Open Access
2019 Fluctuation theory for Lévy processes with completely monotone jumps
Mateusz Kwaśnicki
Electron. J. Probab. 24: 1-40 (2019). DOI: 10.1214/19-EJP300


We study the Wiener–Hopf factorization for Lévy processes $X_{t}$ with completely monotone jumps. Extending previous results of L.C.G. Rogers, we prove that the space-time Wiener–Hopf factors are complete Bernstein functions of both the spatial and the temporal variable. As a corollary, we prove complete monotonicity of: (a) the tail of the distribution function of the supremum of $X_{t}$ up to an independent exponential time; (b) the Laplace transform of the supremum of $X_{t}$ up to a fixed time $T$, as a function of $T$. The proof involves a detailed analysis of the holomorphic extension of the characteristic exponent $f(\xi )$ of $X_{t}$, including a peculiar structure of the curve along which $f(\xi )$ takes real values.


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Mateusz Kwaśnicki. "Fluctuation theory for Lévy processes with completely monotone jumps." Electron. J. Probab. 24 1 - 40, 2019.


Received: 25 November 2018; Accepted: 24 March 2019; Published: 2019
First available in Project Euclid: 12 April 2019

zbMATH: 1412.60067
MathSciNet: MR3940770
Digital Object Identifier: 10.1214/19-EJP300

Primary: 30E20 , 47A68 , 60G51

Keywords: complete Bernstein function , fluctuation theory , Lévy process , Nevanlinna–Pick function , Wiener–Hopf factorisation

Vol.24 • 2019
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