Distributional properties of solutions of dVt = Vt-dUt + dLt with Lévy noise

Abstract

For a given bivariate Lévy process (U_t, L_t)_t≥0, distributional properties of the stationary solutions of the stochastic differential equation dV_t = V_t-dU_t + dL_t are analysed. In particular, the expectation and autocorrelation function are obtained in terms of the process (U, L) and in several cases of interest the tail behavior is described. In the case where U has jumps of size -1, necessary and sufficient conditions for the law of the solutions to be (absolutely) continuous are given.