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

TL;DR

This paper derives precise asymptotic probabilities for the maximum of a Lévy process exceeding a high level within a finite time, under conditions of exponential moments and large thresholds.

Contribution

It provides the first exact Cramér-type asymptotics for finite-time first passage probabilities of general Lévy processes.

Findings

Exact asymptotics for $P( ext{sup}_{u extless t}X(u) > x)$ as $x,t o \infty$ with $x/t$ constant.

Utilizes renewal theory and a two-dimensional renewal theorem for proofs.

Applicable to Lévy processes with exponential moments.

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).