Cram\'{e}r asymptotics for finite time first passage probabilities of general L\'{e}vy processes

Abstract

We derive the exact asymptotics of $P(\sup_{u\leq t}X(u) > x)$ if $x$ and $t$ tend to infinity with $x/t$ constant, for a L\'{e}vy process $X$ that admits exponential moments. The proof is based on a renewal argument and a two-dimensional renewal theorem of H\"{o}glund (1990).