Stationary solutions of the stochastic differential equation with Lévy noise

For a given bivariate Lévy process ( U t , L t ) t ≥ 0 , necessary and sufficient conditions for the existence of a strictly stationary solution of the stochastic differential equation d V t = V t − d U t + d L t are obtained. Neither strict positivity of the stochastic exponential of U nor independence of V 0 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.