Convergence and stability of balanced methods for stochastic delay integro-differential equations

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Abstract

This paper deals with a family of balanced implicit methods for the stochastic delay integro-differential equations. It is shown that the balanced methods, which own the implicit iterative scheme in the diffusion term, give strong convergence rate of at least 1/2. It proves that the mean-square stability for the stochastic delay integro-differential equations is inherited by the strong balanced methods and the weak balanced methods with sufficiently small stepsizes. Several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean-square stabilities.

Introduction

Stochastic integro-differential equations have come to play an important role in a variety of application areas including economy [1], [2], biology [3], population dynamics [4] and so on. The class of stochastic delay integro-differential equations (SDIDEs) that admits explicit solutions is rather limited. Thus, appropriate numerical methods are needed to apply in practice and to study their properties.

The numerical analysis of stochastic delay differential equations (SDDEs) is well studied, for instance, Baker and Buckwar [5], [6], U. Ku¨chler and Platen [7], Mao et al. [8], Liu et al. [9], Wang et.al [10] and so on. As for SDIDEs, there has been much less research of numerical schemes. Mao [12] considered the stability of SDIDEs. Tan et al. [13] investigated the convergence and mean-square stability of the split-step backward Euler method for SDIDEs. Li et al. [14] discussed the mean-square exponential stability of stochastic theta methods for nonlinear SDIDEs.

However, it is already known that the majority of the numerical methods for SDIDEs are explicit or semi-implicit methods which are only implicit in the drift coefficient. These drift-implicit methods are well adapted for stiff systems with small stochastic noise intensity or additive noise. In those case the system with large multiplicative noise, the application of fully implicit methods is unavoidable. One of the most important fully implicit methods are the balanced implicit methods, which were firstly introduced by Milstein, Platen and Schurz [16] and used to solve stiff systems. Some recent papers which consider the balanced implicit methods for the stochastic differential equations include [15], [17], [18], [19].

Consider the following scalar linear stochastic delay integro-differential equationdx(t)=(αx(t)+βt-τtx(s)ds)dt+ηx(t-τ)dW(t),t>0,x(t)=φ(t),t[-τ,0],where α,β,η R,τ is a positive fixed delay, φ(t)C([-τ,0];R), W(t) is a scalar Brownian motion which defined on an appropriate complete probability space (Ω,F,{Ft}t0,P), with a filtration {Ft}t0 satisfying the usual conditions (i.e. it is increasing and right-continuous while F0 contains all P-null sets).

Lemma 1.1 [11]

For any given T>0, there exist positive numbers H1 and H2 such that the solution x(t) of (1) satisfiesEsup-τst|x(s)|2H1[1+Eφ2],for all t[-τ,T],E|x(t)-x(s)|2H2(t-s),for any 0s<tT,t-s<1,Eαx(t)+βt-τtx(s)dsM,for any t[0,T].

Regarding numerical analysis of (1), Ding et al. [11] discussed the convergence and stability of the semi-implicit Euler method for SDIDEs. However, to the best of our knowledge, there are no stability results of balanced methods for the system (1). The aim of this paper is to investigate the strong mean-square convergence and mean-square stability of the balanced implicit methods of the system (1). The rest of the paper is organized as follows. In the subsequent section, we establish Theorem 2.1, showing the strong balanced implicit methods are convergent with strong order 1/2. Section 3 and Section 4 deal with linear mean-square stabilities of the strong balanced implicit methods and the weak balanced implicit methods. In Section 5, some numerical results are reported to demonstrate the conclusions. Finally, conclusion is made in Section 6.

Section snippets

Convergence of the balanced implicit methods

Given a stepsize h=τ/m>0, a version of strong balanced methods for (1) is given byYn+1=Yn+αYn+βY^nh+ηYn-mΔWn+Dn(Yn-Yn+1),n0,Yn=φ(nh),n=-m,-m+1,,0,where Yn is an approximation to x(tn) with tn=nh,ΔWn=W(tn+1)-W(tn). Here Y^n is given byY^n=hk=0m-1Yn-m+k.Here Dn is given byDn=D0nYn,Yn-m,Y^nh+D1nYn,Yn-m,Y^n|ΔWn|=D0nh+D1n|ΔWn|,where the D0n=D0nYn,Yn-m,Y^n,D1n=D1nYn,Yn-m,Y^n are called control functions. In order to obtain our main results in this paper, we assume that D0n,D1n in the Eq. (7) are

Mean-square stability of the strong balanced implicit methods

We investigate the mean-square stability of the strong balanced methods in this section.

The following lemma give the sufficient condition on the stability for the analytic solution of the system (1).

Lemma 3.1

[11]

Assume that α,β,η satisfyα+|β|τ+12η2<0,then the solution of (1) is asymptotically stable in the mean-square, that is, limnE(x(tn))2=0.

Given parameters α,β,η and stepsize h, we say the balanced implicit methods are mean-square stable if limnE(Yn)2=0 for any Y0. The following theorem will show the

Mean-square stability of the weak balanced implicit methods

In this section, we will investigated the mean-square stability of the weak balanced implicit methods equipped with two-point random variables for the driving process.

Given a stepsize h>0, the weak balanced implicit methods are defined byYn+1=Yn+1+D^n-1αYn+βY^nh+ηYn-mΔW^n,n0,Yn=φ(nh),n=-m,-m+1,,0,where D^n=D0nh+D1n|ΔW^n|,PΔW^n=h=PΔW^n=-h=1/2.

It is not difficult to find thatEΔW^n=0,EΔW^n2=h.

The following theorem will show that the weak balanced implicit methods (55) can preserve the

Numerical experiments

In this section several numerical examples are given to illustrate our theoretical results in the previous sections. Consider the scalar linear equationdx(t)=(αx(t)+βt-τtx(s)ds)dt+ηx(t-τ)dW(t),t>0,x(t)=1,t[-τ,0].

Denoting Yni as the numerical approximation to xi(tn) at step point tn in ith simulation of all K=2000 simulations. We use 1Ki=1K|Yni|2 to approximate E|Yn|2. For simply, we choose D0j=1;D1j=1(j=0,1,). All the figures are drawn with the vertical axis scaled logarithmically.

We

Conclusion

In this work, we have examined the convergence and the mean-square stability of the balanced methods for the stochastic delay integro-differential equations. It is shown that the strong balanced implicit methods give strong convergence rate of at least 1/2. The forgoing results show that both the strong balanced methods and weak balanced methods can reproduce the mean-square stability of the system with sufficiently small stepsize h. The theory result and the numerical experiment show that

Acknowledgment

This work was supported by: 1. NSF of China (No. 11326238, 11326138 and 11101101); 2. the Project sponsored by the Science Foundation of Jiangxi Provincial Department of Education (No. gjj13374 and gjj13658); 3. the Project sponsored by the Youth Natural Science Foundation of Jiangxi Province (20132BAB211007).

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