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Deterministic implicit two-step Milstein methods for stochastic differential equations

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  • Ren, Quanwei
  • Tian, Hongjiong
  • Tian, Tianhai

Abstract

In this paper, we propose a class of deterministic implicit two-step Milstein methods for solving Itô stochastic differential equations. Theoretical analysis is conducted for the convergence and stability properties of the proposed methods. We derive sufficient conditions such that these methods have the mean-square(M-S) convergence of order one, as well as sufficient and necessary conditions for linear M-S stability of the implicit two-step Milstein methods. Stability analysis shows that our proposed implicit two-step Milstein methods have much better stability property than those of the corresponding two-step explicit or semi-implicit Milstein methods. Numerical results using two test equations confirm our theoretical analysis results.

Suggested Citation

  • Ren, Quanwei & Tian, Hongjiong & Tian, Tianhai, 2021. "Deterministic implicit two-step Milstein methods for stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 179(C).
    • Handle: RePEc:eee:stapro:v:179:y:2021:i:c:s016771522100170x
    DOI: 10.1016/j.spl.2021.109208
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