Stochastic Taylor Methods for Stochastic Delay Differential Equations

Abstract

This paper demonstrates a systematic derivation of high order numerical methods from stochastic Taylor expansion for solving stochastic delay differential equations (SDDEs) with a constant time lag, r > 0 . The stochastic Taylor expansion of SDDEs is truncated at certain terms to achieve the order of convergence of numerical methods for SDDEs. Three different numerical schemes of Euler method, Milstein scheme and stochastic Taylor method of order 1.5 have been derived. The performance of Euler method, Milstein scheme and stochastic Taylor method of order 1.5 are investigated in a simulation study. Keywords: Numerical Solution; Stochastic Delay Differential Equations; Stochastic Taylor Expansion. 2010 Mathematics Subject Classification: 62L20; 65C20