Modified Split-Step Theta Milstein Methods For M-Dimensional Stochastic Differential Equation With Respect To Poisson-Driven Jump

Recently, split-step techniques have been integrated with a Milstein scheme to improve the fundamental analysis ofnumerical solutions of stochastic differential equations (SDEs). Unfortunately, we note that stability conditions of these methods haverestrictions on parameters and step-size to preserve mean-square stability and A-stability of SDEs.We construct new general modifiedspit-step theta Milstein (MSSTM) methods for using on multi-dimensional SDEs in order to overcome these restrictions.We investigatethat the numerical methods are mean-square (MS) stable with no restrictions on parameters for all step-size h > 0 when q ∈ [1/2,1]and it is proved that the methods with q ≥ 1/2 are stochastically A-stable. Furthermore, there is a gap in discussing the split-stepMilstein type methods for SDEs with Jump in the literature. Here, we extend the new general methods for SDEs with jump calledcompensated MSSTM (CMSSTM) methods. The unconditional MS-stability results of CMSSTM methods are proved for SDEs withPoisson-driven jump.