Levy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes

The generalised Ornstein-Uhlenbeck process constructed from a bivariate Levy process (xi(t),eta(t))(t >= 0) is defined as V-t = e(-xi t)(integral(t)(0) e(xi s-)d eta(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 Levy integral integral(infinity)(0)e(-xi t-)d eta(t). We make precise this relation in the general case, showing that the conditions are not in general equivalent, though they are for example xi and eta are independent. Characterisations are expressed in terms of the Levy measure of (xi,eta). 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. (c) 2005 Elsevier B.V. All rights reserved.