Derivative formulae for SDEs driven by multiplicative αα-stable-like processes

By using Bismut’s approach to the Malliavin calculus with jumps, we establish a derivative formula of Bismut–Elworthy–Li’s type for SDEs driven by multiplicative Lévy noises, whose Lévy measure satisfies some order conditions. In particular, αα-stable-like noises are allowed. Moreover, we also obtain the sharp gradient estimate in short time for the corresponding transition semigroup provided α∈(1,2)α∈(1,2).