Path-dependent rough differential equations

TL;DR

This paper introduces a method using C^1-approximate flows to establish well-posedness of path-dependent rough differential equations driven by rough paths and Holder paths, expanding the theoretical understanding of such equations.

Contribution

It demonstrates how the machinery of C^1-approximate flows can be effectively applied to prove well-posedness of path-dependent rough differential equations.

Findings

Proves well-posedness for a class of path-dependent rough differential equations.

Extends the theory to equations driven by Holder weak geometric p-rough paths.

Provides a framework for analyzing equations with path-dependent vector fields.

Abstract

We show in this work how the machinery of C^1-approximate flows introduced in our previous work "Flows driven by rough paths", provides a very efficient tool for proving well-posedness results for path-dependent rough differential equations on flows of the form d\phi = V h(dt) + F X(dt), for smooth enough path-dependent vector fields V,F = (V_1,...,V_\ell), any Holder weak geometric p-rough path X and any a-Holder path h, with a+1/p>1.