Transportation inequalities for non-globally dissipative SDEs with jumps via Malliavin calculus and coupling

TL;DR

This paper develops new transportation and concentration inequalities for non-globally dissipative SDEs with jumps by employing advanced coupling techniques and Malliavin calculus, extending previous results to more general settings.

Contribution

It introduces novel coupling methods and bounds for Malliavin derivatives in jump-diffusion SDEs, broadening the scope of transportation inequalities beyond globally dissipative cases.

Findings

Extended transportation inequalities to non-globally dissipative SDEs with jumps.

Developed a new method for bounding Malliavin derivatives involving coupling techniques.

Unified jump and Brownian noise cases for SDEs with improved bounds.

Abstract

By using the mirror coupling for solutions of SDEs driven by pure jump L\'evy processes, we extend some transportation and concentration inequalities, which were previously known only in the case where the coefficients in the equation satisfy a global dissipativity condition. Furthermore, by using the mirror coupling for the jump part and the coupling by reflection for the Brownian part, we extend analogous results for jump diffusions. To this end, we improve some previous results concerning such couplings and show how to combine the jump and the Brownian case. As a crucial step in our proof, we develop a novel method of bounding Malliavin derivatives of solutions of SDEs with both jump and Gaussian noise, which involves the coupling technique and which might be of independent interest. The bounds we obtain are new even in the case of diffusions without jumps.