Rate of convergence for Wong-Zakai-type approximations of It\^o stochastic differential equations

TL;DR

This paper analyzes the convergence rate of Wong-Zakai-type approximations for Itô stochastic differential equations driven by Brownian motion, showing that the convergence speed matches that of the noise smoothing process.

Contribution

It introduces a detailed analysis of convergence rates for Wong-Zakai-type approximations in both Stratonovich and Itô cases, highlighting the relation to noise regularization.

Findings

Convergence speed matches noise smoothing rate.

Approximate solutions satisfy modified equations with correction terms.

Results apply to both Stratonovich and Itô approximations.

Abstract

We consider a class of stochastic differential equations driven by a one dimensional Brownian motion and we investigate the rate of convergence for Wong-Zakai-type approximated solutions. We first consider the Stratonovich case, obtained through the point-wise multiplication between the diffusion coefficient and a smoothed version of the noise; then, we consider It\^o equations where the diffusion coefficient is Wick-multiplied by the regularized noise. We discover that in both cases the speed of convergence to the exact solution coincides with the speed of convergence of the smoothed noise towards the original Brownian motion. We also prove, in analogy with a well known property for exact solutions, that the solutions of approximated It\^o equations solve approximated Stratonovich equations with a certain correction term in the drift.