Small-time expansions for the transition distributions of Lévy processes

Abstract

Let $X$ be a Lévy process with absolutely continuous Lévy measure $\nu$. Small time polynomial expansions of order $n$ in $t$ are obtained for the tails $P(X_t\ge y)$ of the process, assuming smoothness conditions on the Lévy density away from the origin. By imposing additional regularity conditions on the transition density $p_t$ of $X_t$, an explicit expression for the remainder of the approximation is also given. As a byproduct, polynomial expansions of order $n$ in $t$ are derived for the transition densities of the process. The conditions imposed on $p_t$ require that its derivatives remain uniformly bounded away from the origin, as $t\to 0$; 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. The expansions seem to correct asymptotics previously reported in the literature.

Relaxes assumptions and includes a few new results