The supremum distribution of a Lévy process with no negative jumps

Let be a process with stationary, independent increments and no negative jumps. Let be this same process modified by a reflecting barrier at zero (a storage process). Assuming that – and denote by ψ( s ) the exponent function of X. A simple formula is derived for the Laplace transform of as a function of W (0). Using the fact that the distribution of M is the unique stationary distribution of the Markov process W , this yields an elementary proof that the Laplace transform of M is µ s /ψ( s ). If it follows that These surprisingly simple formulas were originally obtained by Zolotarev using analytical methods.