Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes

The generalised Ornstein-Uhlenbeck process constructed from a bivariate Lévy process ([xi]t,[eta]t)t[greater-or-equal, slanted]0 is defined aswhere V0 is an independent starting random variable. The stationarity of the process is closely related to the convergence or divergence of the Lévy integral . We make precise this relation in the general case, showing that the conditions are not in general equivalent, though they are for example if [xi] and [eta] are independent. Characterisations are expressed in terms of the Lévy 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.