Numerical Analysis Of Split-step ? Methods For Stochastic Differential Delay Equations

Stochastic differential delay equations were applied to many areas,including physics,biology,finance,control theory,medicine and many other fields as an important mathematics model.This kind of equations consider the effect of delay on the system,at the same time,consider the impact of external environment on the system properties.Therefore,stochastic differential delay equations simulate the natural life more accurately.In practice,it is difficult to solve the exact solution of stochastic differential delay equations.Hence investigating appropriate numerical methods and studying the properties of numerical solutions are very important in application and theory.Recently numerous authors studied the SDDEs with constant delays and numerical methods for this kind of equations.Only a few authors studied the SDDEs with variable delays.In this paper,we investigate the convergence and stability of split-step ? method of SDDEs with variable delays.In this paper,we investigate the convergence and stability of diffused split-step ?method and drifting split-step ? method respectively.First,we analyze when the coefficient of stochastic differential delay equations satisfy the global Lipschitz and linear growth condition,we study that diffused split-step ? method is convergent in mean-square sense,and the convergence order is 1/2.Next we discuss the stability of diffused split-step ? method,we obtain that the numerical method is exponential mean-square stable with limited step size in linear growth and monotonous condition.Secondly,when the coefficient of stochastic differential delay equations satisfy the global Lipschitz and linear growth condition,we study that drifting split-step ? method is convergent in mean-square sense,and the convergence order is 1/2;at the same time,we discuss the stability of drifting split-step ? method in linear growth and monotonous condition,we obtain when ??(1/2,1],the numerical method is exponential mean-square stable for all step size;when ??[0,1/2],there is an h0,when h?(0,h0),drifting split-step ? method is exponential mean-square stable.