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A Wiener-Hopf type factorization for the exponential functional of Lévy processes

par Pardo Milan, Juan Carlos;Patie, Pierre ;Savov, Mladen
Référence Journal of the London Mathematical Society, 86, 3, page (930-956)
Publication Publié, 2012-12

Article révisé par les pairs

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Résumé :

For a Lévy process =( t) t≥0 drifting to -∞, we define the so-called exponential functional as follows: Under mild conditions on , we show that the following factorization of exponential functionals: holds, where × stands for the product of independent random variables, H - is the descending ladder height process of and Y is a spectrally positive Lévy process with a negative mean constructed from its ascending ladder height process. As a by-product, we generate an integral or power series representation for the law of I for a large class of Lévy processes with two-sided jumps and also derive some new distributional properties. The proof of our main result relies on a fine Markovian study of a class of generalized Ornstein-Uhlenbeck processes, which is itself of independent interest. We use and refine an alternative approach of studying the stationary measure of a Markov process which avoids some technicalities and difficulties that appear in the classical method of employing the generator of the dual Markov process. © 2012 London Mathematical Society.