Densities for Ornstein-Uhlenbeck processes with jumps

TL;DR

This paper studies the density properties of Ornstein-Uhlenbeck processes driven by Lévy noise, showing existence and smoothness of densities under certain controllability and measure conditions, especially for stable Lévy processes.

Contribution

It establishes conditions for the existence and smoothness of densities of Ornstein-Uhlenbeck processes with Lévy jumps, including degenerate cases and stable processes.

Findings

Density exists for all t > 0 under controllability and Lévy measure conditions.

Density is infinitely differentiable for α-stable Lévy processes with α in (0,2).

Results apply even when Gaussian component of Lévy noise is degenerate or absent.

Abstract

We consider an Ornstein-Uhlenbeck process with values in R^n driven by a L\'evy process (Z_t) taking values in R^d with d possibly smaller than n. The L\'evy noise can have a degenerate or even vanishing Gaussian component. Under a controllability condition and an assumption on the L\'evy measure of (Z_t), we prove that the law of the Ornstein-Uhlenbeck process at any time t>0 has a density on R^n. Moreover, when the L\'evy process is of $\alpha$-stable type, $\alpha \in (0,2)$, we show that such density is a $C^{\infty}$-function.