Recently, split-step techniques have been integrated with a Milstein scheme to improve the fundamental analysis of numerical solutions of stochastic differential equations (SDEs). Unfortunately, we note that stability conditions of these methods have restrictions on parameters and step-size to preserve mean-square stability and A-stability of SDEs. We construct new general modified spit-step theta Milstein (MSSTM) methods for using on multi-dimensional SDEs in order to overcome these restrictions. We investigate that the numerical methods are mean-square (MS) stable with no restrictions on parameters for all step-size h > 0 when θ ∈ [1/2, 1] and it is proved that the methods with θ ≥ 1/2 are stochastically A-stable. Furthermore, there is a gap in discussing the split-step Milstein type methods for SDEs with Jump in the literature. Here, we extend the new general methods for SDEs with jump called compensated MSSTM (CMSSTM) methods. The unconditional MS-stability results of CMSSTM methods are proved for SDEs with Poisson-driven jump. Finally, several examples are given to show the effectiveness of the proposed method in approximation of one and two dimensional SDEs compared to some existing methods.
Digital Object Identifier (DOI)
A. Eissa, Mahmoud; Tian, Fenglin; and Tian, Boping
"Modified Split-Step Theta Milstein Methods for M-Dimensional Stochastic Differential Equation With Respect To Poisson-Driven Jump,"
Applied Mathematics & Information Sciences: Vol. 14:
6, Article 21.
Available at: https://digitalcommons.aaru.edu.jo/amis/vol14/iss6/21