On It\^o differential equation in rough path theory

TL;DR

This paper establishes the existence and uniqueness of solutions to Itô rough differential equations driven by continuous local martingales, linking them to classical stochastic differential equations and providing a pathwise reconstruction method.

Contribution

It introduces a rigorous framework for Itô rough differential equations driven by local martingales, connecting them with classical SDEs and offering a pathwise solution recovery technique.

Findings

Unique a.s. solution exists for Lip(eta) vector fields with eta > 1.

Itô solutions coincide with the signatures of classical SDE solutions.

Pathwise reconstruction of Itô solutions via concatenated Stratonovich solutions.

Abstract

The solution of rough differential equation, driven by the It\^o signature of a continuous local martingale, exists uniquely a.s. when the vector field is Lip(\beta) for \beta > 1, and coincides a.s. with the It\^o signature of the solution of parallel stochastic differential equation. Moreover, the It\^o solution can be recovered pathwisely by concatenating discounted Stratonovich solutions.