On the absolute continuity of multidimensional Ornstein-Uhlenbeck processes

TL;DR

This paper establishes a precise criterion for when the distribution of a multidimensional Ornstein-Uhlenbeck process at time one is absolutely continuous, linking it to the geometric properties of the driving Lévy process's jumps.

Contribution

It provides an optimal and weaker condition called the exhaustion property that characterizes absolute continuity for these processes, solving a complex problem in multivariate infinitely divisible distributions.

Findings

Absolute continuity holds if and only if the exhaustion property is satisfied.

The criterion is weaker than conditions on the Lévy process alone.

Addresses a challenging problem for certain non-Gaussian distributions.

Abstract

Let $X$ be a $n$-dimensional Ornstein-Uhlenbeck 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 L\'evy process without Gaussian part. We show that when $A$ is non-singular, 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 L\'evy process $B$, which might be very singular and sometimes even have a one-dimensional discrete jumping measure. It also solves a difficult problem for a certain class of multivariate Non-Gaussian infinitely divisible distributions.