# Stochastic differential equations with noise perturbations and Wong-Zakai approximation of fractional Brownian motion

## TL;DR

This paper investigates how small noise perturbations affect solutions of differential equations driven by H"older continuous functions, specifically fractional Brownian motion, and introduces a Wong-Zakai approximation with proven convergence and rate results.

## Contribution

It introduces a Wong-Zakai type stationary approximation for fractional Brownian motion applicable for all Hurst parameters and establishes its convergence and rate of convergence.

## Key findings

- Convergence of the Wong-Zakai approximation in a suitable space.
- Sharp results on the rate of convergence in the p-norm.
- Applicability of the approximation for all H in (0,1).

## Abstract

In this article we study effects that small perturbations in the noise have to the solution of differential equations driven by H\"older continuous functions of order $H>\frac12$. As an application, we consider stochastic differential equations driven by a fractional Brownian motion. We introduce a Wong--Zakai type stationary approximation to the fractional Brownian motions and prove that it converges in a suitable space. Moreover, we provide sharp results on the rate of convergence in the $p$-norm. Our stationary approximation is suitable for all values of $H\in (0,1)$.