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

Pardo, J. C., Patie, P. and Savov, M. (2012) A Wiener-Hopf type factorization for the exponential functional of Levy processes. Journal of the London Mathematical Society, 86 (3). pp. 930-956. ISSN 1469-7750

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To link to this item DOI: 10.1112/jlms/jds028

Abstract/Summary

For a Lévy process ξ=(ξt)t≥0 drifting to −∞, we define the so-called exponential functional as follows: Formula Under mild conditions on ξ, we show that the following factorization of exponential functionals: Formula holds, where × stands for the product of independent random variables, H− is the descending ladder height process of ξ 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 Iξ 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.

Item Type:Article
Refereed:Yes
Divisions:Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:32957
Publisher:Oxford Journals

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