Skip to main content

\(L^{p}\) \((p\geq2)\)-strong convergence in averaging principle for multivalued stochastic differential equation with non-Lipschitz coefficients

Abstract

We investigate the averaging principle for multivalued stochastic differential equations (MSDEs) driven by a random process under non-Lipschitz conditions. We consider the convergence of solutions in \(L^{p}~(p\geq2)\) and in probability between the MSDEs and the corresponding averaged MSDEs.

Introduction

Most systems in science and industry are perturbed by some random environmental effects described by stochastic differential equations with (fractional) Brownian motion, Lévy process, Poisson process, and so on. A series of useful theories and methods have been proposed to explore stochastic differential equations, such as invariant manifolds [13], averaging principle [312], homogenization principle, and so on. All these theories and methods develop to extract an effective dynamics from these stochastic differential equations, which is more effective for analysis and simulation. Averaging principle is often used to approximate dynamical systems with random fluctuations and provides a powerful tool for simplifying nonlinear dynamical systems. The essence of averaging principle is to establish an approximation theorem for a simplified stochastic differential equation that replaces the original one in some sense and the corresponding optimal order convergence. The theory of stochastic averaging principle has a long and rich history. It was first introduced by Khasminskii [13] in 1968, and since then, the principle for stochastic differential equations was intensively and extensively studied. Stoyanov and Bainov [11] investigated the averaging method for a class of stochastic differential equations with Poisson noise, proving that under some conditions the solutions of averaged systems converge to the solutions of the original systems in mean square and in probability. Xu, Duan, and Xu [4] established an averaging principle for stochastic differential equations with general non-Gaussian Lévy noise. Quite recently, \(L^{2}\) (mean square) strong averaging principle for multivalued stochastic differential equations with Brownian motion was established by Xu and Liu [14]. Note that all the works mentioned are under the Lipschitz condition; however, in the real world, the Lipschitz condition seems to be exceedingly harsh when discussing various applications. So it is necessary and significant to consider some non-Lipschitz cases; see [15].

In [16], the author discussed the existence and uniqueness of a solution in \(L^{p}\) (pth moment) sense for some multivalued stochastic differential equations under a non-Lipschitz condition. From the dynamic view, we concern the p-moment averaging principle for a multivalued stochastic differential equation under a non-Lipschitz condition, which is different from [14] under the Lipschitz condition. Although the method used in our paper is similar to that of [14], compared with the results of article [14], our conclusion is more general, as it is well known that the \(L^{2}\)-strong convergence does not imply \(L^{p} (p\geq2)\) strong convergence in general. Meanwhile, results for higher-order moments are needed that possess a good robustness and can be applied in computations in statistic, finance, and other aspects.

Recently, many authors considered multivalued and set-valued stochastic differential equations; see, for example, [1720]. In this article, we study the averaging principle for MSDEs of the form

$$\begin{aligned} dX_{t}+\mathcal{A}(X_{t})\,dt\ni f(t,X_{t})\,dt+g(t,X_{t})\,dB_{t},\qquad X_{0}=x \in \overline{D(\mathcal{A})}, \end{aligned}$$
(1.1)

with \(t\in[0, T]\), where \(\mathcal{A}\) is a multivalued maximal monotone operator, which we introduce in the next section, \(f:[0,T]\times R^{d}\rightarrow R^{d}\) and \(g:[0,T]\times R^{d}\rightarrow R^{d}\) are measurable functions and satisfy non-Lipschitz conditions with respect to x. To derive the averaging principle for non-Lipschitz multivalued stochastic differential equation, we need some assumptions given in the next section.

This paper is organized as follows. In Section 2, we give some assumptions for our theory and then introduce the definition of a multivalued maximal monotone operator and related results. The convergence of solutions in \(L^{p}\ (p\geq2)\) and in probability between the MSDEs and the corresponding averaged MSDEs are considered in Section 3.

Throughout this paper, the letter C will denote positive constants with values changing in different occasions. When necessary, we will explicitly write the dependence of constants on parameters.

Framework and preliminaries

Basic hypothesis

In this paper, we impose the following assumptions.

\(\mathbf{H1}\) Non-Lipschitz condition: Suppose that f and b are bounded and satisfy the following conditions:

For any \(x,y\in \mathbb{R}^{d}\) and \(t\in[0,T]\),

$$\begin{aligned} \bigl\Vert g(t,x)-g(t,y) \bigr\Vert ^{2}\leq \rho^{2}_{2,\eta}\bigl( \Vert x-y \Vert \bigr) \quad\mbox{and}\quad \bigl\Vert f(t,x)-f(t,y) \bigr\Vert \leq\rho _{1,\eta}\bigl( \Vert x-y \Vert \bigr). \end{aligned}$$
(2.1)

For \(0<\eta<\frac{1}{e}\), let \(\rho_{1,\eta},\rho_{2,\eta}\) be two concave functions defined by

$$ \rho_{j,\eta}(x):= \textstyle\begin{cases} x[\log x^{-1}]^{\frac{1}{j}},& x\leq\eta,\\ ([\log\eta^{-1}]^{\frac{1}{j}}-\frac{1}{j}[\log\eta^{-1}]^{\frac {1}{j}-1})x+\frac{1}{j}[\log\eta^{-1}]^{\frac{1}{j}-1}\eta ,&x>\eta. \end{cases} $$

Let \(\bar{f}:R^{d}\rightarrow R^{d}\) and \(\bar{g}:R^{d}\rightarrow R^{d}\) be measurable functions satisfying the Lipschitz conditions with respect to x as \(f(t,x)\) and \(g(x,t)\). Moreover, we assume that \(f(t,x),\bar{f}(x)\), \(g(x,t)\), and \(\bar{g}(x)\) satisfied the following conditions:

\(\mathbf{H2}\)

$$\begin{aligned} \frac{1}{T} \int_{0}^{T} \bigl\Vert f(s,x)-\bar{f}(x) \bigr\Vert ^{2} \,ds\leq\varphi _{1}(T) \bigl(1+ \Vert x \Vert ^{2}\bigr), \end{aligned}$$
(2.2)

\(\mathbf{H3}\)

$$\begin{aligned} \frac{1}{T} \int_{0}^{T} \bigl\Vert g(s,x)-\bar{g}(x) \bigr\Vert ^{2} \,ds\leq\varphi _{2}(T) \bigl(1+ \Vert x \Vert ^{2}\bigr), \end{aligned}$$
(2.3)

where \(\varphi_{i}(T), i=1,2\), are positive bounded functions; moreover, if T is fixed, then \(\varphi_{i}(T)\) is a constant, which means that \(\varphi_{i}(\cdot)\) only depends on time.

\(\mathbf{H4}\) The operator \(\mathcal{A}\) is a maximal monotone operator with \(D({\mathcal{A}})=\mathbb{R}^{d}\).

Multivalued operators and MSDEs

A map \(\mathcal{A}:\mathbb{R}^{d}\rightarrow2^{\mathbb{R}^{d}}\) is called a multivalued operator. Define the domain and image of \(\mathcal {A}\) as

$$D(\mathcal{A}):=\bigl\{ x\in\mathbb{R}^{d}: \mathcal{A}(x)\neq \varnothing \bigr\} ,\qquad \operatorname{Im}(\mathcal{A}):=\bigcup_{x\in D(\mathcal{A})} \mathcal{A}(x), $$

and the graph of \(\mathcal{A}\) is

$$\operatorname{Gr}(\mathcal{A}):=\bigl\{ (x,y)\in\mathbb{R}^{2d}:x\in \mathbb{R}^{d}, y\in \mathcal{A}(x)\bigr\} . $$

Definition 2.1

(1) A multivalued operator \(\mathcal{A}\) is called monotone if

$$\langle y_{1}-y_{2}, x_{1}-x_{2} \rangle\geq0 \quad\mbox{for all } (x_{1},y_{1}),(x_{2},y_{2}) \in \operatorname{Gr}(\mathcal{A}). $$

(2) A monotone operator \(\mathcal{A}\) is called maximal monotone if and only if

$$(x_{1}, y_{1})\in \operatorname{Gr}(\mathcal{A})\quad\Leftrightarrow\quad\bigl\{ \langle y_{1}-y_{2}, x_{1}-x_{2}\rangle \geq0 \mbox{ for all } (x_{2}, y_{2})\in \operatorname{Gr}(\mathcal{A})\bigr\} . $$

Now, we give a precise definition of the solution to equation (1.1).

Definition 2.2

A pair of continuous and \(\mathcal{F}_{t}\)-adapted processes \((X, K)\) is called a strong solution of equation (1.1) if:

  • \(X_{0}=x\), and \(X(t)\in\overline{D(\mathcal{A})}\) a.s.;

  • \(K=\{K(t),\mathcal{F}_{t}; t\in R^{+}\}\) is of finite variation, and \(K(0)=0\) a.s.;

  • \(dX_{t}=f(t,X_{t})\,dt+g(t,X_{t})\,dZ_{t}-dK_{t}, t\in R^{+}\), a.s.;

  • for any continuous processes \((\alpha(t),\beta(t))\) satisfying

    $$\bigl(\alpha(t),\beta(t)\bigr)\in \operatorname{Gr}(\mathcal{A}), \quad t\in R^{+}, $$

    the measure

    $$\bigl\langle X(t)-\alpha(t), dK(t)-\beta(t)\,dt\bigr\rangle \geq0. $$

Also, we need the following lemma from [21].

Lemma 2.1

Let \(\mathcal{A}\) be a multivalued maximal monotone operator, let \(t\mapsto(X(t), K(t))\) and \(t\mapsto(\tilde{X}(t),\tilde{K}(t))\) be continuous functions with \(X(t),\tilde{X}(t)\in\overline{D(\mathcal {A})}\), and let \(t\mapsto K(t),\tilde{K}(t)\) be of finite variation. Let \((\alpha(t),\beta(t))\) be continuous functions satisfying

$$\bigl(\alpha(t),\beta(t)\bigr)\in \operatorname{Gr}(\mathcal{A}),\quad t\in R^{+}. $$

If

$$\bigl\langle X(t)-\alpha(t), dK(t)-\beta(t)\,dt\bigr\rangle \geq0 $$

and

$$\bigl\langle \tilde{X}(t)-\alpha(t),d \tilde{K}(t)-\beta(t)\,dt\bigr\rangle \geq0, $$

then

$$\bigl\langle X(t)-\tilde{X}(t), dK(t)-d\tilde{K}(t)\bigr\rangle \geq0. $$

Lemma 2.2

([22])

Under \(\mathbf{H1}\) and \(\mathbf{H4}\), let the initial condition satisfy \(E \Vert x \Vert ^{2p}<+\infty\). For any \(p\geq1\) and \(0\leq t\leq T\), equation (1.1) has a unique solution satisfying

$$E\Bigl[\sup_{0\leq t\leq T} \Vert X_{t} \Vert ^{2p}\Bigr]\leq C_{T}^{(p, \Vert x \Vert )}< +\infty. $$

The following example and two lemmas are taken from [22].

Lemma 2.3

Let \(\rho:R^{+}\rightarrow R^{+}\) be a continuous nondecreasing function. If \(g(s)\) and \(q(s)\) are two strictly positive functions on \(R^{+}\) such that

$$g(t)\leq g(0)+ \int_{0}^{t} q(s)\rho\bigl(g(s)\bigr)\,ds,\quad t\geq0, $$

then

$$ g(t)\leq f^{-1}\biggl(f\bigl(g(0)\bigr)+ \int_{0}^{t} q(s)\,ds\biggr), $$
(2.4)

where \(f(x):=\int_{x_{0}}^{x}\frac{1}{\rho(y)}\,dy\) is well-defined for some \(x_{0}>0\).

Example 2.1

For \(0<\eta<\frac{1}{e}\), define a concave function as

$$ \rho_{\eta}(x):=\textstyle\begin{cases} x\log x^{-1}, &x\leq\eta,\\ \eta\log\eta^{-1}+(\log\eta^{-1}-1)(x-\eta),&x>\eta. \end{cases} $$

Choosing \(x_{0}=\eta\), we have

$$\begin{aligned} &f(x)=\log\biggl(\frac{\log\eta}{\log x}\biggr),\quad 0< x< \eta, \\ &f^{-1}(x)=\exp\bigl\{ \log\eta\cdot \exp\{-x\}\bigr\} ,\quad x< 0. \end{aligned}$$

If \(g(0)<\eta\), then substituting these into (2.4), we obtain

$$ g(t)\leq \bigl(g(0)\bigr)^{\exp\{-\int_{0}^{t} q(s)\,ds\}}. $$
(2.5)

Lemma 2.4

  • For \(j=1,2\), \(\rho_{j,\eta}\) is decreasing in η, that is, \(\rho_{j,\eta_{1}}\leq\rho_{j,\eta_{2}}\) if \(1>\eta_{1}>\eta_{2}\).

  • For any \(p\geq0\) and η sufficiently small, we have

    $$x^{p} \rho^{j}_{j,\eta}(x)\leq\frac{1}{j+p} \rho_{1,\eta^{j+p}} \bigl(x^{j+p}\bigr),\quad j=1,2. $$

Averaging principle for MSDEs

In this section, we prove an averaging principle for multivalued stochastic differential equations (MSDEs) driven by a random process under non-Lipschitz conditions. We consider the convergence of solutions in \(L^{p}\ (p\geq2)\) and in probability between the MSDEs and the corresponding averaged MSDEs.

For \(t\in[0, T]\), consider

$$\begin{aligned} dX^{\epsilon}_{t}+\epsilon\mathcal{A} \bigl(X^{\epsilon}_{t}\bigr)\,dt\ni\epsilon f\bigl(t,X^{\epsilon}_{t} \bigr)\,dt+\sqrt{\epsilon}g\bigl(t,X^{\epsilon}_{t} \bigr)\,dB_{t},\qquad X^{\epsilon}_{0}=x\in\overline{D( \mathcal{A})}. \end{aligned}$$
(3.1)

The standard form of (3.1) is defined as

$$\begin{aligned} X^{\epsilon}_{t}=X^{\epsilon}(0)+\epsilon \int_{0}^{t}f\bigl(s,X^{\epsilon }(s)\bigr)\,ds+ \sqrt{\epsilon} \int_{0}^{t}g\bigl(s,X^{\epsilon}_{s} \bigr)\,dB_{s}-\epsilon K(t),\quad t\in[0, T], \end{aligned}$$
(3.2)

and the corresponding averaged MSDEs of (3.2) are defined as

$$\begin{aligned} Y^{\epsilon}_{t}=Y^{\epsilon}(0)+\epsilon \int_{0}^{t}\overline {f}\bigl(Y^{\epsilon}(s) \bigr)\,ds+\sqrt{\epsilon} \int_{0}^{t}\overline {g}\bigl(Y^{\epsilon}_{s} \bigr)\,dB_{s}-\epsilon \overline{K}(t),\quad t\in[0, T]. \end{aligned}$$
(3.3)

Here \(\overline{f}:R^{d}\rightarrow R^{d}\) and \(\overline {g}:R^{d}\rightarrow R^{d}\) are measurable functions satisfying the non-Lipschitz conditions with respect to x as \(f(t,x)\) and \(g(t,x)\), \(Y^{\epsilon}(0)=X^{\epsilon}(0)=x\), and \(f,\overline{f}\), \(g,\overline{g}\) satisfy \(\mathbf{H2}\) and \(\mathbf{H3}\).

Now, we are in the position to investigate the relationship between the processes \(X^{\epsilon}_{t}\) and  \(Y^{\epsilon}_{t}\) .

Theorem 3.1

Suppose that conditions \(\mathbf{H1}\)-\(\mathbf{H4}\) hold. Then, for a given arbitrarily small number \(\delta>0\) and for \(\alpha\in(0, \frac{1}{2})\), there exists a number \(\tilde{\epsilon}\in(0, \epsilon_{0}]\) \((\epsilon_{0}=\frac{1}{16p^{2}})\) such that, for all \(\epsilon\in(0, \tilde{\epsilon})\) and \(p\geq1\), we have

$$E\Bigl(\sup_{t\in[0, \epsilon^{\alpha-\frac{1}{2}}(1-4p\sqrt{\epsilon})]} \bigl\Vert X^{\epsilon }_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}\Bigr)\leq \delta. $$

Proof

Consider the difference \(X^{\epsilon}_{t}-Y^{\epsilon}_{t}\). From (3.2) and (3.3) we have

$$ \begin{aligned} X^{\epsilon}_{t}-Y^{\epsilon}_{t}={}& \epsilon \int_{0}^{t}\bigl[f\bigl(s,X^{\epsilon}(s) \bigr) -\overline{f}\bigl(Y^{\epsilon}(s)\bigr)\bigr]\,ds \\ &{}+\sqrt{\epsilon} \int_{0}^{t}\bigl[g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline {g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr]\,dB_{s}-\epsilon \bigl[K(t)-\overline{K}(t)\bigr]. \end{aligned} $$

By Itô’s formula [23],

$$ \begin{aligned} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}={}&{-}\epsilon2p \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2} \bigl\langle X^{\epsilon}_{s}-Y^{\epsilon}_{s}, dK(s)-d\overline {K}(s)\bigr\rangle \\ &{}+\epsilon2p \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle f\bigl(s,X^{\epsilon}(s)\bigr) - \overline{f}\bigl(Y^{\epsilon}(s)\bigr),X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\rangle \,ds \\ &{}+\sqrt{\epsilon}2p \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon }_{s}\bigr),X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\rangle \,dB_{s} \\ &{}+\epsilon p \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2} \bigl\Vert g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\,ds \\ &{}+2\epsilon p(p-1) \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-4}\bigl\langle X^{\epsilon}_{s}-Y^{\epsilon}_{s},g \bigl(s,X^{\epsilon}_{s}\bigr)-\overline {g}\bigl(Y^{\epsilon}_{s} \bigr)\bigr\rangle ^{2}\,ds. \end{aligned} $$

By Definition 2.2 and Lemma 2.1 we get

$$ \begin{aligned} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}\leq{}&\epsilon2p \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle f\bigl(s,X^{\epsilon}(s)\bigr) - \overline{f}\bigl(Y^{\epsilon}(s)\bigr),X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\rangle \,ds \\ &{}+\sqrt{\epsilon}2p \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon }_{s}\bigr),X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\rangle \,dB_{s} \\ &{}+\epsilon p \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2} \bigl\Vert g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\,ds \\ &{}+2\epsilon p(p-1) \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-4}\bigl\langle X^{\epsilon}_{s}-Y^{\epsilon}_{s},g \bigl(s,X^{\epsilon}_{s}\bigr)-\overline {g}\bigl(Y^{\epsilon}_{s} \bigr)\bigr\rangle ^{2}\,ds. \end{aligned} $$

Then

$$ \begin{aligned} &E\sup_{0\leq t\leq T} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}\\ &\quad\leq \epsilon2pE\sup_{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle f\bigl(s,X^{\epsilon}(s)\bigr) - \overline{f}\bigl(Y^{\epsilon}(s)\bigr),X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\rangle \,ds \biggr\vert \\ &\qquad{}+\sqrt{\epsilon}2pE\sup_{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline {g}\bigl(Y^{\epsilon}_{s}\bigr),X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\rangle \,dB_{s} \biggr\vert \\ &\qquad{}+\epsilon pE\sup_{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2} \bigl\Vert g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline {g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\,ds \biggr\vert \\ &\qquad{}+2\epsilon p(p-1)E\sup_{0\leq t\leq T} \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-4}\bigl\langle X^{\epsilon}_{s}-Y^{\epsilon}_{s},g \bigl(s,X^{\epsilon}_{s}\bigr)-\overline {g}\bigl(Y^{\epsilon}_{s} \bigr)\bigr\rangle ^{2}\,ds. \\ &\quad= I_{1}+I_{2}+I_{3}+I_{4}. \end{aligned} $$

We now estimate \(I_{1},I_{2},I_{3}, I_{4}\) separately.

Estimate of \(I_{1}\). Using the trigonometric inequality, we have

$$ \begin{aligned} I_{1}={}&2\epsilon pE\sup_{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle f\bigl(s,X^{\epsilon}(s)\bigr) - \overline{f}\bigl(Y^{\epsilon}(s)\bigr),X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\rangle \,ds \biggr\vert \\ \leq{}&2\epsilon pE\sup_{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle f\bigl(s,X^{\epsilon }(s)\bigr)-f \bigl(s,Y^{\epsilon}(s)\bigr),X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\rangle \,ds \biggr\vert \\ &{}+2\epsilon pE\sup_{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle f\bigl(s,Y^{\epsilon}(s)\bigr) - \overline{f}\bigl(Y^{\epsilon}(s)\bigr),X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\rangle \,ds \biggr\vert \\ ={}& I_{11}+I_{12}. \end{aligned} $$

For \(I_{11}\), using the non-Lipschitz condition of f and the Cauchy-Schwarz inequality, we have

$$ \begin{aligned} I_{11}\leq{}& 2\epsilon pE\sup _{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-1}\rho_{1,\eta}\bigl( \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds \biggr\vert \\ \leq{}& 2\epsilon p \int_{0}^{T}E \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-1}\rho_{1,\eta }\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds. \end{aligned} $$

For \(I_{12}\), using the Hölder and Young inequalities, we deduce

$$ \begin{aligned} I_{12}={}&2\epsilon pE\sup_{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle f\bigl(s,Y^{\epsilon}(s)\bigr) - \overline{f}\bigl(Y^{\epsilon}(s)\bigr),X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\rangle \,ds \biggr\vert \\ \leq{}&\epsilon pE\sup_{0\leq t\leq T}\biggl| \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl( \bigl\Vert f\bigl(s,Y^{\epsilon}(s) \bigr) -\overline{f}\bigl(Y^{\epsilon}(s)\bigr) \bigr\Vert ^{2}+ \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2}\bigr)\,ds\biggr| \\ \leq{}& \epsilon pE\sup_{0\leq t\leq T} \int_{0}^{t}\biggl(\frac{2p-2}{2p} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}+\frac{1}{p}\biggr)\bigl( \bigl\Vert f\bigl(s,Y^{\epsilon}(s) \bigr) -\overline{f}\bigl(Y^{\epsilon}(s)\bigr) \bigr\Vert ^{2}\bigr)\,ds \\ &{}+ \epsilon p \int_{0}^{t}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\,ds \\ \leq{}& \epsilon\bigl[(p-1)C_{T}^{(p, \Vert x \Vert )}+1\bigr]E\biggl(\sup _{0\leq t\leq T}t\frac {1}{t} \int_{0}^{t} \bigl\Vert f\bigl(s,Y^{\epsilon}(s) \bigr) -\overline{f}\bigl(Y^{\epsilon}(s)\bigr) \bigr\Vert ^{2}\,ds \biggr) \\ &{}+ \epsilon p \int_{0}^{t}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\,ds. \end{aligned} $$

Taking condition \(\mathbf{H 2}\), Lemma 2.2, and the Young inequality into account, we have

$$ \begin{aligned} I_{12}\leq{}& \epsilon\bigl[(p-1)C_{T}^{(p, \Vert x \Vert )}+1 \bigr]\sup_{0\leq t\leq T}\Bigl\{ t\varphi_{1}{(t)}\Bigl[1+E \Bigl(\sup_{0\leq s\leq T} \bigl\Vert Y^{\epsilon}_{s} \bigr\Vert ^{2p}\Bigr)\Bigr]\Bigr\} \\ &{}+ \epsilon p \int_{0}^{t}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\,ds \\ \leq{}& \epsilon p \int_{0}^{t}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\,ds+\epsilon C\bigl(p,T, \Vert x \Vert \bigr) T. \end{aligned} $$

Finally, we have

$$ \begin{aligned} I_{1}\leq{}& \epsilon T C\bigl(p,T, \Vert x \Vert \bigr)+\epsilon p \int_{0}^{t}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\,ds \\ &{}+2\epsilon p \int_{0}^{T}E\sup_{0\leq u\leq s} \bigl\Vert X^{\epsilon}_{u}-Y^{\epsilon}_{u} \bigr\Vert ^{2p-1}\rho_{1,\eta}\bigl( \bigl\Vert X^{\epsilon}_{u}-Y^{\epsilon}_{u} \bigr\Vert \bigr)\,ds. \end{aligned} $$

Estimate of \(I_{2}\). Using the Burkholder-Davis-Gundy and Young inequalities, we have

$$ \begin{aligned} I_{2}={}&\sqrt{\epsilon}2pE\sup _{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl\langle g\bigl(s,X^{\epsilon }_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon}_{s}\bigr),X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\rangle \,dB_{s} \biggr\vert \\ \leq{}& \sqrt{\epsilon}8p E\biggl\{ \int_{0}^{T} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{4p-4} \bigl\vert \bigl\langle g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline {g}\bigl(Y^{\epsilon}_{s}\bigr),X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\rangle \bigr\vert ^{2}\,ds\biggr\} ^{\frac{1}{2}} \\ \leq{}& \sqrt{\epsilon}8p E\biggl\{ \int_{0}^{T}\sup_{0\leq s\leq T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\Vert ^{2p-2} \bigl\Vert g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\,ds\biggr\} ^{\frac{1}{2}} \\ \leq{}& \sqrt{\epsilon}4p E\sup_{0\leq t\leq T} \bigl\Vert X^{\epsilon }_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}+ \sqrt{\epsilon}4pE \int_{0}^{T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2} \bigl\Vert g\bigl(s,X^{\epsilon }_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\,ds \\ ={}&I_{21}+I_{22}. \end{aligned} $$

In the following, we estimate

$$ \begin{aligned} I_{22}=\sqrt{\epsilon}4pE \int_{0}^{T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\Vert ^{2p-2} \bigl\Vert g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\,ds. \end{aligned} $$

Using conditions \(\mathbf{H1}\) and \(\mathbf{H3}\) and the Young inequality, we get:

$$ \begin{aligned} I_{22}\leq{}&\sqrt{\epsilon}8pE \int_{0}^{T} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\bigl( \bigl\Vert g\bigl(s,X^{\epsilon}_{s} \bigr)-g\bigl(s,Y^{\epsilon }_{s}\bigr) \bigr\Vert ^{2}+ \bigl\Vert g\bigl(s,Y^{\epsilon}_{s}\bigr)-\overline{g} \bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\bigr)\,ds \\ \leq{}& \sqrt{\epsilon}8pE \int_{0}^{T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\Vert ^{2p-2}\bigl(\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)+ \bigl\Vert g\bigl(s,Y^{\epsilon}_{s}\bigr)- \overline{g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\bigr)\,ds \\ \leq{}&\sqrt{\epsilon}8pE \int_{0}^{T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds \\ &{}+\sqrt{\epsilon}8pE \int_{0}^{T}\biggl[\frac{2p-2}{2p} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}+\frac{1}{p}\biggr]\bigl( \bigl\Vert g \bigl(s,Y^{\epsilon }_{s}\bigr)-\overline{g}\bigl(Y^{\epsilon}_{s} \bigr) \bigr\Vert ^{2}\bigr)\,ds \\ \leq{}&\sqrt{\epsilon}8pE \int_{0}^{T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds \\ &{}+\sqrt{\epsilon}C\bigl(p,T, \Vert x \Vert \bigr)E \biggl(\sup _{0\leq t\leq T}t\frac {1}{t} \int_{0}^{t} \bigl\Vert g\bigl(s,Y^{\epsilon}_{s} \bigr)-\overline{g}\bigl(Y^{\epsilon }_{s}\bigr) \bigr\Vert ^{2}\,ds\biggr) \\ \leq{}&\sqrt{\epsilon} T C_{2}\bigl(p,T, \Vert x \Vert \bigr)+\sqrt{ \epsilon}8pE \int_{0}^{T} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds. \end{aligned} $$

Combing the estimates of \(I_{21}\) and \(I_{22}\), we conclude that

$$ \begin{aligned} I_{2}\leq{}& \sqrt{\epsilon}4p E\sup _{0\leq t\leq T} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}+\sqrt{\epsilon} T C_{2}\bigl(p,T, \Vert x \Vert \bigr) \\ &{}+\sqrt{\epsilon}8pE \int_{0}^{T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds. \end{aligned} $$

Estimate of \(I_{3}\). Note that

$$ \begin{aligned} I_{3}=\epsilon pE\sup_{0\leq t\leq T} \biggl\vert \int_{0}^{t} \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2} \bigl\Vert g\bigl(s,X^{\epsilon}_{s} \bigr)-\overline {g}\bigl(Y^{\epsilon}_{s}\bigr) \bigr\Vert ^{2}\,ds \biggr\vert . \end{aligned} $$

Using the same estimate as for \(I_{22}\), we have

$$I_{3}\leq\epsilon TC_{3}\bigl(p,T, \Vert x \Vert \bigr)+\epsilon2pE \int_{0}^{T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds. $$

Estimate of \(I_{4}\). Using the Cauchy-Schwarz inequality, the term \(I_{4}\) has the same form with \(I_{3}\) with a different constant:

$$I_{4}\leq\epsilon TC_{4}\bigl(p,T, \Vert x \Vert \bigr)+\epsilon4p(p-1)E \int_{0}^{T} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds. $$

Combing the estimates of \(I_{1},I_{2} \mbox{ and } I_{3},I_{4}\), we have

$$ \begin{aligned} &E\sup_{0\leq t\leq T} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p} \\ &\quad\leq \epsilon TC_{1}\bigl(p,T, \Vert x \Vert \bigr)+\epsilon C_{1}(p) \int_{0}^{T}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\,dt \\ &\qquad{}+2\epsilon p \int_{0}^{T}E \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p-1}\rho_{1,\eta}\bigl( \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert \bigr)\,dt \\ &\qquad{}+\sqrt{\epsilon}4p E\sup_{0\leq t\leq T} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}+\sqrt{\epsilon} C_{2}\bigl(p,T, \Vert x \Vert \bigr) \\ &\qquad{}+\sqrt{\epsilon}8p \int_{0}^{T}E \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds \\ &\qquad{}+\epsilon TC_{3}\bigl(p,T, \Vert x \Vert \bigr)+\epsilon 2p \int_{0}^{T}E \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho ^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds \\ &\qquad{}+\epsilon TC_{4}\bigl(p,T, \Vert x \Vert \bigr)+\epsilon 4p(p-1) \int_{0}^{T}E \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho ^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds. \end{aligned} $$

Taking \(\sqrt{\epsilon}4p<1\), that is, \(\epsilon<\frac{1}{16p^{2}}\), we have

$$ \begin{aligned}& E\sup_{0\leq t\leq T} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}\\ &\quad\leq \frac{\sqrt{\epsilon} TC_{5}(p, \Vert x \Vert ,T,\epsilon)}{1-\sqrt{\epsilon }4p}+\frac{\epsilon C_{1}(p)}{1-\sqrt{\epsilon}4p} \int_{0}^{T}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\,dt \\ &\qquad{}+\frac{2\epsilon p}{1-\sqrt{\epsilon}4p} \int_{0}^{T}E \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon }_{t} \bigr\Vert ^{2p-1}\rho_{1,\eta}\bigl( \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert \bigr)\,dt \\ &\qquad{}+\frac{\sqrt{\epsilon}8p}{1-\sqrt{\epsilon}4p} \int_{0}^{T}E \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds \\ &\qquad{}+\frac{\epsilon 2p}{1-\sqrt{\epsilon}4p} \int_{0}^{T}E \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon }_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds \\ &\qquad{}+\frac{\epsilon 4p(p-1)}{1-\sqrt{\epsilon}4p} \int_{0}^{T}E \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p-2}\rho^{2}_{2,\eta}\bigl( \bigl\Vert X^{\epsilon }_{s}-Y^{\epsilon}_{s} \bigr\Vert \bigr)\,ds. \end{aligned} $$

By Lemma 2.4 and the concavity of the function \(\rho_{1,\eta}\) we have

$$ \begin{aligned} &E\sup_{0\leq t\leq T} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p} \\ &\quad\leq\frac{\sqrt{\epsilon} TC_{5}(p, \Vert x \Vert ,T,\epsilon)}{1-\sqrt{\epsilon }4p}+\frac{\epsilon C_{1}(p)}{1-\sqrt{\epsilon}4p} \int_{0}^{T}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\,dt \\ &\qquad{}+\frac{\sqrt{\epsilon}C_{6}(p, \Vert x \Vert ,T,\epsilon)}{1-\sqrt{\epsilon }4p} \int_{0}^{T}\rho_{1,\eta}\Bigl(E\sup _{0\leq s\leq t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\Bigr)\,dt \\ &\quad\leq \frac{\sqrt{\epsilon} TC_{5}(p, \Vert x \Vert ,T,\epsilon)}{1-\sqrt{\epsilon }4p} \\ &\qquad{}+\frac{\sqrt{\epsilon}C_{7}(p, \Vert x \Vert ,T,\epsilon)}{1-\sqrt{\epsilon }4p} \int_{0}^{T}E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}+\rho_{1,\eta}\Bigl(E\sup_{0\leq s\leq t} \bigl\Vert X^{\epsilon}_{s}-Y^{\epsilon}_{s} \bigr\Vert ^{2p}\Bigr)\,dt. \end{aligned} $$

Note that, for sufficiently small ϵ, we have \(g(0)=\frac{\sqrt{\epsilon}C_{5}(p, \Vert x \Vert ,T,\epsilon)}{1-\sqrt {\epsilon}4p}\leq\eta<\frac{1}{e}\), and from Lemma 2.3 and Example 2.1 we get the following estimate:

$$ E\sup_{0\leq t\leq T} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}\leq \frac{\sqrt{\epsilon} TC_{5}(p, \Vert x \Vert ,T,\epsilon)}{1-\sqrt{\epsilon}4p} \exp^{(1-\ln\eta)\exp\{-\frac{\sqrt{\epsilon} TC_{7}(p, \Vert x \Vert ,T,\epsilon)}{1-\sqrt{\epsilon}4p}\}}. $$

Choose \(\alpha\in(0,\frac{1}{2})\) such that, for every \(t\in[0, \epsilon^{\alpha-\frac{1}{2}}(1-4p\sqrt{\epsilon})]\subseteq[0, T]\), we have

$$E \Bigl(\sup_{t\in[0, \epsilon^{\alpha-\frac{1}{2}}(1-4p\sqrt {\epsilon})]} \bigl\Vert X^{\epsilon}_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p} \Bigr)\leq C\epsilon^{\alpha}, $$

where

$$ C=C_{5}\bigl(p, \Vert x \Vert ,T,\epsilon\bigr) \exp^{\{(1-\ln\eta)\exp\{-\epsilon ^{\alpha}(C_{7}(p, \Vert x \Vert ,T,\epsilon)\}\}}. $$

Consequently, given any number \(\delta>0\), we can choose \(\tilde{\epsilon}\in(0, \epsilon_{0}]\) (\(\epsilon_{0}=\frac{1}{16p^{2}}\)) such that, for each \(\epsilon\in(0, \tilde{\epsilon})\) and every \(t\in[0, \epsilon^{\alpha-\frac{1}{2}}(1-4p\sqrt{\epsilon})]\),

$$E\Bigl(\sup_{t\in[0, \epsilon^{\alpha-\frac{1}{2}}(1-4p\sqrt{\epsilon})]} \bigl\Vert X^{\epsilon }_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}\Bigr)\leq \delta, $$

which completes the proof of the theorem. □

Using the Chebyshev-Markov inequality, we can also get the convergence in probability.

Theorem 3.2

Suppose that conditions \(\mathbf{H1}\)-\(\mathbf{H4}\) hold. Then, for a given arbitrarily small number \(\theta>0\) and for \(\alpha\in(0, \frac{1}{2})\), there exists a number \(\tilde{\epsilon}\in(0, \epsilon_{0}]\) \((\epsilon_{0}=\frac{1}{16p^{2}})\) such that, for all \(\epsilon\in(0, \tilde{\epsilon})\) and \(p\geq 1\), we have

$$\lim_{\epsilon\rightarrow0}\mathbb{P}\Bigl(\sup_{t\in[0, \epsilon ^{\alpha-\frac{1}{2}}(1-4p\sqrt{\epsilon})]} \bigl\Vert X^{\epsilon }_{t}-Y^{\epsilon}_{t} \bigr\Vert ^{2p}>\theta\Bigr)=0. $$

References

  1. 1.

    Duan, J, Lu, K, Schmalfuss, B: Smooth stable and unstable manifolds for stochastic evolutionary equations. J. Dyn. Differ. Equ. 16(4), 949-972 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Duan, J, Lu, K, Schmalfuss, B: Invariant manifolds for stochastic partial differential equations. Ann. Probab. 31(4), 2109-2135 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Duan, J, Wang, W: Effective Dynamics of Stochastic Partial Differential Equations. Elsevier, Amsterdam (2014)

    Google Scholar 

  4. 4.

    Xu, Y, Duan, J, Xu, W: An averaging principle for stochastic dynamical systems with Lévy noise. Physica D 240, 1395-1401 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Wang, W, Roberts, AJ: Average and deviation for slow-fast stochastic differential equations. J. Differ. Equ. 253, 1265-1286 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Cerrai, S: Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab. Theory Relat. Fields 125, 271-304 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Cerrai, S: A Khasminkii type averaging principle for stochastic reaction-diffusion equations. Ann. Appl. Probab. 19(3), 899-948 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Cerrai, S, Freidlin, MI: Averaging principle for a class of stochastic reaction-diffusion equations. Probab. Theory Relat. Fields 144, 137-177 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Fu, H, Liu, J: Strong convergence in stochastic averaging for two time-scales stochastic partial differential equations. J. Math. Anal. Appl. 384, 70-86 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Liu, D: Strong convergence of principle of averaging for multiscale stochastic dynamical systems. Commun. Math. Sci. 8(4), 999-1020 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Stoyanov, IM, Bainov, DD: The averaging method for a class of stochastic differential equations. Ukr. Math. J. 26(2), 186-194 (1974)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Pei, B, Xu, Y: \(L^{p}(p>2)\)-strong convergence in averaging principle for two time-scales stochastic evolution equations driven by Lévy process. arXiv:1511.03438v3

  13. 13.

    Khasminskii, RZ: On the principle of averaging the Itô stochastic differential equations. Kibernetika 4, 260-279 (1968)

    Google Scholar 

  14. 14.

    Xu, J, Liu, J: An averaging principle for multivalued stochastic differential equations. Stoch. Anal. Appl. 32(6), 962-974 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Tian, L, Lei, D: The averaging method for stochastic differential delay equations under non-Lipschitz conditions. Adv. Differ. Equ. 2013, 38 (2013)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Xu, S: Multivalued stochastic differential equations with non-Lipschitz coefficients. Chin. Ann. Math. 30B(3), 321-332 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Malinowski, MT: The narrowing set-valued stochastic integral equations. Dyn. Syst. Appl. 24(4), 399-419 (2015)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Malinowski, MT: Fuzzy and set-valued stochastic differential equations with local Lipschitz condition. IEEE Trans. Fuzzy Syst. 23(5), 1891-1898 (2015)

    Article  Google Scholar 

  19. 19.

    Malinowski, MT: Set-valued and fuzzy stochastic differential equations in M-type 2 Banach spaces. Tohoku Math. J. 67(3), 349-381 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Malinowski, MT, Agarwal, RP: On solutions set of a multivalued stochastic differential equation. Czechoslov. Math. J. 67(1), 11-28 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Cépa, E: Équations différentielles stochastiques multivoques. In: Séminaire de Probabilités XXIX. Lecture Notes in Math., vol. 1631, pp. 86-107. Springer, Berlin (1995)

    Google Scholar 

  22. 22.

    Ren, J, Zhang, X: Stochastic flows for SDEs with non-Lipschitz coefficients. Bull. Sci. Math. 127(8), 739-754 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Chow, PL: Stochastic Partial Differential Equations. Chapman Hall/CRC, New York (2007)

    Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous referee for careful reading of the manuscript and valuable comments and suggestions.

This work was supported by NSFs of China (No. 11526196) and the Fundamental Research Funds for the Central Universities (SCUEC: CZQ17005; CZW15124).

Author information

Affiliations

Authors

Contributions

All authors read and approved the final manuscript.

Corresponding author

Correspondence to Zhongkai Guo.

Ethics declarations

Competing interests

The author declares that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Guo, Z. \(L^{p}\) \((p\geq2)\)-strong convergence in averaging principle for multivalued stochastic differential equation with non-Lipschitz coefficients. Adv Differ Equ 2017, 386 (2017). https://doi.org/10.1186/s13662-017-1442-5

Download citation

Keywords

  • multivalued stochastic differential equations
  • non-Lipschitz
  • averaging principle
  • \(L^{p}\) \((p\geq2)\)-strong convergence