Upper bounds for the density of solutions of stochastic differential equations driven by fractional Brownian motions

TL;DR

This paper establishes upper bounds for the density of solutions to stochastic differential equations driven by fractional Brownian motion, demonstrating Gaussian and sub-Gaussian bounds depending on the Hurst parameter and geometric conditions.

Contribution

It provides new upper bounds for the density of solutions to fractional SDEs, including Gaussian and sub-Gaussian bounds, under specific geometric conditions.

Findings

Density satisfies log-Sobolev inequality for H > 1/2

Density admits upper Gaussian bound for H > 1/2

Density admits upper sub-Gaussian bound for H > 1/3

Abstract

In this paper we study upper bounds for the density of solution of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H > 1/3. We show that under some geometric conditions, in the regular case H > 1/2, the density of the solution satisfy the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case H > 1/3 and under the same geometric conditions, we show that the density of the solution is smooth and admits an upper sub-Gaussian bound.