Densities for Ornstein–Uhlenbeck processes with jumps

We consider an Ornstein–Uhlenbeck process with values in ℝn driven by a Lévy process (Zt) taking values in ℝd with d possibly smaller than n. The Lévy noise can have a degenerate or even vanishing Gaussian component. Under a controllability rank condition and a mild assumption on the Lévy measure of...

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Bibliographic Details
Published in:The Bulletin of the London Mathematical Society 2009-02, Vol.41 (1), p.41-50
Main Authors: Priola, Enrico, Zabczyk, Jerzy
Format: Article
Language:English
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Summary:We consider an Ornstein–Uhlenbeck process with values in ℝn driven by a Lévy process (Zt) taking values in ℝd with d possibly smaller than n. The Lévy noise can have a degenerate or even vanishing Gaussian component. Under a controllability rank condition and a mild assumption on the Lévy measure of (Zt), we prove that the law of the Ornstein–Uhlenbeck process at any time t > 0 has a density on ℝn. Moreover, when the Lévy process is of α-stable type, α ∈ (0, 2), we show that such density is a C∞-function.
ISSN:0024-6093
1469-2120
DOI:10.1112/blms/bdn099