Gradient estimates for SDEs driven by multiplicative Lévy noise

Gradient estimates are derived, for the first time, for the semigroup associated to a class of stochastic differential equations driven by multiplicative Lévy noise. In particular, the estimates are sharp for α-stable type noises. To derive these estimates, a new derivative formula of Bismut–Elworthy–Li's type is established for the semigroup by using the Malliavin calculus and a finite-jump approximation argument.