Convergence Analysis of a Splitting Method for Stochastic Differential Equations

(1) { dy(t) = f(y(t))dt + g(y(t))dW (t), 0 ≤ t ≤ T y(0) = y0 where T > 0 is the terminal time, y(t) : [0, T ]× Ω → R, f(y) : R → R, g(y) : R → Rm×d, and W (t) = (W1(t), · · · ,Wd(t))∗ is a standard d-dimensional Brownian motion defined on a complete, filtered probability space (Ω,F , P, {Ft}0≤t≤T ). Stochastic differential equations are used in many fields, such as stock market, financial mathematics, stochastic controls, dynamic system, biological science, chemical reactive kinetics and hydrology, and so on. Thus, it is of importance to study the solution of SDEs. However, it is often very difficult or impossible to find the analytic solutions of SDEs, as a consequence, numerical methods for finding approximate solutions of SDEs have attracted much attentions. There have been a lot of publications in which numerical methods for stochastic differential equations and their applications were studied and discussed. For instance, the Itô-Taylor type method proposed in [11] that makes use of the socalled Itô Taylor expansion to discretize the SDEs; the linearization type methods suggested in [3, 12, 17], that first linearize the drift and diffusion coefficients of the SDEs and then solve the pruned linear SDEs instead; the Runge-Kutta type methods [4,5,16,20], in which the Runge-Kutta methods for solving ordinary differential equations are extended to solve the SDEs. Concerning the stability of the methods,