Differential equations driven by Besov-Orlicz paths

TL;DR

This paper extends rough path theory to exponential Besov-Orlicz spaces, enabling the analysis of nonlinear differential equations driven by complex stochastic paths such as fractional Brownian motion.

Contribution

It introduces a new framework for rough paths in Besov-Orlicz spaces and proves existence and uniqueness of solutions for differential equations driven by these paths.

Findings

Extended rough path theory to exponential Besov-Orlicz spaces.

Proved existence and uniqueness of solutions for driven differential equations.

Applicable to paths like fractional Brownian motion and local martingales.

Abstract

In the article, the rough path theory is extended to cover paths from the exponential Besov-Orlicz space \[B^\alpha_{\Phi_\beta,q}\quad\mbox{ for }\quad \alpha\in (1/3,1/2],\,\quad \Phi_\beta(x) \sim \mathrm{e}^{x^\beta}-1\quad\mbox{with}\quad \beta\in (0,\infty), \quad\mbox{and}\quad q\in (0,\infty],\] and the extension is used to treat nonlinear differential equations driven by such paths. The exponential Besov-Orlicz-type spaces, rough paths, and controlled rough paths are defined and analyzed, a sewing lemma for such paths is given, and the existence and uniqueness of the solution to differential equations driven by these paths is proved. The results cover equations driven by paths of continuous local martingales with Lipschitz continuous quadratic variation (e.g.\ the Wiener process) or by paths of fractionally filtered Hermite processes in the $n$\textsuperscript{th} Wiener chaos with Hurst parameter $H\in (1/3,1/2]$ (e.g.\ the fractional Brownian motion).