On the absolute continuity of Lévy processes with drift
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Ivan Nourdin [1] ; Thomas Simon [2]
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[1] Henri Poincaré Institute
Henri Poincaré Institute
París, Francia
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[2] University of Évry Val d'Essonne
University of Évry Val d'Essonne
Arrondissement d'Évry, Francia
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 34, Nº. 3, 2006, págs. 1035-1051
- Idioma: inglés
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Resumen
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.
