CONVERGENCE AND STABILITY OF MODIFIED FULLY IMPLICIT MILSTEIN SCHEME FOR STOCHASTIC DIFFERENTIALEQUATIONS

Abstract. In this paper we discuss implicit Taylor methods for It^o stochastic differ-ential equations (SDEs). Based on the relationship between It^o stochastic integrals andbackward stochastic integrals, presented two implicit Taylor methods: the implicit Euler-Taylor method with strong order p = 0:5, and the implicit Milstein-Taylor method withstrong order p = 1. The main purpose of this paper is to study the convergence andmean-square stability of a new class of modi ed fully implicit Milstein (MFIM) methodfor solving systems of It^o SDEs. This paper concludes that the MFIM method withtwo parameters θ, ηЄ [0; 1] converge strongly to the exact solution with order p = 1, alsoinvestigates mean-square stability properties of these two implicit Taylor and the MFIMmethods. We combine analytical and numerical techniques to get insights into the stabil-ity properties. Finally, numerical results are reported to illustrate the convergence andstability results.

Keywords:

Mean-square stability , Convergence , Modi ed fully implicit Milstein method , Stochastic differential equation