On Time Uniform Wong-Zakai Approximation Theorems

TL;DR

This paper investigates the long-term accuracy of Wong-Zakai approximations for stochastic differential equations, providing new time-uniform error bounds under spectral conditions, which are crucial for applications requiring reliable long-term simulations.

Contribution

The paper introduces the first time-uniform mean error bounds for Wong-Zakai approximations under spectral conditions, addressing exponential error growth issues.

Findings

Mean error estimates can grow exponentially with time.

Spectral conditions enable time-uniform convergence results.

First known results of this type for Wong-Zakai diffusion approximations.

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.