^{1,2}

^{1}

^{1}

^{2}

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

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 [

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 [

The paper is organized as follows. In Section

In this paper, unless otherwise specified, let

Let

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

To be precise, let us state the following conditions.

(H1) The local Lipschitz condition: for every integer

(H2) Linear growth condition: there exists a positive constant

(H3) Monotone condition: there exists a positive constant

(H4) The bounded

Let us first give the definition of the solution.

An

Equation (

We will show the strong convergence of the EM method on (

Under the condition (H1), let

For

Under the conditions (H1) and (H4), the EM approximate solution converges to the exact solution of (

Fix a

We will show the strong convergence of the EM method on (

Under the linear growth conditions (H2), there exists a positive constant

It follows from (

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

Under the linear growth conditions (H2), there exists a positive constant

By the inequality

According to Theorem

Under the conditions (H1) and (H2), the EM approximate solution converges to the exact solution of (

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:

In this section, we give the convergence of the EM method to (

Under the monotone condition (H3), there exists a positive constant

By Itô formula, for all

According to Theorem

Under the conditions (H1) and (H3), the EM approximate solution converges to the exact solution of (

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.