Stability analysis of split-step θ-Milstein method for a class of n-dimensional stochastic differential equations

Abstract

In this paper, we introduce a split-step theta Milstein (SSTM) method for n-dimensional stochastic delay differential equations (SDDEs). The exponential mean-square stability of the numerical solutions is analyzed, and in accordance with previous findings, we prove that the method is exponentially mean-square stable if the employed time-step is smaller than a given and easily computable upper bound. In particular, according to our investigation, larger time-steps can be used in the case θ∈(12,1] than in the case θ∈[0,12]. Numerical results are presented which reveal that the SSTM method is conditionally mean-square stable and that in the case θ∈(12,1] the interval of time-steps for which the SSTM method is theoretically shown to be mean-square stable is significantly larger than in the case θ∈[0,12]. It is worth mentioning that the SSTM method has never been employed or analyzed for the numerical approximation of SDDEs, at least to the very best of our knowledge.