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

• Published in: Journal of the London Mathematical Society 2012-12, Vol.86 (3), p.930-956
• Main Authors: Pardo, J. C., Patie, P., Savov, M.
• Format: Article
• Language: English
• Subjects: Byproducts; Drift; Factorization; Ladders; Markov processes; Mathematical analysis; Power series; Spectra
• ISSN: 0024-6107; 1469-7750
• DOI: 10.1112/jlms/jds028

For a Lévy process ξ=(ξt)t⩾0 drifting to −∞, we define the so‐called exponential functional as follows: Iξ=∫0∞eξtdt. Under mild conditions on ξ, we show that the following factorization of exponential functionals: Iξ=dIH−×IY 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.