Fluctuation theory for Lévy processes with completely monotone jumps

2018/11/15 by Kwaśnicki, Mateusz
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1811.06617

Abstract

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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