Local Malliavin calculus for Lévy processes and applications

The Malliavin derivative operator for the Poisson process introduced by Carlen and Pardoux [Differential calculus and integration by parts on a Poisson space, in Stochastics, Algebra and Analysis in Classical and Quantum Dynamics, S. Albeverio et al. (eds), Kluwer, Dordrecht, 1990, pp. 63-73] is extended to Lévy processes. It is a true derivative operator (in the sense that it satisfies the chain rule), and we deduce a sufficient condition for the absolute continuity of functionals of the Lévy process. As an application, we analyse the absolute continuity of the law of the solution of some stochastic differential equations with jumps. © 2013 Taylor & Francis.