Densities for Ornstein-Uhlenbeck processes with jumps

2008/12/10 by Enrico Priola, Jerzy Zabczyk · 7 citations
Economics, Econometrics and Finance · Mathematics · #Stochastic processes and financial applications #Spectral Theory in Mathematical Physics #Stochastic processes and statistical mechanics

paper · doi:10.1112/blms/bdn099

Abstract

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.

Citations

• Ornstein–Uhlenbeck–Cauchy process

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