On the absolute continuity of Lévy processes with drift

Abstract

We consider the problem of absolute continuity for the one-dimensional SDE X_t=x+∫₀^ta(X_s) ds+Z_t, where Z is a real Lévy process without Brownian part and a a function of class with bounded derivative. Using an elementary stratification method, we show that if the drift a is monotonous at the initial point x, then X_t is absolutely continuous for every t>0 if and only if Z jumps infinitely often. This means that the drift term has a regularizing effect, since Z_t itself may not have a density. We also prove that when Z_t is absolutely continuous, then the same holds for X_t, in full generality on a and at every fixed time t. These results are then extended to a larger class of elliptic jump processes, yielding an optimal criterion on the driving Poisson measure for their absolute continuity.