Malliavin calculus on the Wiener-Poisson space and its application to canonical SDE with jumps
Consider a so-called canonical stochastic differential equation driven by a multidimensional Lévy process \(Z\). The aim is to find conditions under which the solution has a \(C^\infty\) density. Classical Malliavin's calculus is devoted to the case where \(Z\) is a Wiener process; some other works are devoted to the case where \(Z\) is a pure jump Lévy process. Here, the two cases are mixed. The authors assume a non-degeneracy condition on the coefficients of the equations, and on the matrix \(A+B\) where \(A\) is the covariance matrix of the Brownian part of \(Z\), and \(B\) is a matrix related to the Lévy measure of \(Z\) (more precisely to the small jumps of \(Z\)). Under these conditions, the existence of a smooth density is proved.
canonical stochastic differential equation
scientific article; zbMATH DE number 5083377