Gaussian-type lower bounds for the density of solutions of SDEs driven by fractional Brownian motions

TL;DR

This paper derives Gaussian-type lower bounds for the density of solutions to SDEs driven by fractional Brownian motion, covering all Hurst parameters in 1D and H>1/2 in multidimensional cases, using pathwise and stochastic analysis methods.

Contribution

It provides new Gaussian lower bounds for SDE solutions driven by fractional Brownian motion, extending to all H in 1D and H>1/2 in higher dimensions.

Findings

Gaussian lower bounds established for 1D SDEs with all H

Lower bounds for multidimensional SDEs valid for H>1/2

Method combines pathwise and stochastic analysis techniques

Abstract

In this paper we obtain Gaussian-type lower bounds for the density of solutions to stochastic differential equations (SDEs) driven by a fractional Brownian motion with Hurst parameter $H$. In the one-dimensional case with additive noise, our study encompasses all parameters $H\in(0,1)$, while the multidimensional case is restricted to the case $H>1/2$. We rely on a mix of pathwise methods for stochastic differential equations and stochastic analysis tools.