Small-time expansions for the transition distributions of Lévy processes

Let X=(Xt)t≥0X=(Xt)t≥0 be a Lévy process with absolutely continuous Lévy measure νν. Small-time expansions of arbitrary polynomial order in tt are obtained for the tails P(Xt≥y)P(Xt≥y), y>0y>0, of the process, assuming smoothness conditions on the Lévy density away from the origin. By imposing additional regularity conditions on the transition density ptpt of XtXt, an explicit expression for the remainder of the approximation is also given. As a byproduct, polynomial expansions of order nn in tt are derived for the transition densities of the process. The conditions imposed on ptpt require that, away from the origin, its derivatives remain uniformly bounded as t→0t→0. Such conditions are then shown to be satisfied for symmetric stable Lévy processes as well as some tempered stable Lévy processes such as the CGMY one. The expansions seem to correct the asymptotics previously reported in the literature.