Composite Milstein method for numerical solutions of stochastic differential equations

In this paper we present new numerical methods for the strong solutions of stochastic differential equations driven by 1-dimensional Wiener process. According to the terminology used by K. Burrage and T. Tian [1], we call these methods composite Milstein methods. These methods are a combination between semi implicit and implicit Milstein schemes. At each step either semi implicit or implicit scheme is used in order to obtain better stability properties. Two criterion for selecting semi implicit or implicit scheme are given for linear and nonlinear stochastic differential equations. The convergence and stability properties are given. The numerical results show that the composite methods are very promissing methods.

Keywords: Stochastic differential equation, Wiener process, composite Milstein method of type 1, composite Milstein method of type 2, stability and convergence properties.

[1] K. Burrage, T.H.Tian, The composite Euler method for stiff stochastic differential equations, Journal of computational and applied Mathematics. 131 (2001) 407-426.
[2] P.E. Kloeden, E. Platen, Numerical solution of stochastic differential equations, Springer, Berlin, 1992.
[3] G.N. Milstein, Numerical Integration of stochastic differential equations, Mathematics and Its application, Kluwer, Dordrecht, 1995.

This is joint work with L. Abbaoui (Department of Mathematics, University of Setif, Algeria)