Small-time expansions for the transition distributions of Le19 evy processes

Let X = (Xt)t≥0 be a Lévy process with absolutely continuous Lévy measure ν. Small time expansions, polynomial in t, are obtained for the tails P (Xt ≥ y) of the process. The conditions imposed on X require for Xt to have a C∞-transition density, whose derivatives remain uniformly bounded away from the origin, as t → 0. Such conditions are shown to be satisfied for symmetric stable Lévy processes as well as for other related Lévy processes of relevance in mathematical finance. Also, under very mild conditions on the Lévy density of the process and using a different methodology, a second order power expansion is obtained by identifying explicitly limt→0 1 t { 1 t P (Xt ≥ y)− ν([y,∞)) } . The resulting limit seems to correct a result previously reported in the literature and hints at the fact that our higher order expansions might be valid under milder conditions. AMS 2000 subject classifications: 60G51, 60F99.