Rough stochastic differential equations

TL;DR

This paper unifies Itô's stochastic differential equations with Lyons' rough differential equations, enabling advanced analysis of stochastic systems with common noise for applications like filtering and control.

Contribution

It introduces a generalized framework that combines stochastic and rough differential equations, broadening the theoretical foundation for stochastic analysis.

Findings

Unified theory of stochastic and rough differential equations

Applicable to pathwise stochastic filtering and control

Facilitates analysis of systems with common noise

Abstract

We establish a simultaneous generalization of It\^o's theory of stochastic and Lyons' theory of rough differential equations. The interest in such a unification comes from a variety of applications, including pathwise stochastic filtering, - control and the conditional analysis of stochastic systems with common noise.