The relation between mixed and rough SDEs and its application to numerical methods

TL;DR

This paper explores the connection between mixed stochastic differential equations and rough path equations driven by Brownian and fractional Brownian motions, providing a correction formula to transfer properties and improve numerical methods.

Contribution

It introduces a correction formula linking mixed and rough SDEs, enabling property transfer and enhancing numerical solution techniques.

Findings

Established a correction formula analogous to Itô-Stratonovich correction.

Demonstrated how properties of one equation type can be transferred to the other.

Applied the correction to improve numerical methods for mixed and rough SDEs.

Abstract

We study the relationship between mixed stochastic differential equations and the corresponding rough path equations driven by standard Brownian motion and fractional Brownian motion with Hurst parameter $H>1/2$. We establish a correction formula, which relates both types of equations, analogously to the It\=o-Stratonovich correction formula. This correction formula allows to transfer properties, which are established for one type of equation to the other, and we will illustrate this by considering numerical methods for mixed and rough SDEs