Transportation inequalities for stochastic differential equations driven by a fractional Brownian motion

TL;DR

This paper proves transportation inequalities for solutions of stochastic differential equations driven by fractional Brownian motion with Hurst parameter greater than 1/2, using the $L^2$ and uniform metrics, and explores their asymptotic behavior.

Contribution

It establishes Talagrand's $T_1$ and $T_2$ inequalities for these solutions, providing new tools for analyzing their probabilistic properties.

Findings

Proves $T_1$ and $T_2$ inequalities for fractional SDE solutions.

Analyzes small-time and large-time asymptotics using Hoeffding-type inequalities.

Applies transportation inequalities to study solution behavior over time.

Abstract

We establish Talagrand's $T_1$ and $T_2$ inequalities for the law of the solution of a stochastic differential equation driven by a fractional Brownian motion with Hurst parameter $H>1/2$. We use the $L^2$ metric and the uniform metric on the path space of continuous functions on $[0,T]$. These results are applied to study small-time and large-time asymptotics for the solutions of such equations by means of a Hoeffding-type inequality.