Cramer asymptotics for finite time first passage probabilities of general Levy processes

摘要

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