Functional inequalities for forward and backward diffusions

TL;DR

This paper establishes Talagrand's $T_2$ inequality on the path space for various stochastic processes using pathwise arguments, expanding the scope of functional inequalities beyond traditional methods.

Contribution

It introduces a Girsanov-free, pathwise approach to derive functional inequalities for a broad class of stochastic processes.

Findings

Proves Talagrand's $T_2$ inequality for solutions of SDEs with measurable drifts

Extends functional inequalities to backward SDEs and optimal stopping value processes

Demonstrates processes are Lipschitz transformations of known inequality-satisfying processes

Abstract

In this article we derive Talagrand's $T_2$ inequality on the path space w.r.t. the maximum norm for various stochastic processes, including solutions of one-dimensional stochastic differential equations with measurable drifts, backward stochastic differential equations, and the value process of optimal stopping problems. The proofs do not make use of the Girsanov method, but of pathwise arguments. These are used to show that all our processes of interest are Lipschitz transformations of processes which are known to satisfy desired functional inequalities.