Functional limit theorems for L\'evy processes satisfying Cram\'er's condition

TL;DR

This paper establishes weak limit theorems for Lévy processes under Cramér's condition, analyzing their paths conditioned on positive maxima and revealing identities related to time-reversal and risk modeling.

Contribution

It provides the first weak limit theorems for Lévy processes conditioned on positive maxima under Cramér's condition, connecting path asymptotics with time-reversal and risk applications.

Findings

Two weak limit theorems for Lévy processes as initial position tends to -infinity.

Identities linking time-reversal, insurance risk, and self-similar processes.

Asymptotic behaviors of paths conditioned on maxima.

Abstract

We consider a L\'evy process that starts from $x<0$ and conditioned on having a positive maximum. When Cram\'er's condition holds, we provide two weak limit theorems as $x\to -\infty$ for the law of the (two-sided) path shifted at the first instant when it enters $(0,\infty)$, respectively shifted at the instant when its overall maximum is reached. The comparison of these two asymptotic results yields some interesting identities related to time-reversal, insurance risk, and self-similar Markov processes.