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For the case of homogeneous processes ξ(t), ξ(0) = 0, t ≥ 0, with independent increments and negative jumps, it is proved in A. V. Skorokhod, Random Processes with Independent Increments, Nauka, Moscow, 1964 that the functional τ(x) = inf {t ≥ 0 : ξ(t) > x} , x ≥ 0, is a nondecreasing process with independent increments with respect to x, and its moment generating function is expressed via the cumulant that satisfies the corresponding Lundberg equation. The corresponding representations of this cumulant are specified and its Lévy characteristics (namely, γ and Lévy’s integral measure N(x)) are evaluated by using some of the results of the author’s work of 2007 for the processes under consideration.

The cumulant representation of the Lundberg root in the case of semicontinuous processes