Abstract
We study the Wiener–Hopf factorization for Levy 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.
Highlights
This is the first in a series of papers, where we study a class of one-dimensional Lévy processes Xt with completely monotone jumps, introduced by L.C.G
The main objective of this article is to provide a detailed description of characteristic (Lévy–Khintchine) exponents f of these Lévy processes and their Wiener–Hopf factors κ+(τ, ξ), κ−(τ, ξ)
We extend the result of [70], which asserts that κ+(τ, ξ) and κ−(τ, ξ) are complete Bernstein functions of ξ
Summary
This is the first in a series of papers, where we study a class of one-dimensional Lévy processes Xt with completely monotone jumps, introduced by L.C.G. Rogers in [70]. Characteristic (Laplace) exponents of bi-variate ladder processes of Xt form an interesting class of bi-variate complete Bernstein functions, which deserves a separate study. Fluctuation theory stimulated the study of potential and spectral theory for symmetric Lévy processes (and in particular those with completely monotone jumps); see, e.g., [8, 9, 10, 11, 13, 15, 25, 26, 27, 34, 35, 36, 37, 38, 39, 40, 42, 54, 55, 56, 73] and the references therein.
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