On the absolute continuity of multidimensional Ornstein-Uhlenbeck processes

2009/08/26 by Thomas Simon, Simon, Thomas
Business, Management and Accounting · Economics, Econometrics and Finance · Mathematics · #60E07 #60H10 #60J75 #93C05 #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and financial applications #math.PR #msc:60E07 #msc:60H10 #msc:60J75 #msc:93C05

paper · pdf · doi:10.48550/arxiv.0908.3736

arxiv created 2009/08/26 · openalex publication_date 2009/08/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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× n matrix and B a Lévy process without Gaussian part. We show that when A is non-singular, the law of X₁ is absolutely continuous in \rⁿ 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 Lévy 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.

Citations

• Stochastic calculus of variations for general Lévy processes and its applications to jump-type SDE's with non-degenerated drift

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