Stochastic differential equation involving Wiener process and fractional Brownian motion with Hurst index $H> 1/2$

TL;DR

This paper studies a mixed stochastic differential equation driven by fractional Brownian motion with Hurst index greater than 1/2 and standard Brownian motion, proving the existence and uniqueness of solutions under mild conditions.

Contribution

It establishes the well-posedness of a new class of mixed stochastic differential equations involving dependent fractional Brownian motion and Brownian motion.

Findings

Unique solution exists under mild regularity assumptions

Handles dependence between fractional Brownian motion and Brownian motion

Extends theory to Hurst index H > 1/2

Abstract

We consider a mixed stochastic differential equation driven by possibly dependent fractional Brownian motion and Brownian motion. Under mild regularity assumptions on the coefficients, it is proved that the equation has a unique solution.