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May 2006 On the absolute continuity of Lévy processes with drift
Ivan Nourdin, Thomas Simon
Ann. Probab. 34(3): 1035-1051 (May 2006). DOI: 10.1214/009117905000000620


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 $\mathcal{C}^{1}$ with bounded derivative. Using an elementary stratification method, we show that if the drift $a$ is monotonous at the initial point $x$, then $X_t$ 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.


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Ivan Nourdin. Thomas Simon. "On the absolute continuity of Lévy processes with drift." Ann. Probab. 34 (3) 1035 - 1051, May 2006.


Published: May 2006
First available in Project Euclid: 27 June 2006

zbMATH: 1099.60045
MathSciNet: MR2243878
Digital Object Identifier: 10.1214/009117905000000620

Primary: 60G51 , 60H10

Keywords: Absolute continuity , Jump processes , Lévy processes

Rights: Copyright © 2006 Institute of Mathematical Statistics


Vol.34 • No. 3 • May 2006
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