On the absolute continuity of Levy processes with drift

• 成果类型: Article
• 署名作者: Nourdin, Ivan; Simon, Thomas
• 署名单位: Universite de Lorraine; Universite Paris Saclay
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000620
• 发表日期: 2006
• 页码: 1035-1051
• 关键词: sdes driven; densities; EXISTENCE; times

摘要

We consider the problem of absolute continuity for the one-dimensional SDE [GRAPHICS] where Z is a real Levy process without Brownian part and a a function of class 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 Zt 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.