A singular stochastic differential equation driven by fractional Brownian motion

TL;DR

This paper investigates a singular stochastic differential equation driven by fractional Brownian motion with Hurst parameter greater than 1/2, establishing existence, uniqueness, moment bounds, and absolute continuity of solutions.

Contribution

It demonstrates the existence and uniqueness of solutions under certain conditions and applies Malliavin calculus to prove their absolute continuity.

Findings

Unique solution with moments of all orders

Solution's law is absolutely continuous for all t>0

Conditions on the drift ensure well-posedness

Abstract

In this paper we study a singular stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter $H>\frac 12$. Under some assumptions on the drift, we show that there is a unique solution, which has moments of all orders. We also apply the techniques of Malliavin calculus to prove that the solution has an absolutely continuous law at any time $t>0$.