Numerical Solutions of Stochastic Differential Equations with Piecewise Continuous Arguments under Khasminskii‐Type Conditions

Minghui Song,Ling Zhang

doi:10.1155/2012/696849

Abstract

The main purpose of this paper is to investigate the convergence of the Euler method to stochastic differential equations with piecewise continuous arguments (SEPCAs). The classical Khasminskii‐type theorem gives a powerful tool to examine the global existence of solutions for stochastic differential equations (SDEs) without the linear growth condition by the use of the Lyapunov functions. However, there is no such result for SEPCAs. Firstly, this paper shows SEPCAs which have nonexplosion global solutions under local Lipschitz condition without the linear growth condition. Then the convergence in probability of numerical solutions to SEPCAs under the same conditions is established. Finally, an example is provided to illustrate our theory.

Highlights

Stochastic modeling has come to play an important role in many branches of science and industry
The main purpose of this paper is to investigate the convergence of the Euler method to stochastic differential equations with piecewise continuous arguments SEPCAs
Such models have been used with great success in a variety of application areas, including biology, epidemiology, mechanics, economics, and finance

Summary

Introduction

Stochastic modeling has come to play an important role in many branches of science and industry. Many authors have studied the problem of SDEs. The classical existence-and-uniqueness theorem requires the coefficients f x t and g x t to satisfy the local Lipschitz condition and the linear growth condition see 1. Differential equations with piecewise continuous arguments EPCAs had attracted much attention, and many useful conclusions were obtained These systems have applications in certain biomedical models, control systems with feedback delay in the work of L. Dai and Liu 11 give the mean-square stability of the numerical solutions of linear stochastic differential equations with piecewise continuous arguments. The convergence in probability of numerical solutions to stochastic differential equations with piecewise continuous arguments under the same conditions is established.

Preliminary Notation and Euler Method

Convergence in Probability of the Euler-Maruyama Method

Numerical Example