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

作者：Pardo J C^*; Patie P; Savov M

来源：Journal of the London Mathematical Society-Second Series, 2012, 86: 930-956.

DOI：10.1112/jlms/jds028

For a Levy process xi=(xi(t))(t >= 0) drifting to -infinity, we define the so-called exponential functional as follows:
I xi = integral(infinity)(0) e(xi t) dt.
Under mild conditions on xi, we show that the following factorization of exponential functionals:
I-xi =(d) I-H - X I-Y
holds, where x stands for the product of independent random variables, H- is the descending ladder height process of xi and Y is a spectrally positive Levy 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-xi for a large class of Levy 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.

• 出版日期2012-12