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Stationary distribution of the Milstein scheme for stochastic differential delay equations with first-order convergence

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  • Gao, Shuaibin
  • Li, Xiaotong
  • Liu, Zhuoqi

Abstract

This paper focuses on the stationary distribution of the Milstein scheme for stochastic differential delay equations. The numerical segment process is constructed, which is proved to be a time homogeneous Markov process. We show that this numerical segment process admits a unique numerical stationary distribution. Then we reveal that the distribution of numerical segment process converges exponentially to the underlying one in the Wasserstein metric. Moreover, the first-order convergence of numerical stationary distribution to exact stationary distribution is presented. Finally, abundant numerical experiments confirm the reliability of theoretical findings.

Suggested Citation

  • Gao, Shuaibin & Li, Xiaotong & Liu, Zhuoqi, 2023. "Stationary distribution of the Milstein scheme for stochastic differential delay equations with first-order convergence," Applied Mathematics and Computation, Elsevier, vol. 458(C).
    • Handle: RePEc:eee:apmaco:v:458:y:2023:i:c:s0096300323003934
    DOI: 10.1016/j.amc.2023.128224
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    References listed on IDEAS

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    3. Hu, Rong, 2020. "Almost sure exponential stability of the Milstein-type schemes for stochastic delay differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    4. Mattingly, J. C. & Stuart, A. M. & Higham, D. J., 2002. "Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 185-232, October.
    5. Yuan, Chenggui & Mao, Xuerong, 2003. "Asymptotic stability in distribution of stochastic differential equations with Markovian switching," Stochastic Processes and their Applications, Elsevier, vol. 103(2), pages 277-291, February.
    6. Weng, Lihui & Liu, Wei, 2019. "Invariant measures of the Milstein method for stochastic differential equations with commutative noise," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 169-176.
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