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Gradient estimates are derived, for the first time, for the semigroupassociated to a class of stochastic differential equations driven bymultiplicative L\'evy noise. In particular, the estimates are sharp for$\alpha$-stable type noises. To derive these estimates, a new derivativeformula of Bismut-Elworthy-Li's type is established for the semigroup by usingthe Malliavin calculus and a finite-jump approximation argument.

Gradient estimates for SDEs Driven by Multiplicative L\'evy Noise