On time uniform Wong-Zakai approximation theorems

Del Moral, Pierre; Hu, Shulan; Jasra, Ajay; Ruzayqat, Hamza; Wang, Xinyu

doi:10.1090/mcom/4112

On time uniform Wong-Zakai approximation theorems
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by Pierre Del Moral, Shulan Hu, Ajay Jasra, Hamza Ruzayqat and Xinyu Wang;

Math. Comp.

DOI: https://doi.org/10.1090/mcom/4112

Published electronically: June 16, 2025

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Abstract:

We consider the long time behavior of Wong-Zakai approximations of stochastic differential equations. These piecewise smooth diffusion approximations are of great importance in many areas, such as those with ordinary differential equations associated to random smooth fluctuations; e.g. robust filtering problems. In many examples, the mean error estimate bounds that have been derived in the literature can grow exponentially with respect to the time horizon. We show in a simple example that indeed mean error estimates do explode exponentially in the time parameter, i.e. in that case a Wong-Zakai approximation is only useful for extremely short time intervals. Under spectral conditions, we present some quantitative time-uniform convergence theorems, i.e. time-uniform mean error bounds, yielding what seems to be the first results of this type for Wong-Zakai diffusion approximations.

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Mathematics of Computation

On time uniform Wong-Zakai approximation theorems
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Abstract: