Stationary solutions of the stochastic differential equation dVt=Vt−dUt+dLt with Lévy noise

For a given bivariate Lévy process (Ut,Lt)t≥0(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 UU nor independence of V0V0 and (U,L)(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 UU and LL to remain semimartingales with respect to the corresponding expanded filtration is given.