Convergence And Stability Of Exponential Euler Method For Stochastic Differential Equations With Piecewise Continuous Arguments

Differential equations with piecewise continuous arguments(EPCA) as importantmodels are applied widely in many fields, such as economical, physical, biologicalsystems and control theory and so on. This equations had attracted much attention,and many useful conclusion was obtained about convergence and stability. However,up to now there is only a few scholars who have considered the influence of noise inthe mathematics model. Actually, the environment and accidental events may greatlyinfluence the systems. Thus analyzing stochastic differential equations with piecewisecontinuous arguments(SEPCA) is an interesting topics both in theory andapplications.The paper focus on the stability and convergence of numerical solutions forstochastic differential equations with piecewise continuous arguments.The paper surveys the recent developments of the study of stochastic differentialequations(SDDE) and EPCA from the old paper to now. We introduce exponentialRunge-Kutta method in Ordinary Differential equations, whose one order schema isexponential Euler method. We have proved the convergence order is0.5whenexponential Euler method is applied in semi-linear stochastic differential equationswith piecewise continuous arguments, giving a numerical example is given to verifythe convergence order of exponential Euler method.Then the stability is one of an important part of analyses, including mean squarestability, p-th moment exponential stability. And a sufficient condition is establishedunder which exponential Euler method is mean square stable when is applied tosemi-linear stochastic differential equations with piecewise continuous arguments.When the given semi-linear is almost sure exponential stable and p-momentexponential stable. Finally, we use numerical approximations to verity the meansquare stability of exponential Euler method. Then gives a numerical experiment, inwhich exponential Euler method is able to preserve the mean square stability of theequation while EM method can’t preserve this stability.