Gradient Estimate for Ornstein-Uhlenbeck Jump Processes

TL;DR

This paper develops explicit gradient estimates for Ornstein-Uhlenbeck jump processes and their extensions to linear SDEs driven by Lévy processes, using bounds on the Lévy measure and perturbation techniques.

Contribution

It introduces new explicit gradient estimates for Lévy-driven Ornstein-Uhlenbeck processes and extends these results to linear SDEs with Lévy noise.

Findings

Explicit gradient estimates for Lévy processes with linear drift

Derivative formulas for conditional distributions of jump processes

Extension of gradient bounds to perturbed Lévy-driven SDEs

Abstract

By using absolutely continuous lower bounds of the L\'evy measure, explicit gradient estimates are derived for the semigroup of the corresponding L\'evy process with a linear drift. A derivative formula is presented for the conditional distribution of the process at time $t$ under the condition that the process jumps before $t$. Finally, by using bounded perturbations of the L\'evy measure, the resulting gradient estimates are extended to linear SDEs driven by L\'evy-type processes.