Cramér asymptotics for finite time first passage probabilities of general Lévy processes

2008/04/19 by Zbigniew Palmowski, Palmowski, Zbigniew, Martijn Pistorius +1
Business, Management and Accounting · Decision Sciences · Mathematics · #60G50 #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.0804.3169

openalex publication_date 2008/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 Lévy process X that admits exponential moments. The proof is based on a renewal argument and a two-dimensional renewal theorem of Höglund (1990).

Related

• Exponential moments of first passage times and related quantities for\n L 'evy processes
• Cramer's estimate for the exponential functional of a Levy process
• Levy processes: Hitting time, overshoot and undershoot II - Asymptotic behaviour
• Functional limit theorems for Lévy processes satisfying Cramér's condition
• Asymptotics for first-passage times of Lévy processes and random walks