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^-ξt (∫₀^t e^ξs- dηn_s + V₀), t ≥ 0, where V₀ 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^-ξt- dη_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.