Transportation-cost inequalities for non-linear Gaussian functionals

TL;DR

This paper extends transportation-cost inequalities to non-linear Gaussian functionals with non-Gaussian tails, applying to complex models like rough volatility and the Parabolic Anderson Model, advancing concentration analysis in stochastic processes.

Contribution

It introduces generalized TCIs for non-linear Gaussian functionals, including those with non-Gaussian tails, using a contraction principle and extending prior results for Gaussian-driven diffusions.

Findings

Proved generalized TCIs for rough differential equations.

Extended TCIs to non-Gaussian tail measures.

Applied results to rough volatility and Parabolic Anderson Model.

Abstract

We study concentration properties for laws of non-linear Gaussian functionals on metric spaces. Our focus lies on measures with non-Gaussian tail behaviour which are beyond the reach of Talagrand's classical Transportation-Cost Inequalities (TCIs). Motivated by solutions of Rough Differential Equations and relying on a suitable contraction principle, we prove generalised TCIs for functionals that arise in the theory of regularity structures and, in particular, in the cases of rough volatility and the two-dimensional Parabolic Anderson Model. In doing so, we also extend existing results on TCIs for diffusions driven by Gaussian processes.