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

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  • Ahmadian, D.
  • Farkhondeh Rouz, O.
  • Ballestra, L.V.

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.

Suggested Citation

  • Ahmadian, D. & Farkhondeh Rouz, O. & Ballestra, L.V., 2019. "Stability analysis of split-step θ-Milstein method for a class of n-dimensional stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 413-424.
    • Handle: RePEc:eee:apmaco:v:348:y:2019:i:c:p:413-424
    DOI: 10.1016/j.amc.2018.10.040
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    References listed on IDEAS

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    1. Kahl Christian & Schurz Henri, 2006. "Balanced Milstein Methods for Ordinary SDEs," Monte Carlo Methods and Applications, De Gruyter, vol. 12(2), pages 143-170, April.
    2. Li, Xiuping & Cao, Wanrong, 2015. "On mean-square stability of two-step Maruyama methods for nonlinear neutral stochastic delay differential equations," Applied Mathematics and Computation, Elsevier, vol. 261(C), pages 373-381.
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