Strong solutions of fractional Brownian sheet driven SDEs with integrable drift

TL;DR

This paper establishes the existence and uniqueness of a Malliavin differentiable strong solution to SDEs driven by fractional Brownian sheets with Hurst parameters less than 1/2, using advanced stochastic calculus techniques.

Contribution

It extends the theory of SDEs driven by fractional Brownian sheets to cases with integrable coefficients and Hurst parameters below 1/2, improving previous bounds.

Findings

Proves existence of unique strong solutions for fractional Brownian sheet-driven SDEs.

Utilizes Malliavin calculus and sectorial local nondeterminism techniques.

Improves bounds on Hurst parameters for such SDEs.

Abstract

We prove the existence of a unique Malliavin differentiable strong solution to a stochastic differential equation on the plane with merely integrable coefficients driven by the fractional Brownian sheet with Hurst parameters less than 1/2. The proof of this result relies on a compactness criterion for square integrable Wiener functionals from Malliavin calculus ([Da Prato, Malliavin and Nualart, 1992]), variational techniques developed in the case of fractional Brownian motion ([Ba\~nos, Nielssen, and Proske, 2020]) and the concept of sectorial local nondeterminism (introduced in [Khoshnevisan and Xiao, 2007]). The latter concept enable us to improve the bound of the Hurst parameter (compare with [Ba\~nos, Nielssen, and Proske, 2020]).