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Strong approximation rate for Wiener process by fast oscillating integrated Ornstein–Uhlenbeck processes

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  • Li, Junlin
  • Fu, Hongbo
  • He, Ziying
  • Zhang, Yiwei

Abstract

In this paper, we use fast oscillating integrated Ornstein–Uhlenbeck (abbreviated as O-U) processes to pathwisely approximate Wiener processes. In physics, such approximation process is known as a colored noise approximation, and is suitable for dealing with stochastic flow problems. Our first result shows that if the drift term of a stochastic differential equation (abbreviated as SDE) satisfies usual Lipschitz constrains and a linear growth condition, then the solution of the SDE can be almost surely approximated with a polynomial rate. Next, we explore the O-U process approximation on the random manifold of stochastic evolution equations with linear multiplicative noise. Our second result shows that if the stochastic evolution equation further satisfies a uniformly hyperbolic condition, then the corresponding random manifold approximation also converges almost surely, with a polynomial rate.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:113:y:2018:i:c:p:314-325
    DOI: 10.1016/j.chaos.2018.05.019
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    References listed on IDEAS

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    1. 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.
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