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Fluctuation theory for Lévy processes with completely monotone jumps
Mateusz Kwaśnicki · 2018-11-16 · via math.PR updates on arXiv.org

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 $f(ξ)$ of $X_t$, including a peculiar structure of the curve along which $f(ξ)$ takes real values.

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