Elsevier

Applied Mathematics and Computation

Volume 251, 15 January 2015, Pages 527-538
Applied Mathematics and Computation

Mean-square stability of analytic solution and Euler–Maruyama method for impulsive stochastic differential equations

https://doi.org/10.1016/j.amc.2014.11.098Get rights and content

Abstract

From the view of algebra, the mean-square stability of analytic solutions and numerical solutions for impulsive stochastic differential equations are considered. By the logarithmic norm, the conditions under which the analytic and numerical solutions for a linear impulsive stochastic differential equation are mean-square stable (MS-stable) respectively are obtained. The conditions are simple and easy to use. Some numerical experiments are given to illustrate the results.

Introduction

Impulsive effects exist widely in many evolution processes in which states are changed abruptly at certain moments of time, involving such fields as medicine, biology, economics, mechanics, electronics and telecommunications, etc. (see [1], [8]). In recent years, there have been intensive studies on the impulsive differential equations (see [9], [10], [11]). However, besides impulsive effects, stochastic effects likewise exist widely in real systems (see [7]). It has received considerable attention. For example, the existence and uniqueness of the analytic solution for stochastic differential equations (SDEs) has been studied by Gard [5] and Mao [13], the stability of Euler–Maruyama (EM) method for SDEs has been studied by Cao et al. [3] and Liu et al. [14], the stability and convergence of composite Milstein method for SDEs has been studied by Omar et al. [15], the convergence of EM method for SDEs has been studied by Buckwar [2], etc.

Therefore, it is necessary to consider the stability of solutions of impulsive stochastic differential equations. It has attracted interests of many researchers. Wu and Sun [17], Yang et al. [19], [20] and Zhao et al. [21] have studied the pth moment stability of impulsive stochastic differential equations (ISDEs). Wu and Ding [18] has studied the convergence and stability of Euler method for ISDEs and Zhao et al. [22] has studied pth moment stability of EM method for ISDEs. Liu at al. [12] has studied the mean-square stability of semi-implicit method for one-dimensional ISDEs.

Like many other equations, explicit solutions can rarely be obtained for the impulsive stochastic differential equations. Thus, it is necessary to develop numerical methods and to study the properties of these methods.

Like as ordinary differential equations, we just discuss the test equations. Unfortunately, the diffusion and drift coefficients can not be diagonalized at the same time.

Hence, in this paper, we consider the stability of both analytic and EM solution of the following equationdX(t)=AX(t)dt+BX(t)dW(t),tτk,t>0,X(τk+)=HkX(τk),t=τk,X(0+)=X0,where A=diag(λ1,λ2,,λn)Rn×n, B=(bij)n×nRn×n, Hk=(Hkij)n×nRn×n(k=1,2,3,) are constant matrices, W(t) is standard Brownian motion, X(t)=(X1(t),,Xn(t))T,X(t+) is the right limit of X(t),0<τ1<τ2<τ3<, limkτk=. Denote τ0=0, N={1,2,}.

By Gard [5] and Mao [13], the Eq. (1.1) has on (0,) unique solutionX(t)=expA-12b2t+BW(t)X0,t0,τ1,X(t)=expA-12b2(t-τk)+B(W(t)-W(τk))tτk,τk+1,kN·i=1kHiexpA-12b2(τi-τi-1)+B(W(τi)-W(τi-1))X0.

In this paper, we always assume that there exist two constants θ1 and θ2 such that0<θ1τk-τk-1θ2<,forallkN.In Section 2, MS-stability of Eq. (1.1) is analyzed. The results obtained in Section 2 are coincide with existing results. Such as, Corollary 2.7 is Theorem 1 in [16], Corollary 2.8 is Theorem 2.1 in [12]. In Section 3, conditions of MS-stability of the Euler–Maruyama scheme corresponding to Section 2 are obtained. The results in Section 3 are also coincide with existing results. Corollary 3.8 is coincide with Theorem 2 in [16]. The main result with θ=0 in Section 3 of [12] is coincide with Corollary 3.9. In Section 4, some numerical experiments confirming our stability analysis in Section 3 are given.

Section snippets

The stability of the analytic solution

Let (Ω,F,P) be a probability space with a filtration (Ft)t0, which satisfies the usual conditions. Let W(t),t0 in Eq. (1.1) be Ft-adapted and independent of F0. We assume X0 to be F0-measurable.

In this section we will give the sufficient conditions under which the analytic solution of (1.1) is MS-stable. First, we will give the definition of MS-stability.

Definition 2.1

[16]

The zero solution of the system (1.1) is MS-stable ifE(|X(t)|2)0ast,where |·| stands for the Euclidean norm, E(·) denotes the expectation

MS-stability of Euler–Maruyama scheme

In this section, we shall investigate the MS-stability of EM scheme with variable step-size for system (1.1). The adaption of EM scheme to Eq. (1.1) leads to a numerical solution of the following typeXkm,l=(I+Ahk+BΔWkm,l-1)Xkm,l-1,k=0,1,,l=1,,m,Xkm,0=HkX(k-1)m,m,X0,0=X0,where hk=τk+1-τkm is a step-size for a positive integer m,I is a unit matrix, the discrete point tkm,l=τk+lhk,0lm,ΔWkm,l=W(tkm,l+1)-W(tkm,l) are independent N(0,hk)-distributed Gaussian random variables, Xkm,lX(tkm,l),1lm

Numerical experiments

Example 4.1

dX(t)=2X(t)dt+X(t)dW(t),tk,X(k+)=HkX(k),t=k,X(0+)=1,where HkR,k=1,2,.

We will confirm the main result in this paper through numerical experiments. First, we choose (4.1) as test equations to illustrate the convergence.

From [5], [13], we can obtain the explicit solution of (4.1).X(t)=exp{1.5t+W(t)}X(0+),fort0,1,exp{1.5(t-k)+(W(t)-W(k))}·i=1kHiexp{1.5+(W(i)-W(i-1))}X(0+)fortk,k+1.

In Fig. 1, we plot the explicit solution of (4.1) together with the EM method for step-size h=1/128. Owing to the

Acknowledgements

We would like to thank the professor Xuerong Mao for his useful suggestions.

The work was supported by the National Nature Science Foundation of China (No. 11101130). The work was supported by the foundation of Jiangsu University of Science and Technology of China (Nos. 35050903, 633051205).

References (22)

  • K. Dekker et al.

    Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations

    (1984)
  • Cited by (0)

    View full text