Abstract and Applied Analysis

Volume 2012 (2012), Article ID 643783, 16 pages

doi:10.1155/2012/643783

Research Article

Convergence of the Euler Method of Stochastic Differential Equations with Piecewise Continuous Arguments

Ling Zhang1,2 and Minghui Song¹

¹Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

²Institute of Mathematical Sciences, Daqing Normal University, Daqing 163712, China

Received 20 June 2012; Accepted 29 October 2012

Academic Editor: Roman Dwilewicz

Copyright © 2012 Ling Zhang and Minghui Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The main purpose of this paper is to investigate the strong convergence of the Euler method to stochastic differential equations with piecewise continuous arguments (SEPCAs). Firstly, it is proved that the Euler approximation solution converges to the analytic solution under local Lipschitz condition and the bounded th moment condition. Secondly, the Euler approximation solution converge to the analytic solution is given under local Lipschitz condition and the linear growth condition. Then an example is provided to show which is satisfied with the monotone condition without the linear growth condition. Finally, the convergence of numerical solutions to SEPCAs under local Lipschitz condition and the monotone condition is established.

1. Introduction

Recently, differential equations with piecewise continuous arguments (EPCAs) have attracted much attention, and many useful conclusions have been obtained. These systems have applications in certain biomedical models, control systems with feedback delay in the work of Cooke and Wiener [1]. The general theory and basic results for EPCAs have by now been thoroughly investigated in the book of Wiener [2]. Song et al. [3] deal with the stability analysis of numerical methods for the solution of advanced differential equations with piecewise continuous arguments. A typical EPCA contains arguments that are constant on certain intervals. The solutions are determined by a finite set of initial data, rather than by an initial function, as in the case of general functional differential equation. A solution is defined as a continuous, sectionally smooth function that satisfies the equation within these intervals. Continuity of a solution at a point joining any two consecutive intervals leads to recursion relations for the solution at such points. Hence, EPCAs represent a hybrid of continuous and discrete dynamical systems and combine the properties of both differential and difference equations.

However, up to now there are few people who have considered the influence of noise to EPCAs. Actually, the environment and accidental events may greatly influence the systems. Thus analyzing SEPCAs is an interesting topic both in theory and applications. There is in general no explicit solution to an SEPCA, hence numerical solutions are required in practice. Numerical solutions to stochastic differential equations (SDEs) have been discussed under the local Lipschitz condition and the linear growth condition by many authors (see [4, 5]). Mao [6] discusses numerical solutions to stochastic differential delay equations (SDDEs) under the local Lipschitz condition and the linear growth condition. Mao and Sabanis [7] discuss numerical solutions to SDDEs with variable delay under the local Lipschitz condition and the linear growth condition. Mao discusses numerical solutions to SDEs and SDDEs under the local Lipschitz condition and the monotone condition (see [8]). Dai and Liu [9] give the mean-square stability of the numerical solutions of linear SEPCAs. However, SEPCAs do not have the convergence results. The main aim of this paper is to establish convergence of numerical solution for SEPCAs under the differential conditions.

The paper is organized as follows. In Section 2, we introduce necessary notations and the Euler method. In Section 3, the strong convergence of the Euler-Maruyama method to SEPCAs under local Lipschitz condition and the bounded th moment condition will be given. In Section 4, the strong convergence of the Euler-Maruyama method to SEPCAs under local Lipschitz condition and the linear growth condition will be presented. In Section 5, an example is provided to show which is satisfied with the monotone condition without the linear growth condition. In Section 6, we obtain the convergence of numerical solutions to SEPCAs under local Lipschitz condition and the monotone condition is established.

2. Preliminary Notation and Euler Method

In this paper, unless otherwise specified, let be the Euclidean norm in . If is a vector or matrix, its transpose is defined by . If is a matrix, its trace norm is defined by . For simplicity, we also have to denote by .

Let be a complete probability space with a filtration , satisfying the usual conditions. and denote the family of all real valued -adapted process , such that for every , . and ., respectively. For any with , denote the family of continuous functions from to with the norm . Denote the family of all bounded -measurable -valued random variables. Let be a -dimensional Brownian motion defined on the probability space.

Throughout this paper, we consider stochastic differential equations with piecewise continuous arguments:

with initial data , where is a vector, and denotes the greatest-integer function. By the definition of stochastic differential, this equation is equivalent to the following stochastic integral equation:

Moreover, we also require the coefficients and to be sufficiently smooth.

To be precise, let us state the following conditions.

 (H1) The local Lipschitz condition: for every integer , there exists a positive constant such that

for those with . (H2) Linear growth condition: there exists a positive constant such that

for all . (H3) Monotone condition: there exists a positive constant such that

for all . (H4) The bounded th moment condition: there exists a pair of constants and such that

Let us first give the definition of the solution.

Definition 2.1 (see [10]). An -valued stochastic process is called a solution of (2.1) on , if it has the following properties:

(1) is continuous on and adapted; (2) and ;(3)Equation (2.2) is satisfied on each interval with integral end points almost surely. A solution is said to be unique if any other solution is indistinguishable from , that is,

Let be a given stepsize with integer and the grid points defined by . For simplicity, we assume . We consider the Euler-Maruyama method to (2.1),

for , where is approximation to the exact solution . Let . The adaptation of the Euler method to (2.1) leads to a numerical process of the following type:

where and are approximations to the exact solution and , respectively. The continuous Euler-Maruyama approximate solution is defined by

where and for . It is not difficult to see that for . For sufficiently large integer , define the stopping times .

3. Convergence of the Euler-Maruyama Method under the Bounded pth Moment

We will show the strong convergence of the EM method on (2.1) under local Lipschitz condition and the bounded th moment condition. The following lemma shows that both and are close to each other.

Lemma 3.1. Under the condition (H1), let be arbitrary. Then

where .

Proof. For , there are two integers and such that . By the Hölder inequality, we compute

This implies that, for any ,

By the Doob martingale inequality, we have

Using the local Lipschitz conditions

where . The proof is completed.

Theorem 3.2. Under the conditions (H1) and (H4), the EM approximate solution converges to the exact solution of (2.1) in the sense that

Proof. Fix a ; let ; it is easy to see that

By the Young inequality , for any , we have

Thus for any , we have

By condition (H4), we have

Similarly, the result is

So that

Using these bounds, then

By the definitions of and , we have

Thus, for any

By the Hölder inequality, condition (H1), and Lemma 3.1, one gets

Similarly, by the Burkhôlder-Davis-Gundy inequality, then

Substituting (3.16) and (3.17) into (3.15) gives

By the Gronwall inequality, we must get

where . So we have

Given any , we can choose sufficiently small for

then choose sufficiently large for

and finally choose sufficiently small, so that

Thus . The proof is completed.

4. Convergence of the Euler-Maruyama Method under Linear Growth Condition

We will show the strong convergence of the EM method on (2.1) under local Lipschitz condition and the linear growth condition. In the following we will show that the linear growth condition (H2) implies the bounded th moment condition (H4).

Lemma 4.1. Under the linear growth conditions (H2), there exists a positive constant such that the solution of (2.1) satisfies

where is a constant independent of .

Proof. It follows from (2.2) that

By the Hölder inequality, we obtain

This implies that, for any ,

By the Burkholder-Davis-Gundy inequality and the Hölder inequality, it is not difficult to show that

where is a constant. Note from the linear growth conditions that

By the Gronwall inequality, we must get

where is a constant independent of .

The following lemma shows that the continuous Euler-Maruyama approximate solution has bounded th moments.

Lemma 4.2. Under the linear growth conditions (H2), there exists a positive constant such that the continuous approximate solution of the Euler-Maruyama (2.10) satisfies

where is a constant independent of .

Proof. By the inequality and (2.10), in the same way as Lemma 4.1, for any , we can obtain

where is a constant. Note from the linear growth conditions that

By the Gronwall inequality, we must get

where is a constant independent of .

According to Theorem 3.2, we have the following theorem.

Theorem 4.3. Under the conditions (H1) and (H2), the EM approximate solution converges to the exact solution of (2.1) in the sense that

5. A Motivating Example

In the above section, we give the strong convergence numerical solution of SEPCAs under the local Lipschitz condition (H1) and the linear growth condition (H2). However, there are many SEPCAs that do not satisfy the linear growth condition, consider the following SEPCA:

Clearly, the equation do not satisfy the linear growth condition (H2). But the example is analyzed under condition (H3) which covers many nonlinear SEPCAs. On the other hand, we have

In other words, the equation satisfies condition (H3). Moreover, we also have

We see clearly that (H3) follows from (H2). Therefore, The following result is more general than Theorem 3.2. Let us now turn to establish the convergence of the Euler-Maruyama method to (2.1) under the conditions (H1) and (H3).

6. Convergence of the Euler-Maruyama Method under Monotone Condition

In this section, we give the convergence of the EM method to (2.1) under the local Lipschitz condition (H1) and the monotone condition (H3). We prove the bounded th moment property of the EM approximate solution and the exact solution to (2.1) under the monotone condition (H3).

Lemma 6.1. Under the monotone condition (H3), there exists a positive constant such that the solution of (2.1) satisfies

where is a constant independent of .

Proof. By Itô formula, for all , we have

For any , we have

By the Burkholder-Davis-Gundy inequality and Exercise 2.5 in [8], it is not difficult to show that

Substituting (6.4) into (6.3) and using the Hölder inequality

So we obtain

By the Gronwall inequality, we must get

where is a constant independent of . Similarly, we can show that

The proof is completed.

According to Theorem 3.2, we obtain the following.

Theorem 6.2. Under the conditions (H1) and (H3), the EM approximate solution converges to the exact solution of (2.1) in the sense that

Acknowledgments

The financial support from the National Natural Science Foundation of China (no. 11071050) and the Youths’ Key Projects of Heilongjiang Provincial Education Department of P. R. China (no. 1155G001) is gratefully acknowledged.

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