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A Wiener-Hopf type factorization for the exponential functional of Lévy processes. (English) Zbl 1272.60027
The authors analyze the so called exponential functional associated to a one-dimensional Lévy process which starts in zero and drifts to $$-\infty$$. The exponential functional has various applications in astrophysics, biology and mathematical finance. Under some mild conditions on the driving process the authors prove that the exponential functional can be written as the product of two independent random variables, namely, the descending ladder height process and a spectrally positive Lévy process. Equality is meant here in the distributional sense. The proof involves a careful analysis of certain generalized Ornstein-Uhlenbeck processes, which is interesting in its own right.

##### MSC:
 60G51 Processes with independent increments; Lévy processes 47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators 60J25 Continuous-time Markov processes on general state spaces 60E07 Infinitely divisible distributions; stable distributions
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