Stochastic differential equations driven by fractional Brownian motions

TL;DR

This paper investigates the existence and uniqueness of stochastic differential equations driven by fractional Brownian motions with any Hurst parameter, using advanced stochastic calculus techniques to extend classical results to the fractional setting.

Contribution

It extends the anticipating Girsanov transformation to fractional Brownian motions and proves well-posedness of associated stochastic differential equations.

Findings

Established existence and uniqueness for a class of fractional SDEs.

Extended Girsanov transformation to fractional Brownian motion.

Proved well-posedness using fractional calculus techniques.

Abstract

In this paper, we study the existence and uniqueness of a class of stochastic differential equations driven by fractional Brownian motions with arbitrary Hurst parameter $H\in (0,1)$. In particular, the stochastic integrals appearing in the equations are defined in the Skorokhod sense on fractional Wiener spaces, and the coefficients are allowed to be random and even anticipating. The main technique used in this work is an adaptation of the anticipating Girsanov transformation of Buckdahn [Mem. Amer. Math. Soc. 111 (1994)] for the Brownian motion case. By extending a fundamental theorem of Kusuoka [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982) 567--597] using fractional calculus, we are able to prove that the anticipating Girsanov transformation holds for the fractional Brownian motion case as well. We then use this result to prove the well-posedness of the SDE.