Gradient formula for transition semigroup corresponding to stochastic equation driven by a system of independent L\'evy processes

TL;DR

This paper derives a gradient formula for the transition semigroup of a stochastic differential equation driven by independent Lévy processes, using Malliavin calculus, and provides sharp estimates for stable processes.

Contribution

It introduces a new gradient formula for the transition semigroup of SDEs driven by Lévy processes, expanding the analytical tools available for such stochastic systems.

Findings

Gradient formula for transition semigroup established

Sharp estimates provided for α-stable Lévy processes

Method applicable to systems with independent Lévy drivers

Abstract

Let $(P_t)$ be the transition semigroup of the Markov family $(X^x(t))$ defined by SDE $$ d X= b(X) dt + d Z, \qquad X(0)=x, $$ where $Z=\left(Z_1, \ldots, Z_d\right)^*$ is a system of independent real-valued L\'evy processes. Using the Malliavin calculus we establish the following gradient formula $$ \nabla P_tf(x)= \mathbb{E}\, f\left(X^x(t)\right) Y(t,x), \qquad f\in B_b(\mathbb{R}^d), $$ where the random field $Y$ does not depend on $f$. Sharp estimates on $\nabla P_tf(x)$ when $Z_1, \ldots , Z_d$ are $\alpha$-stable processes, $\alpha \in (0,2)$, are also given.