On the absolute continuity of Lévy processes with drift
| dc.creator | Nourdin, Ivan | |
| dc.creator | Simon, Thomas | |
| dc.date | 2006-06-30 | |
| dc.date.accessioned | 2026-07-07T07:17:53Z | |
| dc.date.available | 2026-07-07T07:17:53Z | |
| dc.description | We consider the problem of absolute continuity for the one-dimensional SDE \[X_t=x+\int_0^ta(X_s) ds+Z_t,\] 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. | |
| dc.description | Published at http://dx.doi.org/10.1214/009117905000000620 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org) | |
| dc.identifier | https://arxiv.org/abs/math/0606783 | |
| dc.identifier | http://arxiv.org/abs/math/0606783 | |
| dc.identifier | Annals of Probability 2006, Vol. 34, No. 3, 1035-1051 | |
| dc.identifier | doi:10.1214/009117905000000620 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/114104 | |
| dc.subject | Probability | |
| dc.subject | 60G51, 60H10 (Primary) | |
| dc.title | On the absolute continuity of Lévy processes with drift | |
| dc.type | text |