An analysis of stability of milstein method for stochastic differential equations with delay

-This paper deals with the adapted Milstein method for solving linear stochastic delay differential equations. It is proved that the numerical method is mean-square (MS) stable under suitable conditions. The obtained result shows that the method preserves the stability property of a class of linear constant-coefficient problems. This is also verified by several numerical examples. (~) 2006 Elsevier Ltd. All rights reserved. Keywords-Stochas t ic delay differential equations, It& stochastic integral, MS-stability, Milstein method, Numerical simulation. 1. I N T R O D U C T I O N Stochastic delay differential equations (SDDE) can be viewed as generalizations of both deterministic delay differential equations (DDE) and stochastic ordinary differential equations (SODE). In many scientific fields, such as finance, biology, mechanics, and ecology, SDDE are often used to model the corresponding systems. In recent years, there has been growing interesting in studying such equations. For the research in theoretical solutions of SDDE, one can refer to Mao's monograph [1] and the references therein. Usually, the solution of a SODE can be obtained as a Markov process [1, Ch. 2]. Unfortunately, it is difficult to get an explicit solution of a SDDE since the models described by SDDEs depend not only on the present but also the history and hence, their solutions cannot be considered as Markovian. Moreover, the presence of a delay term could change a system's dynamic properties such as stability, oscillation, bifurcation, chaos, etc. Therefore, there are many differences between the two kinds of equations. Up to now, the research for SDDE is far fl'om complete because of the complexities originating from both noise and delay. In view of the above causes, it becomes impor tant to construct numerical methods to solve SDDE. This project is supported by NSFC (No. 10571066) and SRF for ROCS, SEM. 0898-1221/06/$ see frout matter © 2006 Elsevier Ltd. All rights reserved. Typeset by A~gg-TEX doi: 10.1016/j.camwa.2006.01.004