<\infty$, q.s.

A difficulty hidden behind is that the G-expectation theory is mainly based on the space of random variables $X = X(\omega )$ which are quasi-continuous with respect to the G-capacity $\hat{c}$. It is not yet clear that the martingale properties still hold for random variables without quasi-continuity condition.

There are still several interesting and fundamentally important issues on G-expectation theory and its applications. It is known that stopping times play a fundamental role in classical stochastic analysis. However, it is often nontrivial to directly apply stopping time techniques in a G-expectation space. The reason is that the stopped process may not belong to the class of processes which are meaningful in the G-framework. Song [160]  considered the properties of hitting times for G- martingale and, moreover the stopped processes. He proved that the stopped processes for G-martingales are still G-martingales and that the hitting times for symmetric G-martingales with strictly increasing quadratic variation processes are quasi-continuous. Hu and Peng [82]  introduced a suitable definition of stopping times and obtained the optional stopping theorem.