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An approximation scheme for reflected stochastic differential equations

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  • Evans, Lawrence Christopher
  • Stroock, Daniel W.

Abstract

In this paper, we consider the Stratonovich reflected SDE in a bounded domain . Letting be the N-dyadic piecewise linear interpolation of Wt, we show that the distribution of the solution to the reflected ODE converges weakly to that of (Xt,Lt). Hence, we prove a distributional version for reflected diffusions of the famous result of Wong and Zakai. In particular, we apply our result to derive some geometric properties of coupled reflected Brownian motion, especially those properties which have been used in the recent work on the "hot spots" conjecture for special domains.

Suggested Citation

  • Evans, Lawrence Christopher & Stroock, Daniel W., 2011. "An approximation scheme for reflected stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1464-1491, July.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:7:p:1464-1491
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    References listed on IDEAS

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    1. CORNET, Bernard, 1983. "Existence of slow solutions for a class of differential inclusions," LIDAM Reprints CORE 539, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Słomiński, Leszek, 2015. "On reflected Stratonovich stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 759-779.
    2. Hu, Ying & Matoussi, Anis & Zhang, Tusheng, 2015. "Wong–Zakai approximations of backward doubly stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(12), pages 4375-4404.
    3. Tao Jiang & Zhongkai Guo & Xingjie Yan, 2022. "Random Perturbation of Invariant Manifolds for Non-Autonomous Dynamical Systems," Mathematics, MDPI, vol. 10(6), pages 1-12, March.
    4. Li, Junlin & Fu, Hongbo & He, Ziying & Zhang, Yiwei, 2018. "Strong approximation rate for Wiener process by fast oscillating integrated Ornstein–Uhlenbeck processes," Chaos, Solitons & Fractals, Elsevier, vol. 113(C), pages 314-325.
    5. Aida, Shigeki, 2015. "Reflected rough differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3570-3595.

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