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

We derive the exact asymptotics of P(supu≤tX(u)>x) if x and t tend to infinity with x/t constant, for a general Lévy process X that admits exponential moments. The proof is based on a renewal argument and a two-dimensional renewal theorem of Höglund [Höglund, T., 1990. An asymptotic expression for the probability of ruin within finite time. Ann. Prob., 18, 378-389].