Weak solutions for stochastic differential equations with additive fractional noise

TL;DR

This paper introduces a novel method using the implicit function theorem and fractional Brownian motion scaling to establish the existence of weak solutions for stochastic differential equations driven by fractional noise.

Contribution

It presents a new approach to prove weak solutions for SDEs with additive fractional noise, expanding the theoretical understanding of such equations.

Findings

Established existence of weak solutions for SDEs with fractional noise

Utilized implicit function theorem and scaling properties of fractional Brownian motion

Provided a framework applicable to equations in Hilbert spaces

Abstract

We give a new approach to prove the existence of a weak solution of \[dx_t = f(t,x_t)dt + g(t)dB^H_t\] where $B^H_t$ is a fractional Brownian motion with values in a separable Hilbert space for suitable functions $f$ and $g$. Our idea is to use the implicit function theorem and the scaling property of the fractional Brownian motion in order to obtain a weak solution for this equation.