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

We derive the exact asymptotics of $P({\sup }_{u\le t}X(u)>x)$ if $x$ and $t$ tend to infinity with $x\mathrm{/}t$ 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).