The Annals of Probability

On the absolute continuity of Lévy processes with drift

Ivan Nourdin and Thomas Simon

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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 $\mathcal{C}^{1}$ 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.

Article information

Ann. Probab. Volume 34, Number 3 (2006), 1035-1051.

First available in Project Euclid: 27 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes 60H10: Stochastic ordinary differential equations [See also 34F05]

Absolute continuity jump processes Lévy processes


Nourdin, Ivan; Simon, Thomas. On the absolute continuity of Lévy processes with drift. Ann. Probab. 34 (2006), no. 3, 1035--1051. doi:10.1214/009117905000000620.

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