Stability And Convergence Of Numerical Methods For Stochastic Delay Differential Equations

Stochastic delay differential equations are often used to describe the dynamicalsystems, which depend on not only present states but also their past history with noisedisturbance. Hence the investigation of stochastic delay differential equations havereceived wide attention at home and abroad based on stochastic differential equations.But it is not easy to obtain the exact solution of stochastic delay differential equations ingeneral, Hence the study of numerical methods for the system is necessary.This paper investigates convergence and stability of numerical methods forstochastic delay differential equations from two parts: The first part investigatesconvergence and mean-square stability of the balanced method for stochastic differentialdelay equations. Comparing to the traditional conditions of stability, a weaker conditionof asymptotic and mean-square stability of the exact solution is given by applying thesemi-martingale convergence theorem. The balanced method reproduces mean-squarestability of the exact solution for stochastic differential delay equations. Finally, anumerical example is given to illustrate the main theory. The second part studies thealmost surely exponentially stable of numerical methods including Euler-Maruyama andback Euler-Maruyama for N-dimensional stochastic delay differential equations by thediscrete and continuous convergence theorems.The main results of this paper are: convergence and stability of stochastic delaydifferential equations under balanced method, It gives a weaker condition of stabilitycomparing to traditional condition of stability; the difference of condition of almost sureexponential stability for stochastic delay differential equation between Euler-Maruyamamethod and backward Euler-Maruyama method. It is different from traditional methodsinvestigating almost sure exponential stability of stochastic delay differential equations.