Gradient formula for transition semigroup corresponding to stochastic equation driven by a system of independent Lévy processes

Abstract

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