Cramér asymptotics for finite time first passage probabilities of general Lévy processes

Abstract

We derive the exact asymptotics of $P({\sup }_{u\le t}X(u)>x)$ if $x$ and $t$ tend to infinity with $x/t$ constant, for a Lévy process $X$ that admits exponential moments. The proof is based on a renewal argument and a two-dimensional renewal theorem of Höglund (1990).