Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters H ∈ (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.