Harnack and log Harnack Inequalities for $G$-SDEs with Multiplicative Noise

TL;DR

This paper establishes Harnack and log Harnack inequalities for G-SDEs with multiplicative noise, extending classical results to the nonlinear G-expectation framework and providing gradient estimates.

Contribution

It introduces new inequalities for G-SDEs with multiplicative noise, extending existing linear expectation results to the nonlinear G-expectation setting.

Findings

Derived Harnack inequalities for G-SDEs with multiplicative noise

Extended inequalities to the nonlinear G-expectation framework

Generalized gradient estimates for these equations

Abstract

The Harnack and log Harnack inequalities for stochastic differential equation driven by $G$-Brownian motion with multiplicative noise are derived by means of coupling by change of mesure. All of the above results extend the existing ones in the linear expectation setting. Moreover, the gradient estimate generalize the nonlinear results appeared in [11].

Equations

$$\text{d}{X}_t=b(t,X_t)\text{d}t+\sum _{i,j=1}^d{h}_{ij}(t,X_t)\text{d}⟨B^i,B^j{⟩}_t+\sum _{i=1}^d{\sigma }_i(t,X_t)\text{d}{B}_t^i,$$

$${\Omega }_T=\{\omega |[0,T]\ni t\mapsto {\omega }_t\in {\mathbb{R}}^d\text{mbox}iscontinuouswith\omega (0)=0\}$$

$$L_{ip}({\Omega }_T)=\{\phi (B_{t_1},\cdots ,B_{t_n}),n\in \mathbb{N},t_1,\cdot \cdot \cdot ,t_n\in [0,T],\phi \in C_{b,lip}({\mathbb{R}}^d\otimes {\mathbb{R}}^n)\},$$

$$\xi (\omega )=\phi (\omega (t_1),\cdot \cdot \cdot ,\omega (t_n)),0=t_0<t_1<\cdot \cdot \cdot <t_n=T,$$

$${\bar{\mathbb{E}}}_t[\xi ]:=u_k(t,\omega (t);\omega (t_1),\cdot \cdot \cdot ,\omega (t_{k-1})),\xi \in L_{ip}({\Omega }_T),t\in [t_{k-1},t_k),k=1,\cdot \cdot \cdot ,n,$$

$$\{\begin{array}{l}{\partial }_t{u}_k+G({\partial }_x^2{u}_k)=0,(t,x)\in [t_{k-1},t_k)\times {\mathbb{R}}^d,k=1,\cdot \cdot \cdot ,n,\\ u_k(t_k,x;x_1,\cdot \cdot \cdot ,x_{k-1})=u_{k+1}(t_k,x;x_1,\cdot \cdot \cdot ,x_{k-1},x_k),k=1,\cdot \cdot \cdot ,n-1,\\ u_n(t_n,x;x_1,\cdot \cdot \cdot ,x_{n-1})=\phi (x_1,\cdot \cdot \cdot ,x_{n-1},x),k=n.\end{array}$$

$$G(A)=\frac{1}{2}\underset{Q\in \Gamma }{\sup }\text{mbox}trace[AQ],A\in {\mathbb{S}}^d.$$

$$G(A)=\frac{1}{2}\underset{\gamma \in [\underline{\sigma },\bar{\sigma }]}{\sup }\text{mbox}trace({\gamma }^{2}A),A\in {\mathbb{S}}^d.$$

$$\bar{\mathbb{E}}[\xi ]=\underset{P\in \mathcal{P}}{\sup }{\mathbb{E}}_P[\xi ]forall\xi \in L_G^1({\Omega }_T).$$

$$c(A)=\underset{P\in \mathcal{P}}{\sup }P(A),A\in \mathcal{B}({\Omega }_T).$$

$${\mathcal{P}}_M:=\{P_h|P_h=P^0\circ X^{-1},X_t={\int }_0^t{h}_s\text{d}{W}_s,h_s\in L_{F^0}^2([0,T];{\Gamma }^{\frac{1}{2}})\}$$

$$\{h|hisanprogressivemeasurableprocessw.r.t.F^{0}and\underline{\sigma }\le |h_s|\le \bar{\sigma }\}.$$

$$L_G^p({\Omega }_T)=\{X\in L^0({\Omega }_T)|\underset{N\to \infty }{\lim }\bar{\mathbb{E}}[|X{|}^p{1}_{|X|\ge N}]=0andXhasaquasi-continuousversion\},$$

$$M_{G}^{p,0}([0,T])=\Bigg{\{}\eta_{t}~{}|~{}\eta_{t}=\sum_{j=0}^{N-1}\xi_{j}I_{[t_{j},t_{j+1})},\xi_{j}\in L_{G}^{p}(\Omega_{t_{j}}),N\in\mathbb{N},\ 0=t_{0}<t_{1}<\cdots<t_{N}=T\Bigg{\}}.$$

$$\Vert \eta {\Vert }_{M_G^p([0,T])}={\left[\bar{\mathbb{E}}\left({\int }_0^T|{\eta }_t{|}^{p}dt\right)\right]}^{\frac{1}{p}},\Vert \eta {\Vert }_{H_G^p([0,T])}={\left[\bar{\mathbb{E}}{\left({\int }_0^T|{\eta }_t{|}^{2}dt\right)}^{\frac{p}{2}}\right]}^{\frac{1}{p}},$$

$${\bar{\mathbb{E}}}_s[X_t]=X_{s}int\in [0,T].$$

$$⟨B{⟩}_t=B_t^i{B}_t^j-{\int }_0^t{B}_s^i\text{d}{B}_s^j-{\int }_0^t{B}_s^j\text{d}{B}_s^i.$$

$$M_t=\exp \left\{{\int }_0^t⟨{\eta }_s,\text{d}{B}_s⟩-\frac{1}{2}{\int }_0^t⟨{\eta }_s,(\text{d}⟨B{⟩}_s{\eta }_s)⟩\right\},$$

$${\hat{B}}_t=B_t-{\int }_0^t(\text{d}⟨B{⟩}_s{\eta }_s),t\in [0,T],$$

$$\bar{\mathbb{E}}\left[\exp \left\{\left(\frac{1}{2}+{ϵ}_0\right){\int }_0^T⟨{\eta }_s,(\text{d}⟨B{⟩}_s{\eta }_s)⟩\right\}\right]<\infty ,$$

$$\gamma \ge {\sigma }_0{I}_{d}forall\gamma \in \Gamma ,$$

$$\hat{\mathbb{E}}[X]=\bar{\mathbb{E}}[X{M}_T],X\in {\hat{L}}_{ip}({\Omega }_T),$$

$$P\{⟨\hat{B}{⟩}_t\ne ⟨B{⟩}_t,t\in [0,T]\}=0.$$

$$c\{⟨\hat{B}{⟩}_t\ne ⟨B{⟩}_t,t\in [0,T]\}=\underset{P\in \mathcal{P}}{\sup }P\{⟨\hat{B}{⟩}_t\ne ⟨B{⟩}_t,t\in [0,T]\}=0,$$

$$\Phi (\bar{P}f(x)))\le \bar{P}\Phi (f(y))e^{\Psi (x,y)},x,y\in {\mathbb{R}}^d,f\in {\mathcal{B}}_b^{+}({\mathbb{R}}^d),$$

$$|b(t,x)-b(t,y)|+|h(t,x)-h(t,y)|+|\sigma (t,x)-\sigma (t,y)|\le K|x-y|,x,y\in \mathbb{R},t>0.$$

$${\bar{P}}_{T}f(x)=\bar{\mathbb{E}}f(X_T^x),f\in C_b^{+}(\mathbb{R}),$$

$${\bar{P}}_T\log f(y)\le \log {\bar{P}}_{T}f(x)+\frac{K\left(2+K+\frac{2}{{\underline{\sigma }}^2}\right)|x-y{|}^2}{2\frac{{\kappa }_1^6}{{\kappa }_2^4}(1-{\text{e}}^{-{\underline{\sigma }}^{2}K\left(2+K+\frac{2}{{\underline{\sigma }}^2}\right)T})}.$$

$$({\bar{P}}_{T}f(y){)}^p\le {\bar{P}}_T{f}^p(x)\exp \left\{\frac{\sqrt{p}(\sqrt{p}-1)K\left(2+K+\frac{2}{{\underline{\sigma }}^2}\right)|x-y{|}^2}{4({\kappa }_2-{\kappa }_1)[{\kappa }_1(\sqrt{p}-1)-C](1-{\text{e}}^{-{\underline{\sigma }}^{2}K\left(2+K+\frac{2}{{\underline{\sigma }}^2}\right)T})}\right\},$$

$$\{X\in \mathcal{H}|(\bar{\mathbb{E}}[|\cdot |])<\infty \},$$

$${\mathcal{H}}^{ext}=\{X\in mF|\min \{{\mathbb{E}}_{P_h}[X^{+}],{\mathbb{E}}_{P_h}[X^{-}]<\infty \}forallh\in L_{F^0}^2([0,T];{\Gamma }^{\frac{1}{2}})\},$$

Full text

Keywords: Harnack inequaity; gradient estimate; multiplicative noise; $G$-Brownian motion; SDEs.

1 Introduction

For the extensive applications in strong Feller property, uniqueness of invariant probability measures, functional inequalities, and heat kernal estimates, Wang’s Harnack inequality has been developed [20]. To establish Harnack inequality, Wang introduced the coupling by change of measures, see [1, 18, 19] and references within for details. However, up to now, most of these papers only focus on the case of linear expectation spaces. Song [14] firstly derived the gradient estimates for nonlinear diffusion semigroups by using the method of Wang’s coupling by change of measure, after Peng [10, 11] established the systematic theory of $G$-expectation theory, $G$-Brownian motion and stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs, in short). Subsequently, Yang [21] generalized the theory of Wang’s Harnack inequality and its applications to nonlinear expectation framework, where the noise is additive. Moreover, Wang’s Harnack inequality and gradient estimates are also proved for the degenerate (functional) case in [6]. An interesting question is whether it can be generalized to the form of multiplicative noise. The answer is positive as some of the results are showed in [14], whereas neither the form of $G$-SDEs with the term of $\text{\rm{d}}\langle B^{i},B^{j}\rangle_{t}$, nor the Harnack inequality studied, where $B_{t}$ is a $d$-dimensional $G$-Brwonian motion, and $\langle B^{i},B^{j}\rangle_{t}$ stands for the mutual variation process of the $i$-th component $B^{i}_{t}$ and the $j$-th component $B^{j}_{t}$. In this paper, we will improve and extend the above assertions to the multiplicative noise. Consider the following $G$-SDE

[TABLE]

where $b,h_{ij}=h_{ji}:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ and $\sigma:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}\otimes\mathbb{R}^{d}$. We aim to establish the Harnack inequality for the $G$-SDE (1.1). In addition, we also prove the gradient estimate. To this end, we firstly recall some basic facts on the $G$-expectation and $G$-Brownian motion.

For a positive integer $d$, let $(\mathbb{R}^{d},\langle\cdot,\cdot\rangle,|\cdot|)$ be the $d$-dimensional Euclidean space, $\mathbb{S}^{d}$ the collection of all symmetric $d\times d$-matrices. For any fixed $T>0$,

[TABLE]

endowed with the uniform form. Let $B_{t}(\omega)=\omega_{t},\omega\in\Omega_{T},$ be the canonical process. Set

[TABLE]

where $C_{b,lip}(\mathbb{R}^{d}\otimes\mathbb{R}^{n})$ denotes the set of bounded Lipschitz functions. Let $G:\mathbb{S}^{d}\rightarrow\mathbb{R}$ be a monotonic, sublinear and homogeneous function; see e.g. [12, p16]. Now we give the construction of $G$-expectation which is also used in [13]. For any $\xi\in L_{ip}(\Omega_{T})$, i.e.,

[TABLE]

the conditional $G$-expectation is defined by

[TABLE]

where $(t,x)\mapsto u_{k}(t,x;x_{1},\cdot\cdot\cdot,x_{k-1})$, $k=1,\cdot\cdot\cdot,n$, solves the following $G$-heat equation

[TABLE]

The corresponding $G$-expectation of $\xi$ is defined by $\bar{\mathbb{E}}[\xi]=\bar{\mathbb{E}}_{0}[\xi]$.

According to [12], there exists a bounded, convex, and closed subset $\Gamma\subset\mathbb{S}^{d}_{+}$ such that

[TABLE]

In particular, fix $\underline{\sigma},\overline{\sigma}\in\mathbb{S}_{+}^{d}$ with $\underline{\sigma}<\overline{\sigma}$, let $\Gamma=[\underline{\sigma}^{2},\bar{\sigma}^{2}],$ then

[TABLE]

Denote $L_{G}^{p}(\Omega_{T})$ be the completion of $L_{ip}(\Omega_{T})$ under the norm $(\bar{\mathbb{E}}[|\cdot|^{p}])^{\frac{1}{p}}$, $p\geq 1.$

Theorem 1.1**.**

([3, 12]) There exists a weakly compact subset $\mathcal{P}\subset M_{1}(\Omega_{T})$, the set of probability measures on $(\Omega_{T},\mathcal{B}(\Omega_{T}))$, such that

[TABLE]

$\mathcal{P}$* is called a set that represents $\bar{\mathbb{E}}$.*

Let $\mathcal{P}$ be a weakly compact set that represents $\bar{\mathbb{E}}$. For this $\mathcal{P}$, we define capacity

[TABLE]

$c$ defined here is independent of the choice of $\mathcal{P}$.

Remark 1.2**.**

• (i)

Let $(\Omega^{0},\mathcal{F}^{0},P^{0})$ be a probability space and $\{W_{t}\}$ be a $d$-dimensional Brownian motion under $P^{0}$. Let $F^{0}=\{\mathcal{F}^{0}_{t}\}_{t\geq 0}$ be the augmented filtration generated by W. **[3]** proved that

[TABLE]

is a set that represents $\bar{\mathbb{E}}$, where $\Gamma^{\frac{1}{2}}:=\{\gamma^{\frac{1}{2}}\ |\ \gamma\in\Gamma\},$ is the set in the representation of $G(\cdot)$ in the formula (1.3) and $L^{2}_{F^{0}}([0,T];\Gamma^{\frac{1}{2}})$ is the set of $F^{0}$-progressive measurable processes with values in $\Gamma^{\frac{1}{2}}$.

• (ii)

For the $1$-dimensional case, $L^{2}_{F^{0}}([0,T];\Gamma^{\frac{1}{2}})$ reduces to the form below:

[TABLE]

Definition 1.1**.**

We say a set $A\subset\Omega_{T}$ is $c$-polar if $c(A)=0$. A property holds quasi-surely ($c$-q.s. for short) if it holds outside a $c$-polar set.

Definition 1.2**.**

• (1)

We say that a map $\xi(\cdot):\Omega_{T}\rightarrow\mathbb{R}$ is quasi-continuous if for all $\epsilon>0$, there exists an open set $G$ with $c(G)<\epsilon$ such that $\xi(\cdot)$ is continuous on $G^{c}$.

• (2)

We say that a process $M_{\cdot}(\cdot):\Omega_{T}\times[0,T]\rightarrow\mathbb{R}$ is quasi-continuous if for all $\epsilon>0$, there exists an open set $G$ with $c(G)<\epsilon$ such that $M_{\cdot}(\cdot)$ is continuous on $G^{c}\times[0,T].$

• (3)

We say that a random variable $X:\Omega_{T}\rightarrow\mathbb{R}$ has a quasi-continuous version if there exists a quasi-continuous function $Y:\Omega_{T}\rightarrow\mathbb{R}$ such that $X=Y$, $c$-q.s.

Remark 1.3**.**

Note that a quasi-continuous process defined here is different from [5].

According to [3],

[TABLE]

where $L^{0}(\Omega_{T})$ denotes the space of all $\mathcal{B}(\Omega_{T})$-measurable real function.

In the paper, we discuss the property of distribution for the solution $X_{t}$ in (1.1), a polar set does not affect the result, so in the following parts, we did not distinguish the quasi-continuous version and itself any more.

Theorem 1.4**.**

(Monotone Convergence Theorem) [3, Theorem 10, Theorem 31] Let $\mathcal{P}$ be weakly compact that represents $\bar{\mathbb{E}}$.

• (1)

Suppose $\{X_{n}\}_{n\geq 1},X\in L^{0}(\Omega_{T}),$ $X_{n}\uparrow X,$ $c$-q.s. and $\mathbb{E}_{P}[X_{1}^{-}]<\infty$ for all $P\in\mathcal{P}$. Then $\bar{\mathbb{E}}[X_{n}]\uparrow\bar{\mathbb{E}}[X]$.

• (1)

Let $\{X_{n}\}_{n=1}^{\infty}\subset L_{G}^{1}(\Omega_{T})$ be such that $X_{n}\downarrow X,$ $c$-q.s.. Then $\bar{\mathbb{E}}[X_{n}]\downarrow\bar{\mathbb{E}}[X]$.

Remark 1.5**.**

We stress that in this theorem $X$ does not necessarily belong to $L_{G}^{1}(\Omega_{T})$.

Let

[TABLE]

For $p\geq 1$, let $M_{G}^{p}([0,T])$ and $H_{G}^{p}([0,T])$ be the completion of $M_{G}^{p,0}([0,T])$ under the following norm

[TABLE]

respectively. Denote by $[M_{G}^{p}([0,T])]^{d}$, $[H_{G}^{p}([0,T])]^{d}$ all $d$-dimensional stochastic processes $\eta_{t}=(\eta_{t}^{1},\cdot\cdot\cdot,\eta_{t}^{d}),$ $\xi_{t}=(\xi_{t}^{1},\cdot\cdot\cdot,\xi_{t}^{d})$, $t\geq 0$ with $\eta_{t}^{i}\in M_{G}^{p}([0,T]),\xi_{t}^{i}\in H_{G}^{p}([0,T])$, respectively.

Definition 1.3**.**

A process $X=\{X_{t}|t\in[0,T]\}$ is called a $G$-martingale if for each $t\in[0,T]$, we have $X_{t}\in L^{1}_{G}(\Omega_{t})$ and

[TABLE]

We call $X$ a symmetric $G$-martingale if both $X$ and $-X$ are $G$-martingales.

Remark 1.6**.**

For $\eta\in M_{G}^{1}([0,T])$, it’s easy to see that the process $\int_{0}^{t}\eta_{s}(\omega)\text{\rm{d}}s$ has a $c$-quasi continuous version. Also, [15] shows that any G-martingale has a c-quasi continuous version.

Let $B_{t}$ be a $d$-dimensional $G$-Brownian motion, then $G(A)=\frac{1}{2}\bar{\mathbb{E}}[\langle AB_{1},B_{1}\rangle]$, $A\in\mathbb{S}^{d}$. In particular, for 1-dimensional $G$-Brownian motion $(B_{t})_{t\geq 0}$, one has $G(a)=(\overline{\sigma}^{2}a^{+}-\underline{\sigma}^{2}a^{-})/2,a\in\mathbb{R},$ where $\overline{\sigma}^{2}:=\bar{\mathbb{E}}[B^{2}_{1}]\geq-\bar{\mathbb{E}}[-B^{2}_{1}]=:\underline{\sigma}^{2}>0$.

Let $\langle B\rangle_{t}=(\langle B^{i},B^{j}\rangle_{t})_{1\leq i,j\leq d},0\leq t\leq T$, which is defined by

[TABLE]

To establish the Wang’s Harnack inequality, $G$-Girsanov’s transform plays a crucial role, the following results is taken from [9, 22]. For $\eta\in[M^{2}_{G}([0,T])]^{d}$, let

[TABLE]

where $({\text{\rm{d}}}\langle B\rangle_{s}\eta_{s})=\left(\sum_{j=1}^{d}\eta_{s}^{j}\text{\rm{d}}\langle B^{i},B^{j}\rangle_{s}\right)_{1\leq i\leq d}.$

Lemma 1.7**.**

([9, 22])* If $\eta\in[M^{2}_{G}([0,T])]^{d}$ satisfies $G$-Novikov’s condition, i.e., for some $\epsilon_{0}>0$, it holds that*

[TABLE]

then the process $M$ is a symmetric $G$-martingale.

Lemma 1.8**.**

([9]) ($G$-Girsanov’s formula)* Assume that there exists $\sigma_{0}>0$ such that*

[TABLE]

and that $M$ is a symmetric $G$-martingale on $(\Omega_{T},L^{1}_{G}(\Omega_{T}),\bar{\mathbb{E}}).$ Define a sublinear expectation $\hat{\mathbb{E}}$ by

[TABLE]

where $\hat{L}_{ip}(\Omega_{T}):=\{\varphi(\hat{B}_{t_{1}},\cdot\cdot\cdot,\hat{B}_{t_{n}}):n\in\mathbb{N},t_{1},\cdot\cdot\cdot,t_{n}\in[0,T],\varphi\in C_{b,lip}(\mathbb{R}^{d}\otimes\mathbb{R}^{n})\}$. Then $\hat{B}_{t}$ is a $G$-Brownian motion on the sublinear expectation space $(\Omega_{T},\hat{L}_{G}^{1}(\Omega_{T}),\hat{\mathbb{E}})$, where $\hat{L}_{G}^{1}(\Omega_{T})$ is the completion of $\hat{L}_{ip}(\Omega_{T})$ under the norm $\hat{\mathbb{E}}[|\cdot|]$.

Remark 1.9**.**

The Girsanov theorem also appeared in [4, Theorem 5.2].

Lemma 1.10**.**

For $\hat{B}$ in (1), then $c$-q.s., $\langle\hat{B}\rangle_{t}=\langle{B}\rangle_{t},$ $t\in[0,T].$

Proof.

For any $P\in\mathcal{P}$, it holds that

[TABLE]

By (1.6), we have

[TABLE]

which implies $c$-q.s., $\langle\hat{B}\rangle_{t}=\langle{B}\rangle_{t},$ $t\in[0,T].$ ∎

We aim to establish the following Harnack-type inequality introduced by Feng-Yu Wang:

[TABLE]

where $\Phi$ is a nonnegative convex function on $[0,\infty)$ and $\Psi$ is a nonnegative function on $\mathbb{R}^{d}\times\mathbb{R}^{d}$. In the setting of $G$-SDEs, we establish this type inequality for the associated nonlinear Markov operator $\bar{P}_{T}$. For simplicity, we consider the case of $d=1$, but our results and methods still hold for the case $d>1$. To get our desired results, we give following assumptions on $b,\sigma,$ and $h$ in (1.1).

• (H1)

There exists a constant $K>0$, such that

[TABLE]

• (H2)

There exist $\kappa_{1},\kappa_{2}$ with $\kappa_{2}\geq\kappa_{1}>0$, such that $\kappa_{1}\leq\sigma(t,x)\leq\kappa_{2},$ $x\in\mathbb{R}$, $t>0.$

From [12, Theorem 1.2], under the assumption of (H1), for any $x\in\mathbb{R}$, (1.1) has a unique solution in $M_{G}^{2}([0,T])$. In what follows, for $T>0$, we define

[TABLE]

where $X_{T}^{x}$ solves (1.1) with initial value $x$.

Remark 1.11**.**

In order to ensure the term $f(X_{T}^{x})\in M_{G}^{2}([0,T])$, we always assume $f\in C_{b}^{+}(\mathbb{R})$.

The remainder of the paper is organized as follows. In Section 2, we characterize the quasi-continuity of hitting time for processes of certain forms. Finally, in Section 3 we present the Harnack and $\log$ Harnack inequalities for $G$-SDE (1.1), so that main results in [18, Theorem 3.4.1, Chap.3] are extended to the present $G$-setting. Moreover, the gradient estimate is showed in this section.

2 Main Results

Now we turn to the main result of this section.

2.1 Harnack and log-Harnack inequalities

Theorem 2.1**.**

Assume (H1)-(H2).

• (1)

For any nonnegative $f\in C_{b}^{+}(\mathbb{R})$ and $T>0,x,y\in\mathbb{R},$ it holds that

[TABLE]

• (2)

For $p>(1+\frac{\kappa_{2}^{3}-\kappa_{1}\kappa_{2}^{2}}{\kappa_{1}^{3}})^{2}$, then

[TABLE]

holds for any $x,y\in\mathbb{R}$ and $f\in C_{b}^{+}(\mathbb{R})$.

To make the proof easy to follow, let us divide the proof into the following aspects.

2.1.1 Martingale convergence

To apply $G$-Girsanov’s formula in Lemma 1.8, we need to check that $M$ is a symmetric $G$-martingale. From Lemma 1.7, we know that $G$-Novikov’s condition is a sufficient condition for $M$ to be a symmetric $G$-martingale. However, if we take this for calculation, the assumptions we impose on $\kappa_{1},\kappa_{2}$ are too strong, thus, we propose the notion of uniform integrability under a nonlinear expectation [2]. We would like to point out [2] discusses the martingale convergence in discrete time, for simplicity, we still use $\bar{\mathbb{E}}$ in this paper instead of the notion in [2].

We define the space $L^{1}$ as the completion under $(\bar{\mathbb{E}}[|\cdot|])$ of the set

[TABLE]

where $\mathcal{H}$ be a vector lattice of real valued functions defined on $\Omega$, namely $c\in\mathcal{H}$ for each constant $c$ and $|X|\in\mathcal{H}$ if $X\in\mathcal{H}$.

Definition 2.1**.**

Let $K\subset L^{1}$. $K$ is said to be uniformly integrable (u.i.) if $\bar{\mathbb{E}}(1_{\{|X|\geq c\}}|X|)$ converges to 0 uniformly in $X\in K$ as $c\rightarrow\infty$.

Lemma 2.2**.**

([2, Corollary 3.1.1]) Let $K\subset L^{1}$. Suppose there is a positive function $f$ defined on $[0,\infty[$ such that $\lim_{t\rightarrow\infty}t^{-1}f(t)=\infty$ and $\sup_{X\in K}\bar{\mathbb{E}}(f\circ|X|)<\infty$. Then $K$ is uniformly integrable.

Let

[TABLE]

where $mF$ is the space of $F_{T}$-measurable $\mathbb{R}\cup{\pm\infty}$-valued functions. According to [2],

[TABLE]

This does not need to restrict our attention to those random variables admitting a quasi-continuous version compared with the structure of $L_{G}^{p}(\Omega_{T})$. It’s clear that $L_{G}^{p}(\Omega_{T})\subset L_{b}^{p}.$

Lemma 2.3**.**

([2, Theorem 3.2]) Suppose $(X_{n})_{n\geq 1}\subset L_{b}^{1}$, and $X\in\mathcal{H}^{ext}$. Then $X_{n}$ converge in $L^{1}$ norm to $X$ if and only if the collection $(X_{n})_{n\geq 1}$ is uniformly integrable and the $X_{n}$ converge in capacity to $X$. Furthermore, in this case, the collection $(X_{n})_{n\geq 1}\cup{X}$ is also uniformly integrable and $X\in L_{b}^{1}.$

Lemma 2.4**.**

([2, Theorem 4.4]) Let $(X_{n})_{n\geq 1}$ be a $G$-submartingale with $\sup_{k}\bar{\mathbb{E}}(|X_{k}|)<\infty$. Then $X_{n}\rightarrow X_{\infty}\in H^{ext}$, q.s..

Lemma 2.5**.**

([2, Theorem 4.5]) Let $(X_{n})_{n\geq 1}$ be a uniformly integrable $G$-submartingale. Then taking $X_{\infty}=lim_{n\rightarrow\infty}X_{n}$, the process $(X_{n})_{n\geq 1\cup\infty}$ is also a uniformly integrable $G$-submartingale. In particular, this implies that $X_{\infty}\in L^{1}_{b}$.

In the following, we aim to extend the convergence theorem for $G$-martingale from discrete time to continuous time.

Theorem 2.6**.**

Let $(X_{s})_{s\in[0,T)}\subset L_{G}^{1}(\Omega_{T})$ be a uniformly integrable $G$-martingale. Then taking $X_{T}=lim_{t\rightarrow T}X_{t}$, the process $(X_{s})_{s\in[0,T]}$ is also a uniformly integrable $G$-martingale. In particular, this implies that $X_{T}\in L_{G}^{1}(\Omega_{T})$.

Proof.

Since $\{X_{T-\frac{T}{n}}\}_{n=1}^{\infty}$ is a sequence of discrete martingale, we have

[TABLE]

For any $s\in[0,T)$, there exists a $n\geq 1$, such that $T-\frac{T}{n}>s$. Moreover,

[TABLE]

where the last step by using the fact that $(X_{s})_{s\in[0,T)}$ is $G$-martingale. This implies that $(X_{s})_{s\in[0,T]}$ is $G$-martingale. Moreover, the collection $(X_{T-\frac{T}{n}})_{n\geq 1}$ is uniformly integrable and the $X_{T-\frac{T}{n}}$ converge in capacity to $X_{T}$, then the $X_{t}$ converge to $X_{T}$ in $(\bar{\mathbb{E}}[|\cdot|])$ norm by Lemma 2.3, which proves that $X_{T}\in L_{G}^{1}(\Omega_{T})$. ∎

To prove Theorem 2.1, we first introduce the construction of coupling by change of measure with multiplicative noise under $G$-setting.

2.1.2 Construction of the coupling

In the sequel, we denote $\hat{\sigma}=\sigma^{\ast}(\sigma\sigma^{\ast})^{-1}.$ We use the coupling by change of measures as explained in [18]. For $\alpha\in(0,\frac{2\kappa_{1}^{2}}{\kappa_{2}^{2}})$, let

[TABLE]

Then $\lambda_{t}^{\alpha}$ is smooth and strictly positive on $[0,T)$ such that

[TABLE]

For convenience, we reformulate (1.1) as

[TABLE]

Consider the equation

[TABLE]

where $g_{t}:=\frac{1}{\lambda_{t}^{\alpha}}\hat{\sigma}(t,X_{t})(X_{t}-Y_{t})$.

2.1.3 Extension of $Y$ to $T$

Let $s\in[0,T)$ be fixed. By (1.1) and (2.6), $X_{t}-Y_{t}$ satisfies the equation below

[TABLE]

Applying Itô’s formula to $|X_{t}-Y_{t}|^{2}$, we obtain

[TABLE]

Combining with the expression (2.4), we have

[TABLE]

Thus,

[TABLE]

Taking expectation $\bar{\mathbb{E}}$ on both sides of (2.9), we obtain

[TABLE]

Since $X_{t},Y_{t}\in M_{G}^{2}([0,T])$, for any $s\in(0,T),$ $g_{t}1_{[0,s]}(t)\in M_{G}^{2}([0,T])$. Note that, for any $s\in(0,T),$

[TABLE]

where $C_{1}$ is a constant.

By the Monotone Convergence Theorem in [1] of Theorem 1.4,

[TABLE]

There exists a $\bar{g}\in M_{G}^{2}([0,T])$ such that $\bar{g}_{s}=g_{s},s\in[0,T).$ In fact, let $g_{t}^{n}=g_{t}1_{[0,T-\frac{1}{n}]}(t)$ $\in M_{G}^{2}([0,T]),$ then it holds that

[TABLE]

where the last step uses the fact of [2] in Theorem 1.4.

Let $\bar{Y}_{t}$ solve the following equation

[TABLE]

Thus, $Y$ can be extended to $[0,T]$ as $\bar{Y}$. In the sequel, we still use $Y$ and $g$ instead $\bar{Y}$ and $\bar{g}$.

2.1.4 Several lemmas

We first prove the following Young inequality under $G$-expectation framework.

Lemma 2.7**.**

(Young Inequality)* For $g_{1},g_{2}\in L_{G}^{1}(\Omega_{T})$ with $g_{1}$, $g_{2}>0$ and ${\mathbb{E}}_{P}[g_{1}]=1$, $\forall\ P\in\mathcal{P},$ then*

[TABLE]

where $\mathcal{P}$ is a weakly compact set that represents $\bar{\mathbb{E}}$.

Proof.

For any $P\in\mathcal{P},$ ${\mathbb{E}}_{P}$ is a linear expectation, it holds that

[TABLE]

Since $\bar{\mathbb{E}}[X]=\sup_{P\in\mathcal{P}}{\mathbb{E}}_{P}[X]$, $X\in L_{G}^{1}(\Omega_{T}),$ then

[TABLE]

where the last step due to the function $\log$ is increasing. ∎

Let

[TABLE]

Following section 3.2.2, we see that ${g}\in M_{G}^{2}([0,T]),$ below we aim to prove

[TABLE]

is a uniformly integrable symmetric $G$-martingale for $s\in[0,T]$.

Lemma 2.8**.**

Assume (H1)-(H2). There holds

[TABLE]

Consequently, $M_{T}:=\lim_{s\uparrow T}M_{s}$ exists and $\{M_{s}\}_{s\in[0,T]}$ is a uniformly integrable symmetric $G$-martingale.

Proof.

Fix $s\in[0,T).$ Applying Itô’s formula to $|X_{t}|^{2}$, we have

[TABLE]

Let

[TABLE]

and

[TABLE]

For any $n\geq 1,$ let $\hat{\tau}_{n}=\inf\{t\in[0,T]||\hat{X}_{t}|+|\hat{Y}_{t}|\geq n\}$. By Lemma 3.3, $\hat{\tau}_{n}$ is quasi-continuous, and $X_{t\wedge\hat{\tau}_{n}}$, $Y_{t\wedge\hat{\tau}_{n}}$ are bounded, which implies $g_{t\wedge\hat{\tau}_{n}}$ is bounded. So for any $n\geq 1$ and by the Girsanov theorem in [4, Theorem 5.2], $(\hat{B}_{t})_{t\in[0,s\wedge\hat{\tau}_{n}]}$ is a $G$-Brownian motion under $\hat{\mathbb{E}}_{n}:=\bar{\mathbb{E}}[\cdot M_{s\wedge\hat{\tau}_{n}}]$.

Moreover, Lemma 1.10 implies $\langle\hat{B}\rangle_{t}=\langle B\rangle_{t}.$ Rewrite (2.5) and (2.11) as

[TABLE]

Substituting $B_{t}=\hat{B}_{t}-\int_{0}^{t}g_{s}\text{\rm{d}}\langle B\rangle_{s}$ in the first equation in (2.8), using the fact of $\langle\hat{B}\rangle_{t}=\langle B\rangle_{t}$, and repeating procedures in (2.8), which yield

[TABLE]

So,

[TABLE]

From (2.4), we know that

[TABLE]

where $\triangle:=2K{\lambda_{t}^{\alpha}}+\frac{2K}{\underline{\sigma}^{2}}\lambda_{t}^{\alpha}+K^{2}{\lambda_{t}^{\alpha}}-\frac{1}{\underline{\sigma}^{2}}(\lambda_{t}^{\alpha})^{\prime}$.

Therefore,

[TABLE]

Since $(\hat{B}_{t})_{t\in{[0,s\wedge\hat{\tau}_{n}]}}$ is a $G$-Brownian motion under $\hat{\mathbb{E}}$, taking expectation $\hat{\mathbb{E}}$ on both sides of (2.14), we obtain

[TABLE]

From the definition of $M_{t}$, $\hat{B}_{t}$ and Lemma 1.10, it holds that

[TABLE]

By (H2), we have

[TABLE]

It follows (2.15) that

[TABLE]

Applying Itô’s formula to $M_{s\wedge\hat{\tau}_{n}}=\text{\rm{e}}^{u_{s\wedge\hat{\tau}_{n}}}$ for the process

[TABLE]

we conclude that

[TABLE]

thus $\{M_{t}\}_{t\in[0,s\wedge\hat{\tau}_{n}]}$ is a symmetric $G$-martingale. From (2.17) and Lemma 2.2, $\{M_{s\wedge\hat{\tau}_{n}}\}_{s\in[0,T)}$ is a uniformly symmetric $G$-martingale, thus $\bar{\mathbb{E}}M_{s}=\lim_{n\rightarrow\infty}\bar{\mathbb{E}}M_{s\wedge\hat{\tau}_{n}}=1$ by Lemma 2.3. So that $\{M_{t}\}_{t\in[0,s)}$ is a symmetric $G$-martingale.

Let $\hat{\mathbb{E}}=\bar{\mathbb{E}}[M_{s}\cdot],\ s\in[0,T)$. Letting $n\rightarrow\infty$, we have $\hat{\tau}_{n}\uparrow T$. By the Fatou lemma,

[TABLE]

Thus

[TABLE]

Using Theorem 2.6 once again, $\{M_{s}\}_{s\in[0,T]}$ is a uniformly symmetric $G$-martingale.

∎

Lemma 2.9**.**

Assume (H1)-(H2). We have $X_{T}=Y_{T}$, c-q.s..

Proof.

Let

[TABLE]

For any $P\in\mathcal{P}$, define $\hat{\mathbb{E}}_{P}=\mathbb{E}_{P}[M_{T}\cdot]$, then $\hat{B}_{t}$ is a martingale under $\hat{\mathbb{E}}_{P}.$ If there exists a $\omega\in\Omega$ such that $\tau(\omega)>T$, then

[TABLE]

So

[TABLE]

holds on the set $\{\omega|\tau(\omega)>T\}$, which is a contradiction with (2.15), thus $\hat{\mathbb{E}}_{P}$-a.s., $\tau(\omega)\leq T$, then

[TABLE]

Similar analysis with Lemma 1.10, we have

[TABLE]

Therefore, $X_{T}=Y_{T}$ under $\hat{\mathbb{E}}$. ∎

Lemma 2.10**.**

Assume (H1)-(H2). Then

[TABLE]

Consequently,

[TABLE]

holds for

[TABLE]

Proof.

Let $\tau_{m}=\inf\{t\in[0,T]\ |\ \int_{0}^{t}(\frac{|X_{s}-Y_{s}|^{2}}{(\lambda_{s}^{\alpha})^{2}}+1)\text{\rm{d}}\langle{B}\rangle_{s}\geq m\}$. Applying Lemma 3.3 for processess $Z_{s}=0,\eta_{s}=0,$ and $\zeta_{s}=\frac{|X_{s}-Y_{s}|^{2}}{(\lambda_{s}^{\alpha})^{2}}+1$, we know that $\tau_{m}$ is quasi-continuous. From (2.10), we know that $\lim_{m\rightarrow\infty}\tau_{m}=T$. By (2.9), (H2), and Lemma 3.4-3.5, for some $\delta>0$, we have

[TABLE]

Taking $\delta=\frac{\alpha^{2}}{8(\kappa_{2}-\kappa_{1})^{2}}$, we arrive at

[TABLE]

Letting $m\rightarrow\infty$, this implies that

[TABLE]

which is (2.19).

Next, let $\tilde{\tau}_{n}=\inf\{t\in[0,T]\ |\int_{0}^{t}(\frac{|X_{s}-Y_{s}|^{2}}{(\lambda_{s}^{\alpha})^{2}}+1)\text{\rm{d}}\langle\hat{B}\rangle_{s}\geq n\}$, similar with $\tau_{m}$, $\tilde{\tau}_{n}$ is quasi-continuous. From (2.15), we know that $\lim_{n\rightarrow\infty}\tilde{\tau}_{n}=T$. Similar with the process of deducing in (2.20), we have

[TABLE]

Moreover,

[TABLE]

From (H2), we have

[TABLE]

Taking $q=1+\sqrt{1+a^{-1}}$, it holds that

[TABLE]

Then,

[TABLE]

Therefore, by recalling the expressions (2.21) – (2.23), we get

[TABLE]

this completes the proof.

∎

2.1.5 Proof to Theorem 2.1

• (1)

Lemma 2.8 ensures that under $\hat{\mathbb{E}}:=\bar{\mathbb{E}}[\cdot M_{T}]$, $\{\hat{B}_{t}\}_{t\in[0,T]}$ is a $G$-Brownian motion, and

[TABLE]

Then by (2.5) and (2.1.4), the coupling $(X_{t},Y_{t})$ is well constructed under $\hat{\mathbb{E}}$ for $t\in[0,T]$. Moreover, due to Lemma 2.9, $X_{T}=Y_{T}$ holds $\hat{\mathbb{E}}$-q.s., which fits well the requirement of coupling by change of measure. Since for all $P\in\mathcal{P}$, ${\mathbb{E}}_{P}[M_{T}]=1$, by Young’s inequality in Lemma 2.7, for any $f\in C_{b}^{+}(\mathbb{R})$, we obtain

[TABLE]

For $\alpha\in(0,\frac{2\kappa_{1}^{2}}{\kappa_{2}^{2}}),$ taking $\alpha=\frac{\kappa_{1}^{2}}{\kappa_{2}^{2}}$, (1) of Theorem 2.1 holds.

• (2)

Taking $\alpha=\frac{2(\kappa_{2}-\kappa_{1})}{{\kappa_{1}}(\sqrt{p}-1)}$ in (2.24) which is in $(0,\frac{2\kappa_{1}^{2}}{\kappa_{2}^{2}})$ for $p>(1+\frac{\kappa_{2}^{3}-\kappa_{1}\kappa_{2}^{2}}{\kappa_{1}^{3}})^{2}$, we have $\frac{p}{p-1}=1+a$, by Lemma 2.10, this leads to

[TABLE]

Thus, due to Hölder’s inequality, for any $f\in C_{b}^{+}(\mathbb{R})$,

[TABLE]

which is the result (2) of Theorem 2.1.

2.2 Gradient Estimate

Due to the lack of additivity of $G$-expectation, neither from the Bismut formula [18, (1.8), (1.14)] by coupling by change of measure to get gradient estimate, nor Malliavin calculus in the $G$-SDEs. Instead, we directly to estimate the local Lipschitz constant defined below. For a real-valued function $f$ defined on a metric sapce $(H,\rho)$, define

[TABLE]

Then $|\nabla f(z)|$ is called the local Lipschitz constant of $f$ at point $z\in H$.

Theorem 2.11**.**

Assume (H1)-(H2). Then for every $f\in{C}_{b}^{+}(\mathbb{R})$, it holds that

[TABLE]

where $\lambda_{0}^{\alpha}$ is defined in (2.3) for $t=0$.

Proof.

By the proof of Theorem 2.1, we have

[TABLE]

Noting that $|x-1|\leq(x+1)|\log x|$ for any $x>0$, then

[TABLE]

From (2.16) and (2.17), it holds that

[TABLE]

Similarly, we obtain

[TABLE]

It follows from (2.27) that

[TABLE]

This together with (2.25) yields

[TABLE]

which implies (2.26).

∎

3 Appendix–The quasi-continuity of stopping times

This part is essentially from [14, 15]. To make the content self-contained, we cite some results from [14, 15] and restated them as follows.

Lemma 3.1**.**

([15, Lemma 3.3]) Let $E$ be a metric space and a mapping $E\times[0,T]\ni(\omega,t)\rightarrow M_{t}(\omega)\in\mathbb{R}$ be continuous on $E\times[0,T]$. Define $\tau_{a}=\inf\{t>0|M_{t}>a\}\wedge T$ and $\underline{\tau}_{a}=\inf\{t>0|M_{t}\geq a\}\wedge T$. Then $-\tau_{a}$ and $\underline{\tau}_{a}$ are both lower semi-continuous.

Lemma 3.2**.**

([15, Lemma 3.4]) For any closed set $F\subset\Omega_{T}$, we have

[TABLE]

where $c$ is the capacity induced by $\bar{\mathbb{E}}$.

The following lemma plays a crucial role in studying the quasi-continuity of stopping times under nonlinear expectation space, which is a dramatic different with classic linear expectation space. For reader’s convenience, we give the proof of the lemma.

Lemma 3.3**.**

([14, Lemma 4.3]) Let $Y_{t}=\int_{0}^{t}\langle Z_{s},\text{\rm{d}}B_{s}\rangle+\int_{0}^{t}\eta_{s}\text{\rm{d}}s+\int_{0}^{t}tr[\zeta_{s}\text{\rm{d}}\langle B\rangle_{s}]$ with $Z\in[H_{G}^{1}([0,T])]^{d}$ and $\eta,\zeta^{i,j}\in M_{G}^{1}([0,T])$. Assume $\int_{0}^{t}\eta_{s}\text{\rm{d}}s+\int_{0}^{t}tr[\zeta_{s}\text{\rm{d}}\langle B\rangle_{s}]$ is non-decreasing and

[TABLE]

is strictly increasing. Then, for $a>0$, $\tau_{a}:=\inf\{t\geq 0|Y_{t}>a\}\wedge T$ is quasi-continuous.

Proof.

Let $\underline{\tau}_{a}=\inf\{t\geq 0|Y_{t}\geq a\}\wedge T$. Since $Y$ is quasi-continuous, then for all $\epsilon>0$, there exists an open set $O_{1}$ with $c(O_{1})<\frac{\epsilon}{2}$ such that $Y_{\cdot}(\cdot)$ is continuous on $O_{1}^{c}\times[0,T].$ Define

[TABLE]

where

[TABLE]

We divide the proof into following five steps.

• (1)

We first prove $[\tau_{a}>\underline{\tau}_{a}]\subset S_{a}(Y)\bigcup\cup_{r\in\mathbb{Q}\bigcap[0,T]}[Y_{r\wedge\tau_{a}}<Y_{r\wedge\underline{\tau}_{a}}]=:A.$

It is equivalent to prove $[\tau_{a}>\underline{\tau}_{a}]\subset S_{a}(Y)+A\setminus S_{a}(Y).$

For any $\omega\in[\tau_{a}>\underline{\tau}_{a}],$ i.e., for any $\omega$ with $\tau_{a}(\omega)>\underline{\tau}_{a}(\omega)$, if $\omega\in S_{a}(Y)$, which ends the proof. If $\omega\notin S_{a}(Y)$, i.e., for any $(r,s)\in Q_{T}$, there exists a $t\in[s,r]$, s.t. $Y_{t}(\omega)\neq a$. Since $\mathbb{Q}$ is dense in $\mathbb{R}$, and $\tau_{a}\geq\underline{\tau}_{a}$, it’s clear that $\omega\in A\setminus S_{a}(Y).$

• (2)

We claim that $c(S_{a}(Y))=0$.

1. (i)

If $Z=0$, then $Y_{t}$ is strictly increasing, thus $\tau_{a}=\underline{\tau}_{a}$, which implies $c(S_{a}(Y))=0$. 2. (ii)

If $Z\neq 0$, since $B_{t}$ with infinite variation, it is impossible for $Y_{t}=a,t\in[s,r],$ then $c(S_{a}(Y))=0$.

• (3)

We claim that $c(A)=0$.

Noting that $Y_{r\wedge\tau_{a}}\leq Y_{r\wedge\underline{\tau}_{a}}$ and

[TABLE]

For $r\leq\underline{\tau}_{a}$ and $r\geq\tau_{a}$, it hold that $\bar{\mathbb{E}}[Y_{r\wedge\tau_{a}}-Y_{r\wedge\underline{\tau}_{a}}]=0.$ For $\underline{\tau}_{a}<r<{\tau}_{a}^{\prime},$ by (3.1), we have

[TABLE]

From the assumption of non-decreasing for $\int_{0}^{t}\eta_{s}\text{\rm{d}}s+\int_{0}^{t}tr[\zeta_{s}\text{\rm{d}}\langle B\rangle_{s}]$, we derive that $\bar{\mathbb{E}}[Y_{r\wedge\tau_{a}}-Y_{r\wedge\underline{\tau}_{a}}]\geq 0.$ By the fact that $Y_{r\wedge\tau_{a}}\leq Y_{r\wedge\underline{\tau}_{a}}$ and $\bar{\mathbb{E}}[Y_{r\wedge\tau_{a}}-Y_{r\wedge\underline{\tau}_{a}}]\geq 0,$ we know that $Y_{r\wedge\tau_{a}}=Y_{r\wedge\underline{\tau}_{a}},$ q.s.. Since $\mathbb{Q}$ is countable, then $c(A)=0$

• (4)

$A\cap O_{1}^{c}$ is an open set under the topology induced by $O_{1}^{c}$.

Since $Y_{\cdot}(\cdot)$ is continuous on $O_{1}^{c}\times[0,T]$, by Lemma 3.1, $\underline{\tau}_{a}$ (${\tau}_{a}$) is lower (upper) semi-continuous on $O_{1}^{c}$, then $Y_{r\wedge\underline{\tau}_{a}}$ ($Y_{r\wedge{\tau}_{a}}$) is lower (upper) semi-continuous on $O_{1}^{c}$, which means that $[Y_{r\wedge\tau_{a}}<Y_{r\wedge\underline{\tau}_{a}}]\cap O_{1}^{c}$ is an open set under the topology induced by $O_{1}^{c}$. Since the union of any collection of open sets in $O_{1}^{c}$ is open, then we prove it.

• (5)

$S_{a}(Y)$ can be covered by countable open sets with capacity small enough.

By the definition of $S_{a}(Y)$, we have

[TABLE]

Since $Y_{\cdot}(\cdot)$ is continuous on $O_{1}^{c}\times[0,T]$, $\{\omega|Y_{t}(\omega)=a\}\cap O_{1}^{c}$ is a closed set under the topology induced by $O_{1}^{c}$ for any $t\in[0,T]$. Moreover, $\{\omega|Y_{t}(\omega)=a\}$ is a closed set as $O_{1}^{c}$ is closed. Then $\bigcap_{t\in[s,r]}\{\omega|Y_{t}(\omega)=a\}$ is closed. By Lemma 3.2 and the fact that $c(S_{a}(Y))=0$, for all $\epsilon>0$, there exists an open set $O_{2}^{s,r}$ with $0\leq c(O_{2}^{s,r})<\frac{\epsilon}{2^{n+1}}$ such that $\bigcap_{t\in[s,r]}\{\omega|Y_{t}(\omega)=a\}\subset O_{2}^{s,r}$. Let $O_{2}=\bigcup_{(r,s)\in Q_{T}}O_{2}^{s,r},$ then

[TABLE]

where $O_{2}$ is open.

Combining (1)–(5), we know that

[TABLE]

where $O_{2}$ is open under topology induced by $\Omega_{T}$ and $A\cap O_{1}^{c}$ is open under the topology induced by $O_{1}^{c}$. So, there exists an open set $O_{3}\subset\Omega_{T}$, such that

[TABLE]

Noting that

[TABLE]

Moreover, $O_{3}=(O_{3}\cap O_{1})\cup(O_{3}\cap O_{1}^{c}),$ by $c(O_{3}\cap O_{1}^{c})=0$ of (3), we have

[TABLE]

Therefore,

[TABLE]

where $c(O_{2}\cup O_{1}\cup O_{3})\leq c(O_{1})+c(O_{2})+c(O_{3})\leq 2\epsilon.$ It is clear that

[TABLE]

thus

[TABLE]

By Lemma 3.1, $\tau_{a}$ is continuous on $[\tau_{a}=\underline{\tau}_{a}]\cap O_{1}^{c}$. Therefore, for all $\epsilon>0$, for the open set, $O_{1}\cup O_{2}\cup O_{3}$, with $c(O_{1}\cup O_{2}\cup O_{3})<2\epsilon,$ $\tau_{a}$ is continuous on $(O_{1}\cup O_{2}\cup O_{3})^{c},$ which implies that $\tau_{a}$ is quasi continuous by Definition 1.2. ∎

Lemma 3.4**.**

([8, Proposition 4.10]) Let $\tau\leq T$ be a quasi-continuous stopping time. Then for each $p\geq 1$, we have $I_{[0,\tau]}\in M_{G}^{p}([0,T]).$

Lemma 3.5**.**

([8, Remark 4.12]) Let $\tau\leq T$ be a quasi-continuous stopping time and $\eta\in M_{G}^{p}([0,T])$. Then for each $p\geq 1$, we have $\eta I_{[0,\tau]}\in M_{G}^{p}([0,T]).$

According to [7], for a stopping time $\tau\leq T$, and $\eta\in M_{G}^{p}([0,T])$, it holds that

[TABLE]

Acknowledgement.

The authors are grateful to Professor Feng-Yu Wang for his guidance and helpful comments, as well as Yongsheng Song and Xing Huang for their patient helps, valuable suggestions and corrections.