Stationary solutions of the stochastic differential equation dV_t =V_t -dU_t +dL_t with Lévy noise

For a given bivariate Lvy process (Ut,Lt)t<0, necessary and sufficient conditions for the existence of a strictly stationary solution of the stochastic differential equation dVt=Vt-dUt+dLt are obtained. Neither strict positivity of the stochastic exponential of U nor independence of V0 and (U,L) is assumed and non-causal solutions may appear. The form of the stationary solution is determined and shown to be unique in distribution, provided it exists. For non-causal solutions, a sufficient condition for U and L to remain semimartingales with respect to the corresponding expanded filtration is given.