Lévy integrals and the stationarity of generalised Ornstein–Uhlenbeck processes

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.