Supremum penalizations for Lévy processes
Shosei Takeda
TL;DR
The paper studies long-time limit theorems for one-dimensional Lévy processes weighted by functions of their supremum, introducing two penalization schemes: an exponential clock and a constant clock. It extends the Brownian supremum penalization framework to general Lévy processes by constructing generalized Azéma–Yor martingales $M^{(f)}$ based on ladder-process machinery, and showing that penalized measures exist as limits under both clocks. Specifically, with an exponential clock and under standard regularity conditions, the limit as $q\downarrow 0$ yields $\mathbb{P}^{(f)}$ via $\lim_{q\downarrow 0} \frac{\mathbb{P}[F_t f(S_{\mathbf{e}_q})]}{\mathbb{P}[f(S_{\mathbf{e}_q})]} = \mathbb{P}^{(f)}[F_t]$ for bounded $F_t$. A parallel result holds for a constant clock under additional absolute-continuity and Spitzer-type regularity assumptions, establishing a unified penalization framework that preserves supremum-related structure under the penalized law and clarifies the excursion/ladder-process viewpoint.
Abstract
Several long-time limit theorems of one-dimensional Lévy processes weighted and normalized by functions of its supremum are studied. The long-time limits are taken via the families of exponential times and that of constant times, called exponential clock and constant clock, respectively.