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Exponential functionals of Lévy processes. (English) Zbl 0979.60038
Barndorff-Nielsen, Ole E. (ed.) et al., Lévy processes. Theory and applications. Boston: Birkhäuser. 39-55 (2001).
This paper investigates the exponential functional of a Lévy process $$(x_t, t\geq 0)$$, $A_t= \int^t_0 e^{x_s} ds,\qquad t\geq 0.\tag{$$*$$}$ The distribution of the terminal value $$A_\infty$$ in $$(*)$$ plays an important role in the fields of mathematical finance and mathematical physics. The main tools used by the authors to obtain the distributional properties of the terminal value $$A_\infty$$ are Lamperti’s transformation (or representation), and the generalized Ornstein-Uhlenbeck processes. Section 2 provides a brief summary on Lamperti’s transformation, showing how it establishes a one-to-one correspondence between 1-dimensional Lévy processes and semi-Markov processes. Lamperti’s representation of the exponential functionals provides a powerful method to determine the law of $$A_\infty$$. Section 3 is devoted to the study of another useful transformation; a 2-dimensional Lévy process is proved to define a homogeneous Markov process called the generalized Ornstein-Uhlenbeck process. Under some mild assumptions, the $$A_\infty$$ distribution for the 2-dimensional Lévy process, if it exists, is proved to be the unique invariant probability law of the generalized Ornstein-Uhlenbeck process associated. Section 4 tries to establish the joint distribution of terminal values $$A_\infty$$ of a 2-dimensional Lévy process. There are discussed within the paper several examples and applications: Dufresne’s perpetuity distribution, the Cauchy process, the deterministic cases of the considered processes, and the risk model.
For the entire collection see [Zbl 0961.00012].

##### MSC:
 60G51 Processes with independent increments; Lévy processes 60K15 Markov renewal processes, semi-Markov processes 91B28 Finance etc. (MSC2000) 91B30 Risk theory, insurance (MSC2010)