Regularization of differential equations by two fractional noises

TL;DR

This paper proves the existence and uniqueness of solutions for stochastic differential equations driven by the sum of two fractional Brownian motions with different Hurst parameters, highlighting the dominant regularization effect of the smaller Hurst index.

Contribution

It introduces a novel analysis of SDEs driven by two fractional noises, demonstrating the regularization effect of the smaller Hurst parameter using fractional calculus and Girsanov theorem.

Findings

Existence and uniqueness of solutions established

The fractional Brownian motion with smaller Hurst index dominates the regularization

Techniques involve fractional calculus and Girsanov theorem

Abstract

In this paper we show the existence and uniqueness of a solution for a stochastic differential equation driven by an additive noise which is the sum of two fractional Brownian motions with different Hurst parameters. The proofs are based on the techniques of fractional calculus and Girsanov theorem. In particular, we show that the regularization effect of the fractional Brownian motion with the smaller Hurst index dominates.