Non-standard approximations of the Ito-map

TL;DR

This paper extends rough path analysis to non-standard approximations of the Ito-map, revealing new drift terms and applying to a broader class of signals beyond Brownian motion, with implications for optimality of rough path estimates.

Contribution

It generalizes the Wong-Zakai theorem to all levels of approximation, including perturbations involving Lie brackets, beyond traditional criteria.

Findings

Applies to McShane and Sussmann approximations.

Includes perturbations on all levels involving Lie brackets.

Demonstrates the approach's applicability beyond Brownian signals.

Abstract

The Wong-Zakai theorem asserts that ODEs driven by "reasonable" (e.g. piecewise linear) approximations of Brownian motion converge to the corresponding Stratonovich stochastic differential equation. With the aid of rough path analysis, we study "non-reasonable" approximations and go beyond a well-known criterion of [Ikeda--Watanabe, North Holland 1989] in the sense that our result applies to perturbations on all levels, exhibiting additional drift terms involving any iterated Lie brackets of the driving vector fields. In particular, this applies to the approximations by McShane ('72) and Sussmann ('91). Our approach is not restricted to Brownian driving signals. At last, these ideas can be used to prove optimality of certain rough path estimates.