Applied Mathematics & Information Sciences

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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.

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