Approximations of the Euler–Maruyama Method of Stochastic Differential Equations with Regime Switching

Abstract

This work focuses on a class of regime-switching diffusion processes with both continuous components and discrete components. Under suitable conditions, we adopt the Euler–Maruyama method to deal with the convergence of numerical solutions of the corresponding stochastic differential equations. More precisely, we first show the convergence rates in the $L^p$-norm of the stochastic differential equations with regime switching under Lipschitz conditions. Then, we also discuss $L^1$ and $L^2$ convergence under non-Lipschitz conditions.

1. Introduction

Recently, regime-switching diffusion processes (RSDPs), in which continuous dynamics and discrete events coexist, have drawn much attention. Owing to the fact that continuous dynamics and discrete events are intertwined in many applications, the independence assumption of the continuous component and the discrete-event process poses restrictions. Such hybrid systems would be able to handle the coupling and dependence of the continuous states and discrete events, which are prevalent in a wide range of applications including ecological and biological modeling [1], control systems and filtering [2], economics and finance [3,4], and networked systems [5], among others. It is well known that SDEs with regime switching have received considerable attention in recent decades. On the one hand, there are some results on the theory of these equations. We know that SDEs with regime switching have a unique solution under Lipschitz and linear growth conditions (see [6]). Xi [7] studied the Feller property and exponential ergodicity of diffusion processes with state-dependent switching. In 2019, Xi, Yin and Zhu [8] discussed regime-switching jump diffusions with non-Lipschitz coefficients and many countable switching states with respect to existence and uniqueness and Feller and strong Feller properties. Li and Xi [9] also studied this topic in 2021 and obtained the boundedness and ergodicity of a switching diffusion process with infinite delays. On the other hand, regime-switching systems are rather complicated; numerical approximations are an important alternative for such systems. Yuan and Mao [10,11] studied the convergence between numerical solutions and exact solutions via the Euler–Maruyama (EM) method for an SDE with Markovian switching under global Lipschitz conditions and non-Lipschitz conditions. In [12], Zhang et al. studied the convergence at a rate of $1/logn$ using the EM method for a CIR model with Markovian switching. The weak convergence of numerical approximations was established by constructing a sequence of discrete-time Markov chains (see [13]). This method is different from the usual time-discretizing EM approximation, and it is difficult to obtain the order of error. Recently, Shao [14] investigated the time-discretizing EM approximation of the state-dependent RSDP and showed its strong convergence in the $L^1$-norm under the Lipschitz condition. In 2024, Chen [15] and their collaborators studied approximations of regime-switching jump diffusion processes using the variable-step Euler–Maruyama method.

Let $(\mathsf{\Omega },\mathcal{F},{\left({\mathcal{F}}_t\right)}_{t\ge 0},\mathbb{P})$ be a complete filtered probability space satisfying the usual conditions (i.e., it is increasing and right continuous and ${\mathcal{F}}_0$ contains all $\mathbb{P}$-null sets). The stochastic processes studied in this work are all defined on this filtered probability space. To formulate our model, let d be a positive integer, and put $\mathbb{S}=\{1,2,\cdots ,N\}$ with $2\le N<\infty$. In this paper, we are interested in the stochastic differential equation (SDE) with regime switching:

$$\mathrm{d}X\left(t\right)=b\left(X\right(t),\Lambda (t\left)\right)\mathrm{d}t+\sigma \left(X\right(t),\Lambda (t\left)\right)\mathrm{d}B\left(t\right),t\in [0,T]$$

(1)

with initial conditions $X\left(0\right)=x\in {\mathbb{R}}^d$ and $\Lambda \left(0\right)=i\in \mathbb{S}.$ Herein, $b:{\mathbb{R}}^d\times \mathbb{S}\to {\mathbb{R}}^d$, $\sigma :{\mathbb{R}}^d\times \mathbb{S}\to {\mathbb{R}}^{d\times d}$ and $B=\left\{B\right(t),t\ge 0\}$ is d-dimensional standard Brownian motion. The state process $X(·)$ takes values in ${\mathbb{R}}^d$ and the switching process $\Lambda (·)$ is a right-continuous jump process on the probability space, taking values in a finite state space $\mathbb{S}=\{1,2,\cdots ,N\}$ with generator $Q\left(x\right)=\left(q_{ij}\left(x\right)\right)$, given by

$$\mathbb{P}\{\Lambda (t+\Delta )=j|\Lambda \left(t\right)=i,X\left(t\right)=x\}=\left\{\begin{array}{cc}{q}_{ij}\left(x\right)\Delta +o(\Delta ),\hfill & \mathrm{if}j\ne i,\hfill \\ 1+q_{ii}\left(x\right)\Delta +o(\Delta ),\hfill & \mathrm{if}j=i,\hfill \end{array}\right.$$

(2)

provided $\Delta \downarrow 0$. As usual, we assume that $Q\left(x\right)={\left(q_{ij}\left(x\right)\right)}_{i,j\in \mathbb{S}}$ is totally stable, irreducible and conservative. Namely, we assume that for each fixed $x\in {\mathbb{R}}^d$ and any $i,j\in \mathbb{S}$, there exist some states $i_1,i_2,...,i_m\in \mathbb{S}$ such that $q_{i{i}_1}\left(x\right)q_{i_1{i}_2}\left(x\right)...q_{i_{m}j}\left(x\right)>0$, and that for each $x\in {\mathbb{R}}^d$ and $i\in \mathbb{S}$, $q_i\left(x\right)=-q_{ii}\left(x\right)={\sum }_{j\ne i}{q}_{ij}\left(x\right)<\infty .$ The process $(\Lambda (t\left)\right)$ also can be expressed by the SDE

$$\begin{array}{c}\hfill \Lambda \left(t\right)=i+{\int }_0^t{\int }_{[0,M]}h(X\left(s\right),\Lambda (s-),z)N_1(\mathrm{d}s,\mathrm{d}z),\end{array}$$

where $\Lambda (s-)$ denotes the left limit of $\Lambda \left(s\right)$, $N_1(\mathrm{d}s,\mathrm{d}z)$ is a Poisson random measure with intensity $\mathrm{d}t\times m\left(\mathrm{d}z\right)$, and $m\left(\mathrm{d}z\right)$ is the Lebesgue measure on $[0,M],$ and the definitions of M and h can be found in [14]. As a standing hypothesis, we assume that both b and $\sigma$ are sufficiently smooth so that SDE (1) has a unique solution. Moreover, when $q_{ij}\left(x\right)$ is independent of x for all $i,j\in \mathbb{S},$$\left(X\right(t),\Lambda (t\left)\right)$ is called a state-independent RSDP or an RSDP with Markovian switching. Otherwise, it is called a state-dependent RSDP.

To the best of our knowledge, there are no more works on the numerical approximation of state-dependent RSDPs due to the close interaction between the continuous component and discrete component. Compared with [14], the novelty of our work is that we first show the convergence rates in the $L^p$-norm of the time-discretizing EM approximation under the Lipschitz condition. Then, we also discuss $L^1$ and $L^2$ convergence under the non-Lipschitz condition. A natural and important difficulty is that the evolution of $(\Lambda (t\left)\right)$ is much more complicated due to its dependence on the continuous-state process $\left(X\right(t\left)\right)$, which makes the transition rate matrices of $(\Lambda (t\left)\right)$ different for every step of jumps. Much care and more techniques are needed for the mixture of $\left(X\right(t\left)\right)$ and $(\Lambda (t\left)\right)$ under non-Lipschitz conditions.

This paper is organized as follows. In Section 2, we state the EM approximation of $\left(X\right(t),\Lambda (t\left)\right)$, some useful lemmas and the main results. In Section 3, we present the proof of the main results. In Appendix A,we show some auxiliary propositions. Let C represent positive constants whose values may change from one place to another.

2. Preliminary and Main Results

2.1. EM Schemes and Preparatory Lemmas

Firstly, we introduce some notation and terminology which will be used later. We use $A^{\prime }$ to denote the transpose, and the Euclidean norm for a row or column vector x is denoted by $\left|x\right|$; $\mathbf{1}$ denotes the indicator function. Throughout the paper, for $x\in {\mathbb{R}}^d$ and $\sigma =\left({\sigma }_{ij}\right)\in {\mathbb{R}}^{d\times d}$, define $|x|=({\sum }_{i=1}^d{{|x_i|}^2)}^{1/2}$ and $\parallel \sigma \parallel =({\sum }_{i,j=1}^d|{\sigma }_{ij}{{|}^2)}^{1/2}$.

Let $\left(X\right(t),\Lambda (t\left)\right)$ be the solution of (1) and (2). In the present paper, we consider the following EM approximation of $\left(X\right(t),\Lambda (t\left)\right)$. For $\delta \in (0,1),t\in [0,T]$, define

$$\begin{array}{c}\hfill \mathrm{d}Y\left(t\right)=b(Y\left(t_{\delta }\right),{\Lambda }^{\prime }\left(t_{\delta }\right))\mathrm{d}t+\sigma (Y\left(t_{\delta }\right),{\Lambda }^{\prime }\left(t_{\delta }\right))\mathrm{d}B\left(t\right),\end{array}$$

(3)

where the second component ${\Lambda }^{\prime }\left(t\right)$ is a continuous time jumping process whose transition rate depends on the process $\left(Y\right(t\left)\right)$, taking value in the finite set $\mathbb{S}$ such that

$$\begin{array}{c}\hfill \mathbb{P}\{{\Lambda }^{\prime }(t+\Delta )=j|{\Lambda }^{\prime }\left(t\right)=i,Y\left(t_{\delta }\right)=y\}=\left\{\begin{array}{cc}{q}_{ij}\left(y\right)\Delta +o(\Delta ),\hfill & \mathrm{if}j\ne i,\hfill \\ 1+q_{ii}\left(y\right)\Delta +o(\Delta ),\hfill & \mathrm{if}j=i,\hfill \end{array}\right.\end{array}$$

(4)

provided $\Delta \downarrow 0.$ Here, $(Y\left(0\right),{\Lambda }^{\prime }\left(0\right))=(X\left(0\right),\Lambda \left(0\right)),$$t_{\delta }=[t/\delta ]\delta ,$ and $[t/\delta ]$ denotes the integer part of $t/\delta .$ Note that the evolution of $Y\left(t\right)$ depends only on the embedded chain ${\left({\Lambda }^{\prime }\left(k\delta \right)\right)}_{k\ge 1}$ of the process $\left({\Lambda }^{\prime }\left(t\right)\right)$. Then, the process $\left({\Lambda }^{\prime }\left(t\right)\right)$ also can be expressed by the SDE

$$\begin{array}{c}\hfill {\Lambda }^{\prime }\left(t\right)=i+{\int }_0^t{\int }_{[0,M]}h(Y\left(s_{\delta }\right),{\Lambda }^{\prime }(s-),z)N_1(\mathrm{d}s,\mathrm{d}z).\end{array}$$

For the transition rate matrix $Q\left(x\right)={\left(q_{ij}\left(x\right)\right)}_{i,j\in \mathbb{S}},$ we shall use the following conditions:

(Q1)

Let $H:={max}_{i\in \mathbb{S}}{sup}_{x\in {\mathbb{R}}^d}{q}_i\left(x\right)<\infty ,$ where $q_i\left(x\right)={\sum }_{j\ne i}{q}_{ij}\left(x\right)$ for $i,j\in \mathbb{S},x\in {\mathbb{R}}^d.$

(Q2)

There exists a constant C so that ${\sum }_{j\ne i}|q_{ij}\left(x\right)-q_{ij}\left(y\right)|\le C|x-y|,$$\forall x,y\in {\mathbb{R}}^d,i,j\in \mathbb{S}.$

Conditions (Q1) and (Q2) also play an important role for the estimation of $\left({\int }_0^t{\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}\mathrm{d}s\right)$.

In order to prove the main results, we cite the following results as lemmas.

Lemma 1

(Lemma 3.2 [16]). Let $\left\{Z\right(t),t\ge 0\}$ be a non-negative stochastic process and set $V\left(t\right)={sup}_{s\le t}Z\left(s\right).$ Assume that for some $p>0,q\ge 1,\varrho \in [1,q],$ and constants $C_1$ and $\vartheta \ge 0$

$$\begin{array}{c}\hfill \mathbb{E}{V}^p\left(t\right)\le C_1\mathbb{E}\left({\int }_0^{t}V\left(s\right)\mathrm{d}s\right)+C_1\mathbb{E}{\left({\int }_0^t{Z}^{\varrho }\left(s\right)\mathrm{d}s\right)}^{p/q}+\vartheta <\infty ,\end{array}$$

(5)

for $t\ge 0$. Then, for each $T\ge 0$, the following statements hold.

(i) If $\varrho =q$, then there is a constant $C_T$ such that

$$\mathbb{E}{V}^p\left(T\right)\le C_T\vartheta .$$

The constant $C_T$ depends only on $C,p,q$ and T and it increases in $T.$

(ii) If $p\ge q$ or both $\varrho <q$ and $p>q+1-\varrho$ hold, then there exists constants $C_2$ and $C_3$, depending on $C_1,T,\varrho ,q$ and p, such that

$$\mathbb{E}{V}^p\left(T\right)\le C_2\vartheta +C_3{\int }_0^T\mathbb{E}Z\left(s\right)\mathrm{d}s.$$

Lemma 2

(Bihari’s inequality [17]). Let $T\in (0,\infty )$ and $C>0$. Let $K:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be a continuous nondecreasing function such that $K\left(t\right)>0$ for all $t>0$. Let $u(·)$ be a Borel measurable bounded non-negative function on $[0,T]$, and let $\upsilon (·)$ be a non-negative integrable function on $[0,T]$. If, for all $0\le t\le T,$

$$u\left(t\right)\le C+{\int }_0^t\upsilon \left(s\right)K\left(u\left(s\right)\right)\mathrm{d}s,$$

then

$$u\left(t\right)\le G^{-1}\left(G\left(C\right)+{\int }_0^t\upsilon \left(s\right)\mathrm{d}s\right)$$

holds for all such $t\in [0,T]$ that $G\left(C\right)+{\int }_0^t\upsilon \left(s\right)\mathrm{d}s\in \mathit{Dom}\left(G^{-1}\right)$, where $G\left(r\right)={\int }_1^r\mathrm{d}s/K\left(s\right)$ on $r>0$, and $G^{-1}$ is the inverse function of G.

2.2. Main Results

In this subsection, we mainly discuss the convergence of an SDE with state-dependent regime switching under Lipschitz and non-Lipschitz conditions. We give the rate of the convergence between the EM numerical solutions and exact solutions.

Theorem 1.

Assume $\left(\mathrm{Q}1\right)$ and $\left(\mathrm{Q}2\right)$ hold if ${\mathbb{E}\left|X\left(0\right)\right|}^p<\infty$ for $p\ge 2$. Let b and σ be bounded, and there exists a constant $C>0,$ such that

$$|b(x,i)-b(y,i)|\vee \parallel \sigma (x,i)-\sigma (y,i)\parallel \le C|x-y|,x,y\in {\mathbb{R}}^d,i\in \mathbb{S}.$$

Then, it holds that

$$\mathbb{E}\left[\underset{0\le t\le T}{sup}{|X\left(t\right)-Y\left(t\right)|}^p\right]\le C{\delta }^{1/4},$$

for some constants $C>0$ depending on $T,p$ that are independent of δ.

Corollary 1.

The assumption is that the conditions of Theorem 1 hold. If ${\mathbb{E}\left|X\left(0\right)\right|}^p<\infty$ for $p\in (1,2)$, then it holds that

$$\mathbb{E}\left[\underset{0\le t\le T}{sup}{|X\left(t\right)-Y\left(t\right)|}^p\right]\le C{\delta }^{p/8},$$

for some constants $C>0$ depending on $T,p$ that are independent of δ.

Remark 1.

We present two results, which are related to the rate of the convergence between the EM numerical solutions and exact solutions for different ranges of p-values, and generalize Shao’s [14] first moment case. In addition, Shao’s method is no longer applicable to the case of p moment, so we adopt the approximation technique of Yamada and Watanabe to prove the results.

Theorem 2.

Assume $\left(\mathrm{Q}1\right)$ and $\left(\mathrm{Q}2\right)$ hold and let b and σ be bounded. Let there exist a positive number ${\delta }_0$ and a nondecreasing, differentiable, concave function $\rho :[0,\infty )\to [0,\infty )$ satisfying

$$\begin{array}{c}\hfill 0<\rho \left(r\right)\le {(1+r)}^2\rho (r/(1+r))\mathit{for}\mathit{all}r>0,\mathit{and}{\int }_0^{\infty }\frac{\mathrm{d}r}{\rho \left(r\right)}=\infty \end{array}$$

(6)

such that for all $i\in \mathbb{S},R>0$ and $x,y\in {\mathbb{R}}^d$ with $\left|x\right|\vee \left|y\right|\le R$ and $|x-y|\le {\delta }_0$,

$$\begin{array}{c}\hfill {|b(x,i)-b(y,i)|+\parallel \sigma (x,i)-\sigma (y,i)\parallel }^2\le {\kappa }_R\rho \left(\right|x-y\left|\right),\end{array}$$

(7)

where ${\kappa }_R$ is a positive constant. Then, for $\delta \in (0,1),t\in [0,T]$, it holds that

$$\begin{array}{c}\hfill \underset{\delta \to 0}{lim}\mathbb{E}\left[\underset{0\le t\le T}{sup}|X\left(t\right)-Y\left(t\right)|\right]=0.\end{array}$$

(8)

Remark 2.

Examples of functions satisfying (6) include $\rho \left(r\right)=r$ and concave and increasing functions such as $\rho \left(r\right)=rlog(1/r),\rho \left(r\right)=rlog(log(1/r\left)\right),$ and $\rho \left(r\right)=rlog(1/r)log(log(1/r\left)\right)$ for $r\in (0,{\epsilon }^{*})$ with ${\epsilon }^{*}>0$ small enough. When $\rho \left(r\right)=r$, (7) is just the usual local Lipschitz conditions.

Theorem 3.

Assume $\left(\mathrm{Q}1\right)$ and $\left(\mathrm{Q}2\right)$ hold. Let there exist a positive number ${\delta }_0$ and a nondecreasing, concave function $\rho :[0,\infty )\to [1,\infty )$ satisfying ${\int }_{0^{+}}\frac{dr}{\rho \left(r\right)}=\infty$ such that for all $i\in \mathbb{S},R>0$ and $x,y\in {\mathbb{R}}^d$ with $\left|x\right|\vee \left|y\right|\le R$ and $|x-y|\le {\delta }_0$,

$$\begin{array}{c}\hfill {|b(x,i)-b(y,i)|}^2{+\parallel \sigma (x,i)-\sigma (y,i)\parallel }^2\le {\kappa }_R{\rho \left(\right|x-y|}^2),\end{array}$$

where ${\kappa }_R$ is a positive constant. Then, it holds that

$$\begin{array}{c}\hfill \underset{\delta \to 0}{lim}\mathbb{E}\left[\underset{0\le t\le T}{sup}{|X\left(t\right)-Y\left(t\right)|}^2\right]=0.\end{array}$$

3. Proof of Main Results

3.1. Proof of Theorem 1

In order to obtain Theorem 1, we first prove the following proposition. Denote

$$Z\left(t\right)=X\left(t\right)-Y\left(t\right)\mathrm{and}U\left(t\right)=|Y\left(t\right)-Y\left(t_{\delta }\right)|.$$

Proposition 1.

Let the conditions of Theorem 1 hold. Then, for any $\xi >1$ and $\epsilon \in (0,1)$, we have

$$\begin{array}{c}\hfill \left|Z\left(t\right)\right|\le C{\int }_0^t\left|Z\left(s\right)\right|\mathrm{d}s+C{\int }_0^{t}U\left(s\right)\mathrm{d}s+V\left(t\right)+M\left(t\right),\end{array}$$

(9)

where

$$\begin{array}{cc}\hfill V\left(t\right)=& \epsilon +C\left(\frac{\xi }{\epsilon log\xi }+\frac{\xi }{\epsilon }\right){\int }_0^t{U}^2\left(s\right)\mathrm{d}s\hfill \\ \hfill & +C\left(\frac{\xi }{\epsilon log\xi }+\frac{\xi }{\epsilon }\right){\int }_0^t\left({\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right)\mathrm{d}s+\frac{C\epsilon }{log\xi },\hfill \end{array}$$

and $M\left(t\right)$ is a martingale. The constants C depend on only $T,x,d.$

Proof. 

The proof is similar to that of Proposition 3.1 in [12]; however, the difference is that we have to deal with ${\mathbf{1}}_{\{\Lambda \left(t\right)\ne {\Lambda }^{\prime }\left(t\right)\}}$ since $X\left(t\right)$ and $\Lambda \left(t\right)$ are dependent here. Inspired by [16], we will use the approximation technique of Yamada and Watanabe (see Theorem 1 in [18]). For each $\xi >1$ and $\epsilon \in (0,1)$, we can define a continuous function ${\psi }_{\xi ,\epsilon }:\mathbb{R}\to {\mathbb{R}}^{+}$ such that ${\int }_{\epsilon /\xi }^{\epsilon }{\psi }_{\xi ,\epsilon }\left(z\right)\mathrm{d}z=1;0\le {\psi }_{\xi ,\epsilon }\left(z\right)\le \frac{2}{zlog\xi },z>0.$ Furthermore, we define a function ${\varphi }_{\xi ,\epsilon }\in C^2(\mathbb{R};\mathbb{R})$ by

$${\varphi }_{\xi ,\epsilon }\left(r\right):={\int }_0^{\left|r\right|}{\int }_0^y{\psi }_{\xi ,\epsilon }\left(z\right)\mathrm{d}z\mathrm{d}y.$$

It is easy to verify that ${\varphi }_{\xi ,\epsilon }$ has the following useful properties:

$$\begin{array}{c}\hfill 0\le |{\varphi }_{\xi ,\epsilon }^{\prime }\left(r\right)|\le 1,|r|\le \epsilon +{\varphi }_{\xi ,\epsilon }\left(\right|r\left|\right),\mathrm{for}\mathrm{any}r\in \mathbb{R},\end{array}$$

(10)

$$\begin{array}{c}\hfill {\varphi }_{\xi ,\epsilon }^{″}\left(\right|r\left|\right)={\psi }_{\xi ,\epsilon }\left(\right|r\left|\right)\le \frac{2}{\left|r\right|log\xi }{\mathbf{1}}_{[\epsilon /\xi ,\epsilon ]}\left(\right|r\left|\right),\mathrm{for}\mathrm{any}r\in \mathbb{R}\setminus \left\{0\right\}.\end{array}$$

Notice also that the derivative of ${\varphi }_{\xi ,\epsilon }$ at the origin is equal to 0. Then, by Itô’s formulas, Equation (10) implies that

$$\begin{array}{cc}\hfill \left|Z\right(t\left)\right|\le & \epsilon +{\varphi }_{\xi ,\epsilon }\left(\right|Z\left(t\right)\left|\right)\hfill \\ \hfill =& \epsilon +{\int }_0^t\langle \nabla {\varphi }_{\xi ,\epsilon }\left(\right|Z\left(s\right)\left|\right),(b(X\left(s\right),\Lambda \left(s\right))-b(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right)))\rangle \mathrm{d}s\hfill \\ \hfill & +\frac{1}{2}{\int }_0^t{\nabla }^2{\varphi }_{\xi ,\epsilon }\left(\right|Z\left(s\right)\left|\right)(\sigma (X\left(s\right),\Lambda \left(s\right))-\sigma (Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))){(\sigma (X\left(s\right),\Lambda \left(s\right))-\sigma (Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right)))}^T\mathrm{d}s\hfill \\ \hfill & +{\int }_0^t\langle \nabla {\varphi }_{\xi ,\epsilon }\left(\right|Z\left(s\right)\left|\right),(\sigma (X\left(s\right),\Lambda \left(s\right))-\sigma (Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right)))\rangle \mathrm{d}B\left(s\right)\hfill \\ \hfill =:& \epsilon +I_1\left(t\right)+J_1\left(t\right)+M\left(t\right).\hfill \end{array}$$

(11)

Note that for any $r\in \mathbb{R}$, $|{\varphi }_{\xi ,\epsilon }^{\prime }\left(r\right)|\le 1,$ one has

$$\begin{array}{cc}\hfill I_1\left(t\right)\le & {\int }_0^t|\nabla {\varphi }_{\xi ,\epsilon }\left(\right|Z\left(s\right)\left|\right)\left|\right|(b(X\left(s\right),\Lambda \left(s\right))-b(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right)))|\mathrm{d}s\hfill \\ \hfill \le & {\int }_0^t[|b(X\left(s\right),\Lambda \left(s\right))-b(Y\left(s\right),\Lambda \left(s\right))|+|b(Y\left(s\right),\Lambda \left(s\right))-b(Y\left(s_{\delta }\right),\Lambda \left(s\right))|\hfill \\ \hfill & +|b(Y\left(s_{\delta }\right),\Lambda \left(s\right))-b(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s\right))|+|b(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s\right))-b(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))|]\mathrm{d}s\hfill \\ \hfill \le & {\int }_0^t\left[C\left|Z\left(s\right)\right|+CU\left(s\right)+2C{\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+2C{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right]\mathrm{d}s\hfill \\ \hfill \le & C{\int }_0^t\left|Z\left(s\right)\right|\mathrm{d}s+C{\int }_0^{t}U\left(s\right)\mathrm{d}s+C{\int }_0^t\left({\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right)\mathrm{d}s.\hfill \end{array}$$

(12)

Since partial differentiations of ${\varphi }_{\xi ,\epsilon }^{\prime }$ give the following: for any $x\in {\mathbb{R}}^d\setminus \left\{0\right\}$ and $i=1,2,\cdots ,d,$

$$\begin{array}{cc}\hfill & {\partial }_i^2{\varphi }_{\xi ,\epsilon }\left(\right|x\left|\right)={\varphi }_{\xi ,\epsilon }^{″}\left(\right|x\left|\right)\frac{x_i^2}{{\left|x\right|}^2}+{\varphi }_{\xi ,\epsilon }^{\prime }\left(\right|x\left|\right)\left(\frac{{\left|x\right|}^2-x_i^2}{{\left|x\right|}^3}\right),\hfill \\ \hfill & {\partial }_i{\partial }_j{\varphi }_{\xi ,\epsilon }\left(\right|x\left|\right)={\varphi }_{\xi ,\epsilon }^{″}\left(\right|x\left|\right)\frac{x_i{x}_j}{{\left|x\right|}^2}-{\varphi }_{\xi ,\epsilon }^{\prime }\left(\right|x\left|\right)\left(\frac{x_i{x}_j}{{\left|x\right|}^3}\right),\hfill \end{array}$$

we can see that

$$\begin{array}{cc}\hfill J_1\left(t\right)=& \frac{1}{2}\sum _{k=1}^d\sum _{i,j=1}^d{\int }_0^t{\partial }_i{\partial }_j{\varphi }_{\xi ,\epsilon }\left(\right|Z\left(s\right)\left|\right)\{{\sigma }_{i,k}(X\left(s\right),\Lambda \left(s\right))-{\sigma }_{i,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))\}\hfill \\ \hfill & ·\{{\sigma }_{j,k}(X\left(s\right),\Lambda \left(s\right))-{\sigma }_{j,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))\}\mathrm{d}s\hfill \\ \hfill =:& A_1\left(t\right)+A_2\left(t\right),\hfill \end{array}$$

(13)

where

$$\begin{array}{cc}\hfill A_1\left(t\right)=& \frac{1}{2}\sum _{k=1}^d\sum _{i,j=1}^d{\int }_0^t{\varphi }_{\xi ,\epsilon }^{″}\left(\right|Z\left(s\right)\left|\right)\frac{Z_i\left(s\right)Z_j\left(s\right)}{{\left|Z\left(s\right)\right|}^2}\{{\sigma }_{i,k}(X\left(s\right),\Lambda \left(s\right))-{\sigma }_{i,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))\}\hfill \\ \hfill & ·\{{\sigma }_{j,k}(X\left(s\right),\Lambda \left(s\right))-{\sigma }_{j,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))\}\mathrm{d}s.\hfill \\ \hfill A_2\left(t\right)=& \frac{1}{2}\sum _{k=1}^d\sum _{i=1}^d{\int }_0^t{\varphi }_{\xi ,\epsilon }^{\prime }\left(\right|Z\left(s\right)\left|\right)\frac{{\left|Z\left(s\right)\right|}^2-{\left|Z_i\left(s\right)\right|}^2}{{\left|Z\left(s\right)\right|}^3}|{\sigma }_{i,k}(X\left(s\right),\Lambda \left(s\right))-{\sigma }_{i,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right)){|}^2\mathrm{d}s\hfill \\ \hfill & -\sum _{k=1}^d\sum _{1<i<j<d}^d{\int }_0^t{\varphi }_{\xi ,\epsilon }^{\prime }\left(\right|Z\left(s\right)\left|\right)\frac{Z_i\left(s\right)Z_j\left(s\right)}{{\left|Z\left(s\right)\right|}^3}\{{\sigma }_{i,k}(X\left(s\right),\Lambda \left(s\right))-{\sigma }_{i,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))\}\hfill \\ \hfill & ·\{{\sigma }_{j,k}(X\left(s\right),\Lambda \left(s\right))-{\sigma }_{j,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))\}\mathrm{d}s.\hfill \end{array}$$

Firstly, by the conditions of Theorem 1 and (A1), we can readily verify that

$$\begin{array}{cc}\hfill A_1\left(t\right)\le & \frac{1}{2}\sum _{k=1}^d\sum _{i,j=1}^d{\int }_0^t{\varphi }_{\xi ,\epsilon }^{″}\left(\right|Z\left(s\right)\left|\right)\frac{|Z_i\left(s\right)\left|\right|Z_j\left(s\right)|}{{\left|Z\left(s\right)\right|}^2}\{|{\sigma }_{i,k}(X\left(s\right),\Lambda \left(s\right))-{\sigma }_{i,k}(Y\left(s\right),\Lambda \left(s\right))|\hfill \\ \hfill & +|{\sigma }_{i,k}(Y\left(s\right),\Lambda \left(s\right))-{\sigma }_{i,k}(Y\left(s_{\delta }\right),\Lambda \left(s\right))|+|{\sigma }_{i,k}(Y\left(s_{\delta }\right),\Lambda \left(s\right))-{\sigma }_{i,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s\right))|\hfill \\ \hfill & +|{\sigma }_{i,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s\right))-{\sigma }_{i,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))\left|\right\}\hfill \\ \hfill & \times \left\{\right|{\sigma }_{j,k}(X\left(s\right),\Lambda \left(s\right))-{\sigma }_{j,k}(Y\left(s\right),\Lambda \left(s\right))|+|{\sigma }_{j,k}(Y\left(s\right),\Lambda \left(s\right))-{\sigma }_{j,k}(Y\left(s_{\delta }\right),\Lambda \left(s\right))|\hfill \\ \hfill & +|{\sigma }_{j,k}(Y\left(s_{\delta }\right),\Lambda \left(s\right))-{\sigma }_{j,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s\right))|+|{\sigma }_{j,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s\right))-{\sigma }_{j,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))\left|\right\}\mathrm{d}s\hfill \\ \hfill \le & \frac{1}{2}C{\int }_0^t{\varphi }_{\xi ,\epsilon }^{″}\left(\right|Z\left(s\right)\left|\right)\left\{{\left|Z\left(s\right)\right|}^2+{|Y\left(s\right)-Y\left(s_{\delta }\right)|}^2+{\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right\}\mathrm{d}s\hfill \\ \hfill \le & C{\int }_0^t\frac{{\mathbf{1}}_{[\epsilon /\xi ,\epsilon ]}\left(\right|Z\left(s\right)\left|\right)}{\left|Z\right(s\left)\right|log\xi }\left\{{\left|Z\left(s\right)\right|}^2+U^2\left(s\right)+{\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right\}\mathrm{d}s\hfill \\ \hfill \le & \frac{C\epsilon }{log\xi }+\frac{C\xi }{\epsilon log\xi }{\int }_0^t{U}^2\left(s\right)\mathrm{d}s+\frac{C\xi }{\epsilon log\xi }{\int }_0^t\left({\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right)\mathrm{d}s.\hfill \end{array}$$

(14)

Then, by the conditions of Theorem 1 and (A1), we get

$$\begin{array}{cc}\hfill A_2\left(t\right)=& \frac{1}{2}\sum _{k=1}^d\sum _{i=1}^d{\int }_0^t{\varphi }_{\xi ,\epsilon }^{\prime }\left(\right|Z\left(s\right)\left|\right)\frac{{\left|Z\left(s\right)\right|}^2-{\left|Z_i\left(s\right)\right|}^2}{{\left|Z\left(s\right)\right|}^3}|{\sigma }_{i,k}(X\left(s\right),\Lambda \left(s\right))-{\sigma }_{i,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right)){|}^2\mathrm{d}s\hfill \\ \hfill & -\sum _{k=1}^d\sum _{1<i<j<d}^d{\int }_0^t{\varphi }_{\xi ,\epsilon }^{\prime }\left(\right|Z\left(s\right)\left|\right)\frac{Z_i\left(s\right)Z_j\left(s\right)}{{\left|Z\left(s\right)\right|}^3}\{{\sigma }_{i,k}(X\left(s\right),\Lambda \left(s\right))-{\sigma }_{i,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))\}\hfill \\ \hfill \le & \frac{1}{2}\sum _{k=1}^d\sum _{i=1}^d{\int }_0^t{\varphi }_{\xi ,\epsilon }^{\prime }\left(\right|Z\left(s\right)\left|\right)\frac{{\left|Z\left(s\right)\right|}^2-{\left|Z_i\left(s\right)\right|}^2}{{\left|Z\left(s\right)\right|}^3}|{\sigma }_{i,k}(X\left(s\right),\Lambda \left(s\right))-{\sigma }_{i,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right)){|}^2\mathrm{d}s\hfill \\ \hfill & +\sum _{k=1}^d\sum _{1<i<j<d}^d{\int }_0^t{\varphi }_{\xi ,\epsilon }^{\prime }\left(\right|Z\left(s\right)\left|\right)\frac{Z_i\left(s\right)Z_j\left(s\right)}{{\left|Z\left(s\right)\right|}^3}\{{\sigma }_{i,k}(X\left(s\right),\Lambda \left(s\right))-{\sigma }_{i,k}(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))\}\hfill \\ \hfill \le & \frac{1}{2}{d}^2{\int }_0^t\frac{{\varphi }_{\xi ,\epsilon }^{\prime }\left(\right|Z\left(s\right)\left|\right)}{\left|Z\right(s\left)\right|}\left\{{\left|Z\left(s\right)\right|}^2+U^2\left(s\right)+{\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right\}\mathrm{d}s\hfill \\ \hfill & +d^2{\int }_0^t\frac{{\varphi }_{\xi ,\epsilon }^{\prime }\left(\right|Z\left(s\right)\left|\right)}{\left|Z\right(s\left)\right|}\left\{{\left|Z\left(s\right)\right|}^2+U^2\left(s\right)+{\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right\}\mathrm{d}s\hfill \\ \hfill \le & C{\int }_0^t\left|Z\left(s\right)\right|\mathrm{d}s+\frac{C\xi }{\epsilon }{\int }_0^t{U}^2\left(s\right)\mathrm{d}s+\frac{C\xi }{\epsilon }{\int }_0^t\left({\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right)\mathrm{d}s.\hfill \end{array}$$

(15)

Substituting (14) and (15) into (13) yields

$$\begin{array}{cc}\hfill J_1\left(t\right)=& A_1\left(t\right)+A_2\left(t\right)\hfill \\ \hfill \le & C{\int }_0^t\left|Z\left(s\right)\right|\mathrm{d}s+\left(\frac{C\xi }{\epsilon log\xi }+\frac{C\xi }{\epsilon }\right){\int }_0^t{U}^2\left(s\right)\mathrm{d}s\hfill \\ \hfill & +\left(\frac{C\xi }{\epsilon log\xi }+\frac{C\xi }{\epsilon }\right){\int }_0^t\left({\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right)\mathrm{d}s+\frac{C\epsilon }{log\xi }.\hfill \end{array}$$

(16)

Inserting (12) and (16) into (11), we have

$$\begin{array}{cc}\hfill \left|Z\right(t\left)\right|\le & \epsilon +I_1\left(t\right)+J_1\left(t\right)+M\left(t\right)\hfill \\ \hfill \le & C{\int }_0^t\left|Z\left(s\right)\right|\mathrm{d}s+C{\int }_0^{t}U\left(s\right)\mathrm{d}s+V\left(t\right)+M\left(t\right),\hfill \end{array}$$

(17)

where

$$\begin{array}{c}\hfill V\left(t\right)=\epsilon +C\left(\frac{\xi }{\epsilon log\xi }+\frac{\xi }{\epsilon }\right){\int }_0^t\left[U^2\left(s\right)+{\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right]\mathrm{d}s+\frac{C\epsilon }{log\xi }.\end{array}$$

(18)

□

Proof of Theorem 1.

Define $Z^{*}\left(t\right)={sup}_{0\le s\le t}|X\left(s\right)-Y\left(s\right)|={sup}_{0\le s\le t}\left|Z\left(s\right)\right|.$ We will check that $Z^{*}\left(t\right)$ satisfies (5), that is, for some $p>0,q\ge 1,\varrho \in [1,q],$ there exist constants C that depend on $T,x,d$ but are independent of $\delta$ and $\vartheta \ge 0$, such that

$$\mathbb{E}{\left(Z^{*}\left(t\right)\right)}^p\le C\mathbb{E}{\left({\int }_0^t{Z}^{*}\left(s\right)\mathrm{d}s\right)}^p+C\mathbb{E}{\left({\int }_0^t{\left|Z\left(s\right)\right|}^{\varrho }\mathrm{d}s\right)}^{p/q}+\vartheta .$$

It follows from (9) that

$$Z^{*}\left(t\right)\le C{\int }_0^t{Z}^{*}\left(s\right)\mathrm{d}s+C{\int }_0^{t}U\left(s\right)\mathrm{d}s+V\left(t\right)+\underset{0\le s\le t}{sup}M\left(s\right).$$

To proceed, the inequality ${|a+b|}^p\le 2^{p-1}{\left(\right|a|}^p+{\left|b\right|}^p)$ for $p\ge 1$ leads to

$$\begin{array}{cc}\hfill {\left(Z^{*}\left(t\right)\right)}^p\le & {\left(C{\int }_0^t{Z}^{*}\left(s\right)\mathrm{d}s+C{\int }_0^{t}U\left(s\right)\mathrm{d}s+V\left(t\right)+\underset{0\le s\le t}{sup}M\left(s\right)\right)}^p\hfill \\ \hfill \le & 4^{p-1}\left[C{\left({\int }_0^t{Z}^{*}\left(s\right)\mathrm{d}s\right)}^p+C{\left({\int }_0^{t}U\left(s\right)\mathrm{d}s\right)}^p+{\left|V\left(t\right)\right|}^p+{\left(\underset{0\le s\le t}{sup}\left|M\left(s\right)\right|\right)}^p\right].\hfill \end{array}$$

Taking expectation of the above inequality, it follows that

$$\begin{array}{c}\hfill \mathbb{E}{\left(Z^{*}\left(t\right)\right)}^p\le C\mathbb{E}{\left({\int }_0^t{Z}^{*}\left(s\right)\mathrm{d}s\right)}^p+C\mathbb{E}{\left({\int }_0^{t}U\left(s\right)\mathrm{d}s\right)}^p+C\mathbb{E}{\left(V\left(t\right)\right)}^p+C\mathbb{E}{\left(\underset{0\le s\le t}{sup}\left|M\left(s\right)\right|\right)}^p.\end{array}$$

(19)

By Proposition A2 and Hölder’s inequality, we get

$$\begin{array}{c}\hfill \mathbb{E}{\left({\int }_0^{t}U\left(s\right)\mathrm{d}s\right)}^p\le T^{p-1}\mathbb{E}\left({\int }_0^t{U}^p\left(s\right)\mathrm{d}s\right)\le T^{p-1}{\int }_0^t\mathbb{E}{U}^p\left(s\right)\mathrm{d}s\le T^{p-1}{\int }_0^{t}C{\delta }^{p/2}\mathrm{d}s\le C{\delta }^{p/2}.\end{array}$$

(20)

From (18), we obtain

$$\begin{array}{cc}\hfill {\left(V\left(t\right)\right)}^p=& \{\epsilon +C\left(\frac{\xi }{\epsilon log\xi }+\frac{\xi }{\epsilon }\right){\int }_0^t{U}^2\left(s\right)\mathrm{d}s\hfill \\ \hfill & +C\left(\frac{\xi }{\epsilon log\xi }+\frac{\xi }{\epsilon }\right){\int }_0^t\left({\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right)\mathrm{d}s{\}}^p\hfill \\ \hfill \le & 3^{p-1}\{{\epsilon }^p+C{\left(\frac{\xi }{\epsilon log\xi }+\frac{\xi }{\epsilon }\right)}^p|{\int }_0^t{U}^2\left(s\right)\mathrm{d}s{|}^p\hfill \\ \hfill & +C{\left(\frac{\xi }{\epsilon log\xi }+\frac{\xi }{\epsilon }\right)}^p\left|{\int }_0^t\left({\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right)\mathrm{d}s{|}^p\right\}.\hfill \end{array}$$

(21)

The Hölder inequality and Proposition A2 imply

$$\begin{array}{cc}\hfill \mathbb{E}|{\int }_0^t{U}^2\left(s\right)\mathrm{d}s{|}^p\le & \mathbb{E}{\left({\left({\int }_0^t{U}^{2p}\left(s\right)\mathrm{d}s\right)}^{1/p}{\left({\int }_0^t{1}^q\mathrm{d}s\right)}^{1/q}\right)}^p\hfill \\ \hfill \le & C\mathbb{E}{\int }_0^t{U}^{2p}\left(s\right)\mathrm{d}s=C{\int }_0^t\mathbb{E}{U}^{2p}\left(s\right)\mathrm{d}s\le C{\delta }^p.\hfill \end{array}$$

(22)

Now, we estimate $\mathbb{E}{\int }_0^t{\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}\mathrm{d}s$ and $\mathbb{E}{\int }_0^t{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\mathrm{d}s$. For $t>0$, set $K=[t/\delta ],t_{\delta }=k\delta$ for $K\le k\le K+1$ and $t_{K+1}=t$. Then, according to (4) and (Q1),

$$\begin{array}{c}\hfill \mathbb{E}{\int }_0^t{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\mathrm{d}s={\int }_0^t\mathbb{E}{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\mathrm{d}s=\sum _{k=0}^K{\int }_{t_k}^{t_{k+1}}\mathbb{P}({\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_k\right))\mathrm{d}s\le C\delta +o\left(\delta \right).\end{array}$$

(23)

Under the Lipschitz condition, according to the proof of Lemma 3.2 in [14], there exists a constant $C>0$ depending on T such that

$$\begin{array}{cc}\hfill \mathbb{E}{\int }_0^t{\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}\mathrm{d}s=& {\int }_0^t\mathbb{E}{\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}\mathrm{d}s={\int }_0^t\mathbb{P}(\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right))\mathrm{d}s\hfill \\ \hfill \le & C{\delta }^{1/2}+C{\int }_0^t\mathbb{E}\left|Z\left(s\right)\right|\mathrm{d}s.\hfill \end{array}$$

(24)

Then, by Hölder’s inequality, (23), (24) and $\mathbb{E}\left[{sup}_{0\le s\le t}|X\left(s\right)-Y\left(s\right)|\right]\le C{\delta }^{1/2}$ in [14], we obtain that

$$\begin{array}{cc}\hfill & \mathbb{E}|{\int }_0^t\left({\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right)\mathrm{d}s{|}^p\le \mathbb{E}\left(T^{p-1}{\int }_0^t{\left({\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right)}^p\mathrm{d}s\right)\hfill \\ \hfill \le & T^{p-1}{2}^{p-1}\left\{\mathbb{E}{\int }_0^t{\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}\mathrm{d}s+\mathbb{E}{\int }_0^t{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\mathrm{d}s\right\}\hfill \\ \hfill \le & C\delta +o\left(\delta \right)+C{\delta }^{1/2}+C{\int }_0^t\mathbb{E}\left|Z\left(s\right)\right|\mathrm{d}s\le C{\delta }^{1/2}.\hfill \end{array}$$

(25)

Taking expectation of the inequality (21), we have from (22) and (25) that

$$\begin{array}{cc}\hfill \mathbb{E}{\left(V\left(t\right)\right)}^p\le & 3^{p-1}\mathbb{E}\{{\epsilon }^p+C{\left(\frac{\xi }{\epsilon log\xi }+\frac{\xi }{\epsilon }\right)}^p|{\int }_0^t{U}^2\left(s\right)\mathrm{d}s{|}^p\hfill \\ \hfill & +C{\left(\frac{\xi }{\epsilon log\xi }+\frac{\xi }{\epsilon }\right)}^p\left|{\int }_0^t\left({\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right)\mathrm{d}s{|}^p\right\}\hfill \\ \hfill \le & 3^{p-1}{\epsilon }^p+C{\left(\frac{\xi }{\epsilon log\xi }+\frac{\xi }{\epsilon }\right)}^p\mathbb{E}|{\int }_0^t{U}^2\left(s\right)\mathrm{d}s{|}^p\hfill \\ \hfill & +C{\left(\frac{\xi }{\epsilon log\xi }+\frac{\xi }{\epsilon }\right)}^p\mathbb{E}|{\int }_0^t\left({\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right)\mathrm{d}s{|}^p\hfill \\ \hfill \le & C{\left(\frac{\xi }{\epsilon log\xi }+\frac{\xi }{\epsilon }\right)}^p{\delta }^p+C{\left(\frac{\xi }{\epsilon log\xi }+\frac{\xi }{\epsilon }\right)}^p{\delta }^{1/2}+C{\epsilon }^p\hfill \\ \hfill \le & C{\left(\frac{\xi }{\epsilon log\xi }+\frac{\xi }{\epsilon }\right)}^p{\delta }^{1/2}+C{\epsilon }^p.\hfill \end{array}$$

(26)

It suffices to show from Proposition A2, BDG’s inequality and $|{\partial }_i{\varphi }_{\xi ,\epsilon }\left(\right|x\left|\right)|\le 1$ that

$$\begin{array}{cc}\hfill \mathbb{E}\left(\underset{0\le s\le t}{sup}{\left|M\left(s\right)\right|}^p\right)=& \mathbb{E}{\left(\underset{0\le s\le t}{sup}\frac{1}{2}|{\int }_0^s\langle \nabla {\varphi }_{\xi ,\epsilon }\left(\right|Z\left(r\right)\left|\right),(\sigma (X\left(r\right),\Lambda \left(r\right))-\sigma (Y\left(r_{\delta }\right),{\Lambda }^{\prime }\left(r_{\delta }\right)))\rangle \mathrm{d}B\left(r\right)|\right)}^p\hfill \\ \hfill \le & \mathbb{E}{\left(\underset{0\le s\le t}{sup}\frac{1}{2}{\int }_0^s|\nabla {\varphi }_{\xi ,\epsilon }\left(\right|Z\left(r\right)\left|\right)\left|\right|\sigma (X\left(r\right),\Lambda \left(r\right))-\sigma (Y\left(r_{\delta }\right),{\Lambda }^{\prime }\left(r_{\delta }\right))|\mathrm{d}B\left(r\right)\right)}^p\hfill \\ \hfill \le & \mathbb{E}{\left(\underset{0\le s\le t}{sup}\frac{1}{2}{\int }_0^s|\sqrt{d}\left|\right|\sigma (X\left(r\right),\Lambda \left(r\right))-\sigma (Y\left(r_{\delta }\right),{\Lambda }^{\prime }\left(r_{\delta }\right))|\mathrm{d}B\left(r\right)\right)}^p\hfill \\ \hfill \le & C\mathbb{E}{\left[{\int }_0^t{|\sigma (X\left(s\right),\Lambda \left(s\right))-\sigma (Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))|}^2\mathrm{d}s\right]}^{p/2}\hfill \\ \hfill \le & C\mathbb{E}{\left({\int }_0^t{\left|Z\left(s\right)\right|}^2\mathrm{d}s\right)}^{p/2}+C{\delta }^{p/2}.\hfill \end{array}$$

(27)

Inserting (20), (26), and (27) into (19), we derive

$$\begin{array}{cc}\hfill \mathbb{E}{\left(Z^{*}\left(t\right)\right)}^p\le & C\mathbb{E}{\left({\int }_0^t{Z}^{*}\left(s\right)\mathrm{d}s\right)}^p+C\mathbb{E}{\left({\int }_0^{t}U\left(s\right)\mathrm{d}s\right)}^p+C\mathbb{E}{\left(V\left(t\right)\right)}^p+C\mathbb{E}{\left(\underset{0\le s\le t}{sup}M\left(s\right)\right)}^p\hfill \\ \hfill \le & C\mathbb{E}{\left({\int }_0^t{Z}^{*}\left(s\right)\mathrm{d}s\right)}^p+C\mathbb{E}{\left({\int }_0^t{\left|Z\left(s\right)\right|}^2\mathrm{d}s\right)}^{p/2}+\vartheta \left(\delta \right),\hfill \end{array}$$

where

$$\vartheta \left(\delta \right)=C{\left(\frac{\xi }{\epsilon log\xi }+\frac{\xi }{\epsilon }\right)}^p{\delta }^{1/2}+C{\delta }^{p/2}+C{\epsilon }^p>0.$$

We choose $\xi =2$ and $\epsilon ={\delta }^{1/4p}$. Letting $q=2$ and $p\ge q=\varrho =2$ from Lemma 1, we conclude

$$\mathbb{E}{\left(Z^{*}\left(t\right)\right)}^p\le C\vartheta \left(\delta \right)\le C{\delta }^{1/4}.$$

This completes the proof. □

Proof of Corollary 1.

In Theorem 1 we have discussed the case when $p\ge 2$. Let us now consider the case of $p\in (1,2)$. This is rather easy if we note that Hölder’s inequality implies

$$\mathbb{E}\left[\underset{0\le t\le T}{sup}{|X\left(t\right)-Y\left(t\right)|}^p\right]\le {\left[\mathbb{E}\left[\underset{0\le t\le T}{sup}{|X\left(t\right)-Y\left(t\right)|}^2\right]\right]}^{p/2}\le C{\delta }^{p/8},$$

which yields the desired conclusion. □

3.2. Proof of Theorem 2

Before proving Theorem 2, we present the following proposition.

Proposition 2.

Let the conditions of Theorem 2 hold. Then, for any $t\in [0,T]$, there exists a positive constant C independent of δ such that

$$\begin{array}{c}\hfill \mathbb{E}\left[{\mathbf{1}}_{\{\Lambda \left(t\right)\ne {\Lambda }^{\prime }\left(t\right)\}}\right]\le C\delta .\end{array}$$

Proof. 

The difference between this proposition and the previous result is that we handle ${\mathbf{1}}_{\{\Lambda \left(t\right)\ne {\Lambda }^{\prime }\left(t\right)\}}$ under the non-Lipschitz condition. In order to overcome difficulties, we will use the technique of stopping times. Define $\tau :=inf\{t\ge 0:\Lambda \left(t\right)\ne {\Lambda }^{\prime }\left(t\right)\}$ and $S_{{\delta }_0}:=inf\{t\ge 0:|Z\left(t\right)|\ge {\delta }_0\}=inf\{t\ge 0:|X\left(t\right)-Y\left(t\right)|\ge {\delta }_0\}.$ For $R>0$, let $T_R:=inf\{t\ge 0:|X\left(t\right)|\vee |Y\left(t\right)|>R\}$. Then, $T_R\to \infty$ as $R\to \infty .$ Moreover, we also use the approximation technique of Yamada and Watanabe, but it is different from the proof of Proposition 1. Thanks to the assumption imposed on the function $\rho$, for each $\xi >1$ and $\epsilon \in (0,1)$, we can define a continuous function ${\rho }^{*}$ on $\mathbb{R}$ with support on $(\epsilon /\xi ,\epsilon )$, so that $0\le {\rho }^{*}\left(r\right)\le {\rho }^{-1}\left(r\right)$ holds for every $r>0$, and ${\int }_{\epsilon /\xi }^{\epsilon }{\rho }^{*}\left(r\right)\mathrm{d}r=1.$

Now, we consider the following function

$$\begin{array}{c}\hfill V_{\epsilon ,\xi }\left(r\right):={\int }_0^{\left|r\right|}{\int }_0^y{\rho }^{*}\left(u\right)\mathrm{d}u\mathrm{d}y,r\in \mathbb{R}.\end{array}$$

We can immediately verify that $V_{\epsilon ,\xi }$ is even and twice continuously differentiable with $|V_{\epsilon ,\xi }^{\prime }\left(r\right)|\le 1$. Furthermore, $\left\{V_{\epsilon ,\xi }(·)\right\}$ is a nondecreasing function. Itô’s formula shows that for $0\le t\le T$,

$$\begin{array}{cc}\hfill & \mathbb{E}[V_{\epsilon ,\xi }\left(\right|Z(t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R)\left|\right)]\hfill \\ \hfill & \le \mathbb{E}[{\int }_0^{t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R}\langle \nabla V_{\epsilon ,\xi }\left(\right|Z\left(s\right)\left|\right),(b(X\left(s\right),\Lambda \left(s\right))-b(Y\left(s_{\delta }\right),\Lambda \left(s\right)))\rangle \mathrm{d}s\hfill \\ \hfill & +\frac{1}{2}{\int }_0^{t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R}\parallel {\nabla }^2{V}_{\epsilon ,\xi }\left(\right|Z\left(s\right)\left|\right)\parallel \parallel \sigma (X\left(s\right),\Lambda \left(s\right))-\sigma (Y\left(s_{\delta }\right),\Lambda \left(s\right)){\parallel }^2\mathrm{d}s]\hfill \\ \hfill & =:I_2+J_2.\hfill \end{array}$$

(28)

Noting that $|\partial V_{\epsilon ,\xi }\left(\right|x\left|\right)/\partial x_i|\le 1,1\le i\le d$, by (7) and Fubini’s theorem, we compute that

$$\begin{array}{cc}\hfill I_2\le & \sqrt{d}\mathbb{E}{\int }_0^{t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R}\left[\right|b(X\left(s\right),\Lambda \left(s\right))-b(Y\left(s\right),\Lambda \left(s\right))|+|b(Y\left(s\right),\Lambda \left(s\right))-b(Y\left(s_{\delta }\right),\Lambda \left(s\right))\left|\right]\mathrm{d}s\hfill \\ \hfill \le & \sqrt{d}\mathbb{E}{\int }_0^{t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R}[{\kappa }_R\rho \left(\right|Z\left(s\right)\left|\right)+{\kappa }_R\rho \left(\right|Y\left(s\right)-Y\left(s_{\delta }\right)\left|\right)]\mathrm{d}s\hfill \\ \hfill \le & \sqrt{d}{\int }_0^{t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R}\mathbb{E}[{\kappa }_R\rho \left(\right|Z\left(s\right)\left|\right)+{\kappa }_R\rho \left(\right|Y\left(s\right)-Y\left(s_{\delta }\right)\left|\right)]\mathrm{d}s\hfill \\ \hfill \le & \sqrt{d}{\int }_0^{t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R}[{\kappa }_R\rho \left(\mathbb{E}\right|Z\left(s\right)\left|\right)+{\kappa }_R\rho \left(\mathbb{E}\right|Y\left(s\right)-Y\left(s_{\delta }\right)\left|\right)]\mathrm{d}s,\hfill \end{array}$$

(29)

where we used the concavity of $\rho$ and Jensen’s inequality to derive the last inequality. On the other hand, since

$$\frac{\partial V_{\epsilon ,\xi }\left(\right|x\left|\right)}{\partial x_i\partial x_j}=V_{\epsilon ,\xi }^{\prime }\left(\right|x\left|\right)({\delta }_{ij}{\left|x\right|}^2-x_i{x}_j{\left)\right|x|}^{-3}+V_{\epsilon ,\xi }^{″}\left(\right|x\left|\right)x_i{x}_j{\left|x\right|}^{-2},$$

where ${\delta }_{ij}=1$ if $i=j$ or otherwise 0, one can see that

$$\left|\frac{\partial V_{\epsilon ,\xi }\left(\right|x\left|\right)}{\partial x_i\partial x_j}\right|\le \left(\frac{1}{\left|x\right|}+\frac{1}{\rho \left(\right|x\left|\right)}\right){\mathbf{1}}_{\{\epsilon /\xi <|x|<\epsilon \}},$$

and hence

$$\parallel {\nabla }^2{V}_{\epsilon ,\xi }\left(\right|x\left|\right)\parallel \le d\left(\frac{1}{\left|x\right|}+\frac{\left|x\right|}{\rho \left(\right|x\left|\right)}\right){\mathbf{1}}_{\{\epsilon /\xi <|x|<\epsilon \}}.$$

Similar to $I_2$, it is easy to see that

$$\begin{array}{cc}\hfill J_2\left(t\right)\le & \mathbb{E}{\int }_0^{t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R}[d\left(\frac{1}{\left|Z\right(s\left)\right|}+\frac{\left|Z\right(s\left)\right|}{\rho \left(\right|Z\left(s\right)\left|\right)}\right){\mathbf{1}}_{\{\epsilon /\xi <|Z\left(s\right)|<\epsilon \}}\hfill \\ \hfill & ·\left[\right|\sigma (X\left(s\right),\Lambda \left(s\right))-\sigma (Y\left(s\right),\Lambda \left(s\right)){|}^2+|\sigma (Y\left(s\right),\Lambda \left(s\right))-\sigma (Y\left(s_{\delta }\right),\Lambda \left(s\right)){|}^2]]\mathrm{d}s\hfill \\ \hfill \le & C\mathbb{E}{\int }_0^{t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R}\left(\frac{\rho \left(\right|Z\left(s\right)\left|\right)}{\left|Z\right(s\left)\right|}+\epsilon \right)\mathrm{d}s+C{\kappa }_{R}T\left(\frac{1}{\rho (\epsilon /\xi )}+\frac{\xi }{\epsilon }\right)\rho \left(\sqrt{\delta }\right).\hfill \end{array}$$

(30)

Plugging (29) and (30) into (28), we obtain

$$\begin{array}{cc}\hfill & \mathbb{E}[V_{\epsilon ,\xi }\left(\right|Z(t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R)\left|\right)]\hfill \\ \hfill \le & C\frac{\epsilon {\kappa }_R}{\xi }{\int }_0^{t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R}\rho \left(\mathbb{E}\right|Z\left(s\right)\left|\right)\mathrm{d}s+C{\kappa }_R\left(\frac{1}{\rho (\epsilon /\xi )}+\frac{\xi }{\epsilon }\right)\rho \left(\sqrt{\delta }\right)+CT\epsilon ,\hfill \end{array}$$

where we can get the last inequality from Proposition A2. However, by the definition of $V_{\epsilon ,\xi }\left(\right|x\left|\right)$ and the property of $V_{\epsilon ,\xi }\left(x\right)$, we have

$$\mathbb{E}|Z(t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R)|\le \epsilon +\mathbb{E}{V}_{\epsilon ,\xi }\left(\right|Z(t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R)\left|\right).$$

Thus,

$$\begin{array}{cc}\hfill \mathbb{E}\left|Z\right(t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R\left)\right|\le & C\frac{\epsilon {\kappa }_R}{\xi }{\int }_0^{t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R}\rho \left(\mathbb{E}\right|Z\left(s\right)\left|\right)\mathrm{d}s+C(\epsilon ,\xi ,\delta ),\hfill \end{array}$$

where $C(\epsilon ,\xi ,\delta )=C{\kappa }_R\left(\frac{1}{\rho (\epsilon /\xi )}+\frac{\xi }{\epsilon }\right)\rho \left(\sqrt{\delta }\right)+CT\epsilon .$ Applying Bihari’s inequality (i.e., Lemma 2), we obtain that

$$\begin{array}{cc}\hfill \underset{0\le t\le T}{sup}\mathbb{E}\left|Z(t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R)\right|\le & G^{-1}\left(G\left(C(\epsilon ,\xi ,\delta )\right)+C\frac{\epsilon {\kappa }_{R}T}{\xi }\right),\hfill \end{array}$$

provided $G\left(C(\epsilon ,\xi ,\delta )\right)+C\frac{\epsilon {\kappa }_{R}T}{\xi }\in \mathrm{Dom}\left(G^{-1}\right),$ where $G\left(r\right)={\int }_1^r\mathrm{d}u/\rho \left(u\right)$ on $r>0$, and $G^{-1}$ is the inverse function of G. Now, we choose $\epsilon ={\delta }^{1/3},\xi =e^{\delta }$ and $\rho \left(\delta \right)=\delta .$ Recalling the condition that ${\int }_0^1\mathrm{d}u/\rho \left(u\right)=\infty ,$ we see that $(-\infty ,0]\subset \mathrm{Dom}\left(G^{-1}\right),G\left(r\right)\to -\infty$ as $r\to 0$ and $G^{-1}\to 0$ as $r\to -\infty$. Thus,

$$\underset{\delta \to 0}{lim}\underset{0\le t\le T}{sup}\mathbb{E}\left|Z(t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R)\right|=0.$$

On the set $\{S_{{\delta }_0}\le t\}$, we have $2R\ge |Z(t\wedge \tau \wedge T_R)|\ge {\delta }_0.$ Thus, it follows that

$$0=\mathbb{E}|Z(t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R)|\ge \mathbb{E}[|Z(t\wedge \tau \wedge S_{{\delta }_0}\wedge T_R)|{\mathbf{1}}_{\{S_{{\delta }_0}\le t\wedge \tau \wedge T_R\}}]\ge {\delta }_0\mathbb{P}\{S_{{\delta }_0}\le t\wedge \tau \wedge T_R\}.$$

Then, we have $\mathbb{P}\{S_{{\delta }_0}\le t\wedge \tau \wedge T_R\}=0.$ This gives

$$\begin{array}{cc}\hfill \mathbb{E}\left|Z\right(t\wedge \tau \wedge T_R\left)\right|=& \mathbb{E}\left[\right|Z(t\wedge \tau \wedge T_R)|{\mathbf{1}}_{\{S_{{\delta }_0}\le t\wedge \tau \wedge T_R\}}]+\mathbb{E}[|Z(t\wedge \tau \wedge T_R)|{\mathbf{1}}_{\{S_{{\delta }_0}>t\wedge \tau \wedge T_R\}}]\hfill \\ \hfill \le & 2R\mathbb{P}\{S_{{\delta }_0}\le t\wedge \tau \wedge T_R\}+\mathbb{E}\left[\right|Z(t\wedge \tau \wedge T_R)|{\mathbf{1}}_{\{S_{{\delta }_0}>t\wedge \tau \wedge T_R\}}]=0.\hfill \end{array}$$

Clearly,

$$\begin{array}{cc}\hfill \mathbb{E}\left|Z\right(t\wedge \tau \left)\right|=& \mathbb{E}\left[\right|Z(t\wedge \tau \wedge T_R)|{\mathbf{1}}_{\{t\wedge \tau \le T_R\}}]+\mathbb{E}[|Z(t\wedge \tau )\left|{\mathbf{1}}_{\{t\wedge \tau \ge T_R\}}\right]\hfill \\ \hfill \le & \mathbb{E}\left[\right|Z(t\wedge \tau \wedge T_R)\left|\right]+\mathbb{E}\left[\right|Z(t\wedge \tau )\left|{\mathbf{1}}_{\{t\wedge \tau \ge T_R\}}\right].\hfill \end{array}$$

By Hölder’s inequality, we have

$$\mathbb{E}\left[\right|Z(t\wedge \tau )|{\mathbf{1}}_{\{t\wedge \tau \ge T_R\}}{]\le (\mathbb{E}\left[\right|Z(t\wedge \tau )|}^2{)}^{1/2}·\mathbb{P}{\{T_R\le t\wedge \tau \}}^{1/2}.$$

Note that:

$${\mathbb{E}\left[\right|Z(t\wedge \tau )|}^2]\le 2\mathbb{E}[\underset{0\le t\le T}{sup}{\left|X(t\wedge \tau )\right|}^2+\underset{0\le t\le T}{sup}{\left|Y(t\wedge \tau )\right|}^2]\le 2C,$$

and

$$\mathbb{P}\{T_R\le t\wedge \tau \}=\mathbb{E}\left[{\mathbf{1}}_{\{T_R\le t\}}\frac{|Y\left(T_R\right){|}^2}{R^2}\right]\le \frac{C}{R},$$

We can get that

$$\mathbb{E}\left|Z(t\wedge \tau )\right|\le \mathbb{E}\left[\right|Z(t\wedge \tau \wedge T_R)\left|\right]+\frac{C}{R},$$

which implies ${lim}_{\delta \to 0}\mathbb{E}\left|Z(t\wedge \tau )\right|=0$ as $R\to \infty .$

Then,

$$\Lambda \left(t\right)-{\Lambda }^{\prime }\left(t\right)={\int }_0^t{\int }_{{\mathbb{R}}_{+}}[h(X\left(s\right),\Lambda (s-),z)-h(Y\left(s_{\delta }\right),\Lambda (s-),z)]N_1(\mathrm{d}s,\mathrm{d}z).$$

Note that $\Lambda \left(t\right)={\Lambda }^{\prime }\left(t\right)$ for all $t<\tau$, $\tau \le t$ if and only if $\Lambda (t\wedge \tau )\ne {\Lambda }^{\prime }(t\wedge \tau )$. Therefore, it follows that

$$\begin{array}{cc}\hfill & \mathbb{P}(\tau \le t)=\mathbb{E}\left[{\mathbf{1}}_{\{\Lambda (t\wedge \tau )\ne {\Lambda }^{\prime }(t\wedge \tau )\}}\right]\hfill \\ \hfill =& \mathbb{E}\left[{\int }_0^{t\wedge \tau }{\int }_{{\mathbb{R}}_{+}}({\mathbf{1}}_{\{\Lambda (s-)+h(X\left(s\right),\Lambda (s-),z)\ne {\Lambda }^{\prime }(s-)+h(Y\left(s_{\delta }\right),\Lambda (s-),z)\}}-{\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}})m\left(\mathrm{d}z\right)\mathrm{d}s\right]\hfill \\ \hfill =& \mathbb{E}\left[{\int }_0^{t\wedge \tau }{\int }_{{\mathbb{R}}_{+}}{\mathbf{1}}_{\{h(X\left(s\right),\Lambda (s-),z)\ne h(Y\left(s_{\delta }\right),{\Lambda }^{\prime }(s-),z)\}}m\left(\mathrm{d}z\right)\mathrm{d}s\right]\hfill \\ \hfill =& \mathbb{E}\left[{\int }_0^{t\wedge \tau }\sum _{l\ne \Lambda (s-)}|q_{\Lambda (s-),l}\left(X\left(s\right)\right)-q_{\Lambda (s-),l}\left(Y\left(s_{\delta }\right)\right)|\mathrm{d}s\right]\hfill \\ \hfill \le & C\mathbb{E}\left[{\int }_0^{t\wedge \tau }|X\left(s\right)-Y\left(s_{\delta }\right)|\mathrm{d}s\right]=C\left[{\int }_0^t\mathbb{E}|X(s\wedge \tau )-Y(s_{\delta }\wedge \tau )|\mathrm{d}s\right]\hfill \\ \hfill \le & C\left[{\int }_0^t\mathbb{E}\left[\right|X(s\wedge \tau )-Y(s\wedge \tau )|+|Y(s\wedge \tau )-Y(s_{\delta }\wedge \tau )\left|\right]\mathrm{d}s\right]\le C\delta ,\hfill \end{array}$$

where the first inequality follows from (Q2), and from the proof of Proposition A2, we have $\mathbb{E}\left[\right|Y(s\wedge \tau )-Y(s_{\delta }\wedge \tau )\left|\right]\le C\delta$. In particular, we have

$$\mathbb{E}{\mathbf{1}}_{\{\Lambda \left(t\right)\ne {\Lambda }^{\prime }\left(t\right)\}}\le \mathbb{P}(\tau \le t)\le C\delta .$$

We obtain the desired result. □

Proof of Theorem 2.

We divide the proof into three steps. As before, C will be used to denote a positive constant independent of $\delta$ whose value may change for different appearances.

Step 1: We first claim that

$$\begin{array}{c}\hfill \underset{\delta \to 0}{lim}\left[\underset{0\le t\le T}{sup}\mathbb{E}|X\left(t\right)-Y\left(t\right)|\right]=0.\end{array}$$

(31)

Similar to ${lim}_{\delta \to 0}\mathbb{E}|X(t\wedge \tau )-Y(t\wedge \tau )|=0$, we can prove (31). Itô’s formula shows that, for $0\le t\le T$,

$$\begin{array}{cc}\hfill & \mathbb{E}{V}_{\epsilon ,\xi }\left(\right|Z(t\wedge S_{{\delta }_0}\wedge T_R)\left|\right)\hfill \\ \hfill \le & \mathbb{E}[{\int }_0^{t\wedge S_{{\delta }_0}\wedge T_R}\langle \nabla V_{\epsilon ,\xi }\left(\right|Z\left(s\right)\left|\right),(b(X\left(s\right),\Lambda \left(s\right))-b(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right)))\rangle \mathrm{d}s\hfill \\ \hfill & +\frac{1}{2}{\int }_0^{t\wedge S_{{\delta }_0}\wedge T_R}\parallel {\nabla }^2{V}_{\epsilon ,\xi }\left(\right|Z\left(s\right)\left|\right)\parallel \parallel \sigma (X\left(s\right),\Lambda \left(s\right))-\sigma (Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right)){\parallel }^2\mathrm{d}s]\hfill \\ \hfill =:& I_3\left(t\right)+J_3\left(t\right).\hfill \end{array}$$

(32)

The remainder of the proof follows in a similar manner to that of $I_2$; we compute

$$\begin{array}{cc}\hfill I_3\left(t\right)\le & \sqrt{d}\mathbb{E}{\int }_0^{t\wedge S_{{\delta }_0}\wedge T_R}|b(X\left(s\right),\Lambda \left(s\right))-b(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))|\mathrm{d}s\hfill \\ \hfill \le & \sqrt{d}\mathbb{E}{\int }_0^{t\wedge S_{{\delta }_0}\wedge T_R}\left[\right|b(X\left(s\right),\Lambda \left(s\right))-b(Y\left(s\right),\Lambda \left(s\right))|+|b(Y\left(s\right),\Lambda \left(s\right))-b(Y\left(s_{\delta }\right),\Lambda \left(s\right))|\hfill \\ \hfill & +|b(Y\left(s_{\delta }\right),\Lambda \left(s\right))-b(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s\right))|+|b(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s\right))-b(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))\left|\right]\mathrm{d}s\hfill \\ \hfill \le & \sqrt{d}{\int }_0^{t\wedge S_{{\delta }_0}\wedge T_R}\left[{\kappa }_R\rho \left(\mathbb{E}\right|Z\left(s\right)\left|\right)+{\kappa }_R\rho \left(\mathbb{E}\right|Y\left(s\right)-Y\left(s_{\delta }\right)\left|\right)\right]\mathrm{d}s\hfill \\ \hfill & +2C_2\sqrt{d}{\int }_0^{t\wedge S_{{\delta }_0}\wedge T_R}\left[\mathbb{E}{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}+\mathbb{E}{\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}\right]\mathrm{d}s\hfill \\ \hfill \le & \sqrt{d}{\int }_0^{t\wedge S_{{\delta }_0}\wedge T_R}{\kappa }_R\rho \left(\mathbb{E}\right|Z\left(s\right)\left|\right)\mathrm{d}s+C(\delta +{\kappa }_R\rho \left(\delta \right)),\hfill \end{array}$$

(33)

and also, from (23) and Propositions 2 and A1, we can obtain the last inequality. On the other hand, similar to $J_2$, we can readily verify that

$$\begin{array}{cc}\hfill J_3\left(t\right)=& \mathbb{E}{\int }_0^{t\wedge S_{{\delta }_0}\wedge T_R}\parallel {\nabla }^2{V}_{\epsilon ,\xi }\left(\right|X\left(s\right)-Y\left(s\right)\left|\right)\parallel \parallel \sigma (X\left(s\right),\Lambda \left(s\right))-\sigma (Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right)){\parallel }^2\mathrm{d}s\hfill \\ \hfill \le & C\mathbb{E}{\int }_0^{t\wedge S_{{\delta }_0}\wedge T_R}[\left(\frac{1}{\left|Z\right(s\left)\right|}+\frac{\left|Z\right(s\left)\right|}{\rho \left(\right|Z\left(s\right)\left|\right)}\right){\mathbf{1}}_{\{\epsilon /\xi <|Z\left(s\right)|<\epsilon \}}\hfill \\ \hfill & ·\left[\right|\sigma (X\left(s\right),\Lambda \left(s\right))-\sigma (Y\left(s\right),\Lambda \left(s\right)){|}^2+|\sigma (Y\left(s\right),\Lambda \left(s\right))-\sigma (Y\left(s_{\delta }\right),\Lambda \left(s\right)){|}^2\hfill \\ \hfill & +|\sigma (Y\left(s_{\delta }\right),\Lambda \left(s\right))-\sigma (Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s\right)){|}^2+|\sigma (Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s\right))-\sigma (Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right)){|}^2]]\mathrm{d}s\hfill \\ \hfill \le & C{\int }_0^{t\wedge S_{{\delta }_0}\wedge T_R}\left(\frac{\rho \left(\right|Z\left(s\right)\left|\right)}{\left|Z\right(s\left)\right|}+\epsilon \right)\mathrm{d}s\hfill \\ \hfill & +C\left(\frac{1}{\rho (\epsilon /\xi )}+\frac{\xi }{\epsilon }\right){\int }_0^{t\wedge S_{{\delta }_0}\wedge T_R}\left[\mathbb{E}{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}+\mathbb{E}{\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+{\kappa }_R\rho \left(\mathbb{E}\right|Y\left(s\right)-Y\left(s_{\delta }\right)\left|\right)\right]\mathrm{d}s\hfill \\ \hfill \le & C\frac{\epsilon {\kappa }_R}{\xi }{\int }_0^{t\wedge S_{{\delta }_0}\wedge T_R}\rho \left(\mathbb{E}\right|Z\left(s\right)\left|\right)\mathrm{d}s+C\left(\frac{1}{\rho (\epsilon /\xi )}+\frac{\xi }{\epsilon }\right)(\delta +{\kappa }_R\rho \left(\delta \right))+CT\epsilon .\hfill \end{array}$$

(34)

Substituting (33) and (34) into (32) yields that

$$\begin{array}{cc}\hfill \mathbb{E}{V}_{\epsilon ,\xi }\left(\right|Z(t\wedge S_{{\delta }_0}\wedge T_R)\left|\right)\le & C\frac{\epsilon {\kappa }_R}{\xi }{\int }_0^{t\wedge S_{{\delta }_0}\wedge T_R}\rho \left(\mathbb{E}\right|Z\left(s\right)\left|\right)\mathrm{d}s\hfill \\ \hfill & +C\left(\frac{1}{\rho (\epsilon /\xi )}+\frac{\xi }{\epsilon }\right)(\delta +{\kappa }_R\rho \left(\delta \right))+CT\epsilon .\hfill \end{array}$$

However, by the definition of $V_{\epsilon ,\xi }\left(x\right)$ and the property of $V_{\epsilon ,\xi }\left(x\right)$, we have $\mathbb{E}|Z(t\wedge S_{{\delta }_0}\wedge T_R)|\le \epsilon +\mathbb{E}{V}_{\epsilon ,\xi }\left(\right|Z(t\wedge S_{{\delta }_0}\wedge T_R)\left|\right).$ Thus,

$$\begin{array}{cc}\hfill \mathbb{E}\left|Z\right(t\wedge S_{{\delta }_0}\wedge T_R\left)\right|\le & C\frac{\epsilon {\kappa }_R}{\xi }{\int }_0^{t\wedge S_{{\delta }_0}\wedge T_R}\rho \left(\mathbb{E}\right|Z\left(s\right)\left|\right)\mathrm{d}s+C(\epsilon ,\xi ,\delta ).\hfill \end{array}$$

Thus, applying Bihari’s inequality,

$$\mathbb{E}|Z(t\wedge S_{{\delta }_0}\wedge T_R)|\le G^{-1}\left(G\left(C(\epsilon ,\xi ,\delta )\right)+C\frac{\epsilon {\kappa }_{R}T}{\xi }\right),$$

where $C(\epsilon ,\xi ,\delta )=C\left(\frac{1}{\rho (\epsilon /\xi )}+\frac{\xi }{\epsilon }\right)(\delta +{\kappa }_R\rho \left(\delta \right))+CT\epsilon ,$ provided that $G\left(C(\epsilon ,\xi ,\delta )\right)+C\frac{\epsilon {\kappa }_{R}T}{\xi }\in \mathrm{Dom}\left(G^{-1}\right),$ where $G\left(r\right)={\int }_1^r\mathrm{d}u/\rho \left(u\right)$ on $r>0$ and $G^{-1}$ is the inverse function of G. Now, we choose $\epsilon ={\delta }^{1/3},\xi =e^{\delta }$ and $\rho \left(\delta \right)=\delta .$ Recalling the condition that ${\int }_0^1\mathrm{d}u/\rho \left(u\right)=\infty ,$ we see that $(-\infty ,0]\subset \mathrm{Dom}\left(G^{-1}\right),G\left(r\right)\to -\infty$ as $r\to 0$ and $G^{-1}\to 0$ as $r\to -\infty$. Thus, we derive that ${lim}_{\delta \to 0}{sup}_{0\le t\le T}\mathbb{E}|X(t\wedge S_{{\delta }_0}\wedge T_R))-Y(t\wedge S_{{\delta }_0}\wedge T_R)\left)\right|=0.$ Recalling the proof of the Proposition 2, we have

$$\underset{\delta \to 0}{lim}\underset{0\le t\le T}{sup}\mathbb{E}|X\left(t\right)-Y\left(t\right)|=0.$$

Step 2: In this step, we shall show that

$$\underset{\delta \to 0}{lim}\mathbb{E}{\int }_0^T|b(X\left(s\right),\Lambda \left(s\right))-b(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))|\mathrm{d}s=0,$$

(35)

$$\underset{\delta \to 0}{lim}\mathbb{E}{\int }_0^T{|\sigma (X\left(s\right),\Lambda \left(s\right))-\sigma (Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))|}^2\mathrm{d}s=0.$$

(36)

In fact, analogous to the discussion in part 1, it is easy to see that

$$\begin{array}{cc}\hfill & \mathbb{E}{\int }_0^T|b(X\left(s\right),\Lambda \left(s\right))-b(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))|\mathrm{d}s\hfill \\ \hfill \le & 4\mathbb{E}{\int }_0^T\left[\right|b(X\left(s\right),\Lambda \left(s\right))-b(Y\left(s\right),\Lambda \left(s\right))|+|b(Y\left(s\right),\Lambda \left(s\right))-b(Y\left(s_{\delta }\right),\Lambda \left(s\right))|\hfill \\ \hfill & +|b(Y\left(s_{\delta }\right),\Lambda \left(s\right))-b(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s\right))|+|b(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s\right))-b(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))\left|\right]\mathrm{d}s\hfill \\ \hfill \le & C{\int }_0^T\left[{\kappa }_R\rho \left(\mathbb{E}\right|X\left(s\right)-Y\left(s\right)\left|\right)+{\kappa }_R\rho \left(\mathbb{E}\right|Y\left(s\right)-Y\left(s_{\delta }\right)\left|\right)\right]\mathrm{d}s\hfill \\ \hfill & +C{\int }_0^T\left[\mathbb{E}{\mathbf{1}}_{\{\Lambda \left(s\right)\ne {\Lambda }^{\prime }\left(s\right)\}}+\mathbb{E}{\mathbf{1}}_{\{{\Lambda }^{\prime }\left(s\right)\ne {\Lambda }^{\prime }\left(s_{\delta }\right)\}}\right]\mathrm{d}s.\hfill \end{array}$$

Thus, by Propositions 2 and A1, (23) and (31) imply (35) immediately. Similarly, we can obtain (36).

Step 3: We can now easily prove the required assertion (8). Indeed, by BDG’s inequality, we have

$$\begin{array}{cc}\hfill \mathbb{E}\left(\underset{0\le t\le T}{sup}|X\left(t\right)-Y\left(t\right)|\right)\le & \mathbb{E}{\int }_0^T|b(X\left(s\right),\Lambda \left(s\right))-b(Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))|\mathrm{d}s\hfill \\ \hfill & +\mathbb{E}{\left({\int }_0^T{|\sigma (X\left(s\right),\Lambda \left(s\right))-\sigma (Y\left(s_{\delta }\right),{\Lambda }^{\prime }\left(s_{\delta }\right))|}^2\mathrm{d}s\right)}^{1/2}.\hfill \end{array}$$

Using (35) and (36), we can readily verify that

$$\underset{\delta \to 0}{lim}\mathbb{E}\left[\underset{0\le t\le T}{sup}|X\left(t\right)-Y\left(t\right)|\right]=0.$$

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3.3. Proof of Theorem 3

This theorem will not be proved in detail, since the method of proof is similar to that of the previous theorem. Under the conditions of Theorem 3, we follow Yamada and Watanabe’s idea, and construct a sequence of smooth functions to control the $L^2$ convergence of two solutions, $X,Y$, up to an appropriately defined stopping time. Then, we use a Bihari’s inequality-type argument to show that such an $L^2$ convergence vanishes if the interval segmentation is enough small.