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Explicit Approximation of Invariant Measure for Stochastic Delay Differential Equations with the Nonlinear Diffusion Term

Author

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  • Xiaoyue Li

    (Tiangong University)

  • Xuerong Mao

    (University of Strathclyde)

  • Guoting Song

    (Northeast Normal University)

Abstract

To our knowledge, existing measure approximation theory requires the diffusion term of the stochastic delay differential equations (SDDEs) to be globally Lipschitz continuous. Our work is to develop a new explicit numerical method for SDDEs with nonlinear diffusion term and establish the measure approximation theory. Precisely, we construct a function-valued explicit truncated Euler–Maruyama segment process and prove that it admits a unique ergodic numerical invariant measure. We also prove that the numerical invariant measure converges to the underlying invariant measure of the SDDE in the Fortet–Mourier distance. Finally, we give an example and numerical simulations to support our theory.

Suggested Citation

  • Xiaoyue Li & Xuerong Mao & Guoting Song, 2024. "Explicit Approximation of Invariant Measure for Stochastic Delay Differential Equations with the Nonlinear Diffusion Term," Journal of Theoretical Probability, Springer, vol. 37(2), pages 1850-1881, June.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:2:d:10.1007_s10959-023-01290-5
    DOI: 10.1007/s10959-023-01290-5
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    References listed on IDEAS

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    1. 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.
    2. Cui, Jianbo & Hong, Jialin & Sun, Liying, 2021. "Weak convergence and invariant measure of a full discretization for parabolic SPDEs with non-globally Lipschitz coefficients," Stochastic Processes and their Applications, Elsevier, vol. 134(C), pages 55-93.
    3. Mei, Hongwei & Yin, George, 2015. "Convergence and convergence rates for approximating ergodic means of functions of solutions to stochastic differential equations with Markov switching," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 3104-3125.
    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.
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