Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$

TL;DR

This paper establishes the existence and uniqueness of local pathwise solutions for stochastic evolution equations driven by fractional Brownian motions with Hurst parameters between 1/3 and 1/2, using a generalized Young integral approach.

Contribution

It introduces a new fractional integration by parts-based integral and a coupled system of path and area equations to analyze solutions driven by fractional Brownian motion.

Findings

Proves local existence and uniqueness of solutions

Develops a new integral framework for H"older continuous signals

Applies results to equations driven by fractional Brownian motion with Hurst in (1/3,1/2]

Abstract

In this article we are concerned with the study of the existence and uniqueness of pathwise mild solutions to evolutions equations driven by a H\"older continuous function with H\"older exponent in $(1/3,1/2)$. Our stochastic integral is a generalization of the well-known Young integral. To be more precise, the integral is defined by using a fractional integration by parts formula and it involves a tensor for which we need to formulate a new equation. From this it turns out that we have to solve a system consisting in a path and an area equations. In this paper we prove the existence of a unique local solution of the system of equations. The results can be applied to stochastic evolution equations with a non-linear diffusion coefficient driven by a fractional Brownian motion with Hurst parameter in $(1/3,1/2]$, which is particular includes white noise.