Conditions for existence and smoothness of the distribution density for an Ornstein-Uhlenbeck process with Levy noise

TL;DR

This paper establishes conditions under which an Ornstein-Uhlenbeck process driven by Lévy noise has an absolutely continuous or smooth distribution density, including necessary and multidimensional non-degeneracy conditions.

Contribution

It provides sufficient and necessary conditions for the existence and smoothness of the distribution density of Lévy-driven Ornstein-Uhlenbeck processes, extending to multidimensional cases.

Findings

Conditions for absolute continuity of the process distribution.

Necessary conditions for smooth density when the drift is non-degenerate.

Introduction of a multidimensional non-degeneracy condition.

Abstract

Conditions are given, sufficient for the distribution of an Ornstein-Uhlenbeck process with L\'evy noise to be absolutely continuous or to possess a smooth density. For the processes with non-degenerate drift coefficient, these conditions are a necessary ones. A multidimensional analogue for the non-degeneracy condition on the drift coefficient is introduced.