Regularity Properties of the Stochastic Flow of a Skew Fractional Brownian Motion

TL;DR

This paper investigates the smoothness of the stochastic flow generated by a fractional Brownian motion-driven SDE with local time, revealing higher order differentiability for small Hurst parameters using advanced Fourier and variational methods.

Contribution

It establishes higher order differentiability of the stochastic flow for small Hurst parameters, incorporating local time into the analysis, which is a novel extension.

Findings

Higher order differentiability for small Hurst parameters

Use of Fourier analysis and variational calculus techniques

Inclusion of local time in the stochastic flow analysis

Abstract

In this paper we prove, for small Hurst parameters, the higher order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process. The proof of this result relies on Fourier analysis based variational calculus techniques and on intrinsic properties of the fractional Brownian motion.