On the absolute continuity of multidimensional OrnsteinUhlenbeck processes
Abstract
Let $X$ be a $n$dimensional OrnsteinUhlenbeck process, solution of the S.D.E. $$\d X_t = AX_t \d t + \d B_t$$ where $A$ is a real $n\times n$ matrix and $B$ a Lévy process without Gaussian part. We show that when $A$ is nonsingular, the law of $X_1$ is absolutely continuous in $\r^n$ if and only if the jumping measure of $B$ fulfils a certain geometric condition with respect to $A,$ which we call the exhaustion property. This optimal criterion is much weaker than for the background driving Lévy process $B$, which might be very singular and sometimes even have a onedimensional discrete jumping measure. It also solves a difficult problem for a certain class of multivariate NonGaussian infinitely divisible distributions.
 Publication:

arXiv eprints
 Pub Date:
 August 2009
 DOI:
 10.48550/arXiv.0908.3736
 arXiv:
 arXiv:0908.3736
 Bibcode:
 2009arXiv0908.3736S
 Keywords:

 Mathematics  Probability;
 60E07;
 60H10;
 60J75;
 93C05