Lévy integrals and the stationarity of generalised Ornstein–Uhlenbeck processes

• Published in: Stochastic processes and their applications 2005-10, Vol.115 (10), p.1701-1722
• Main Authors: Lindner, Alexander, Maller, Ross
• Format: Article
• Language: English
• Subjects: Autocovariance function; Exact sciences and technology; Generalised Ornstein-Uhlenbeck process Lévy integral Stochastic integral Strict stationarity Autocovariance function Heavy-tailed behaviour; Generalised Ornstein–Uhlenbeck process; Heavy-tailed behaviour; Limit theorems; Lévy integral; Markov processes; Mathematics; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Stochastic analysis; Stochastic integral; Stochastic processes; Strict stationarity
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/j.spa.2005.05.004

The generalised Ornstein–Uhlenbeck process constructed from a bivariate Lévy process ( ξ t , η t ) t ⩾ 0 is defined as V t = e - ξ t ∫ 0 t e ξ s - d η s + V 0 , t ⩾ 0 , where V 0 is an independent starting random variable. The stationarity of the process is closely related to the convergence or divergence of the Lévy integral ∫ 0 ∞ e - ξ t - d η t . We make precise this relation in the general case, showing that the conditions are not in general equivalent, though they are for example if ξ and η are independent. Characterisations are expressed in terms of the Lévy measure of ( ξ , η ) . Conditions for the moments of the strictly stationary distribution to be finite are given, and the autocovariance function and the heavy-tailed behaviour of the stationary solution are also studied.