Stochastic Differential Equations Driven by Fractional Brownian Motion and Standard Brownian Motion

TL;DR

This paper establishes existence and uniqueness of solutions for multidimensional stochastic differential equations driven by both fractional Brownian motion with H>1/2 and standard Brownian motion, using fractional integration and Ito calculus.

Contribution

It provides the first combined analysis of SDEs driven simultaneously by fractional and standard Brownian motions with a rigorous proof of existence and uniqueness.

Findings

Proves existence and uniqueness of solutions for the combined SDEs.

Develops a priori estimates using fractional integration methods.

Utilizes Yamada-Watanabe theorem for the existence proof.

Abstract

We prove an existence and uniqueness theorem for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter H>1/2 and a multidimensional standard Brownian motion. The proof relies on some a priori estimates, which are obtained using the methods of fractional integration, and the classical Ito stochastic calculus. The existence result is based on the Yamada-Watanabe theorem.