On the absolute continuity of Lévy processes with drift
Ivan Nourdin and Thomas Simon
Source: Ann. Probab. Volume 34, Number 3 (2006), 1035-1051.
Abstract
We consider the problem of absolute continuity for the one-dimensional SDE
Xt=x+∫0ta(Xs) ds+Zt,
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 Xt 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 Zt itself may not have a density. We also prove that when Zt is absolutely continuous, then the same holds for Xt, 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.
Primary Subjects: 60G51, 60H10
Keywords: Absolute continuity; jump processes; Lévy processes
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1151418492
Digital Object Identifier: doi:10.1214/009117905000000620
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