Harnack Inequalities for SDEs Driven by Time-Changed Fractional Brownian Motions

TL;DR

This paper proves Harnack inequalities for SDEs driven by time-changed fractional Brownian motions with Hurst parameter less than 1/2, using coupling and regularization techniques, with results depending on drift conditions.

Contribution

It introduces new Harnack inequalities for a class of SDEs driven by time-changed fractional Brownian motions, extending existing results to Hurst parameters below 1/2.

Findings

Harnack inequalities established for H<1/2

Dimension-free inequalities under one-sided Lipschitz drift

Constants depend on space dimension without the Lipschitz condition

Abstract

We establish Harnack inequalities for stochastic differential equations (SDEs) driven by a time-changed fractional Brownian motion with Hurst parameter $H\in(0,1/2)$. The Harnack inequality is dimension-free if the SDE has a drift which satisfies a one-sided Lipschitz condition, otherwise we still get Harnack-type estimates, but the constants will, in general, depend on the space dimension. Our proof is based on a coupling argument and a regularization argument for the time-change.