Local time penalizations with various clocks for Lévy processes

2022/03/16 by Shosei Takeda, Takeda, Shosei, Kouji Yano +1 · 1 citation
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60F05 (Primary) 60G44 #60G51 (Secondary) #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · doi:10.48550/arxiv.2203.08428

openalex publication_date 2022/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Several long-time limit theorems of one-dimensional Lévy processes weighted and normalized by functions of the local time are studied. The long-time limits are taken via certain families of random times, called clocks: exponential clock, hitting time clock, two-point hitting time clock and inverse local time clock. The limit measure can be characterized via a certain martingale expressed by an invariant function for the process killed upon hitting zero. The limit processes may differ according to the choice of the clocks when the original Lévy process is recurrent and of finite variance.

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