A Lévy process on the real line seen from its supremum and max-stable processes

2014/05/14 by Sebastian Engelke, Engelke, Sebastian, Jevgeņijs Ivanovs +1
Business, Management and Accounting · Decision Sciences · Mathematics · #Advanced Queuing Theory Analysis #FOS: Mathematics #Primary 60G51 #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics #secondary 60G70

paper · pdf · doi:10.48550/arxiv.1405.3443

openalex publication_date 2014/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a process Z on the real line composed from a Lévy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of Z seen from its supremum, the supremum Z and the time T at which the supremum occurs. In fact, it is closely related to the laws of the original and the tilted Lévy processes conditioned to stay negative and positive. The result is used to derive a new representation of stationary particle systems driven by Lévy processes. In particular, this implies that a max-stable process arising from Lévy processes admits a mixed moving maxima representation with spectral functions given by the conditioned Lévy processes.

Citations

• Wiener-Hopf Factorization for Lévy Processes Having Positive Jumps with Rational Transforms

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