Strong existence and higher order Fr\'echet differentiability of stochastic flows of fractional Brownian motion driven SDE's with singular drift

TL;DR

This paper introduces a novel method for constructing strong solutions to SDEs with integrable drift driven by fractional Brownian motion with H<1/2, and proves higher order Fréchet differentiability of their stochastic flows for small H.

Contribution

It presents a new approach combining Malliavin calculus and local time variational calculus to establish strong solutions and higher order differentiability for SDEs with singular drift driven by fractional Brownian motion.

Findings

Existence of strong solutions for SDEs with integrable drift and fractional Brownian motion.

Higher order Fréchet differentiability of stochastic flows for small Hurst parameter.

Potential applicability to stochastic partial differential equations driven by rough paths.

Abstract

In this paper we present a new method for the construction of strong solutions of SDE's with merely integrable drift coefficients driven by a multidimensional fractional Brownian motion with Hurst parameter H < 1/2. Furthermore, we prove the rather surprising result of the higher order Frechet differentiability of stochastic flows of such SDE's in the case of a small Hurst parameter. In establishing these results we use techniques from Malliavin calculus combined with new ideas based on a "local time variational calculus". We expect that our general approach can be also applied to the study of certain types of stochastic partial differential equations as e.g. stochastic conservation laws driven by rough paths.