The Convergence And Stability Of Exponential EULER Method For Stochastic Differential Equations

The development of stochastic differential equations (SDEs) has been more than 60 years, and since the Japanese mathematician Ito founded the stochastic calculus in the 1940's, the theory of stochastic differential equation has developed rapidly. It has been applied widely in the economic, biological, physical, telecommunications, automation and other fields. According different models stochastic differential equations have been classified for delay differential equations, neutral differential equations, and the backward stochastic differential equations. However, in a very long period of time computer calculation power is not strong enough, so people simplified the problems by omitting the stochastic factors in practical application. But in recent years, with the rapid development of computer technology and numerical methods, people have constructed a number of numerical methods for stochastic differential equations, which means some stochastic models can be studied via the simulation of high speed compute. Because of the complexity of the stochastic system, the expression of analytical solution is difficult to obtain. Therefore, the construction of numerical methods plays an important role in the numerical approximation. In this paper, we constructed the exponential Euler numerical scheme and then discuss the convergence and stability when it is applied in SDEs.In this paper, on the base of the background of SDEs, Firstly, we introduce several numerical methods which are commonly used and the exponential Runge-Kutta method in ordinary differential equations, whose one order schema is exponential Euler method. We have proved the convergence order is 0.5 when exponential Euler method is applied in semi-linear stochastic differential equation and a numerical example is given to verify the convergence order of exponential Euler method. Then the stability is analysis, including mean square stability, almost sure exponential stability, p-th moment exponential stability. Mean square stability region of exponential Euler method is calculated in the scale linear differential equations. We found that mean square stability region of exponential Euler method contains that of EM method, and contains that of the equation itself. Then gives a numerical experiment, in which exponential Euler method is able to preserve the mean square stability of the equation while EM method can't preserve this stability. And a sufficient condition is established under which exponential Euler method is mean square stable when is applied to semi-linear stochastic differential equation. When the given semi-linear SDE is almost sure exponential stable and p-moment exponential stable there exists a sufficiently small step such that exponential Euler method can preserve these two properties. Finally, we use numerical approximations to verity the almost sure stability of exponential Euler method.