Distributional properties of solutions of d<i>V</i>_<i>t</i> = <i>V</i>_<i>t</i>-d<i>U</i>_<i>t</i> + d<i>L</i>_<i>t</i> 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 d V t = V t - d U t + d L 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.