The Malliavin calculus for SDE with jumps and the partially hypoelliptic problem
Let \(x_t\in{\mathbf R}^m\) be the solution of a stochastic differential equation driven by a Brownian motion and an \(\alpha\)-stable rotationally invariant Lévy process. Let \(\pi\) be a smooth mapping from \({\mathbf R}^m\) into \({\mathbf R}^n\) for \(m\geq n\). It is proved that under some conditions on the coefficients of the equation, the law of \(\pi(x_T)\) has a smooth density; this is called a partial hypoellipticity property. Conditions on the coefficients are technical, but in the continuous case, they reduce to classical assumptions and the result was proved by Taniguchi. The general result relies on a Malliavin calculus on the space of càdlàg paths which was previously worked out by the author and Komatsu.
stochastic differential equations with jumps
scientific article; zbMATH DE number 1836034