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On the distribution and asymptotic results for exponential functionals of Lévy processes. (English) Zbl 0905.60056
Yor, Marc (ed.), Exponential functionals and principal values related to Brownian motion. A collection of research papers. Madrid: Univ. Autónoma de Madrid, Departamento de Matemáticas. Biblioteca de la Revista Matemática Iberoamericana. 73-126 (1997).
Summary: The aim of this paper is to study the distribution and the asymptotic behavior of the exponential functional $$A_t:= \int^t_0 e^{\xi_s} ds$$, where $$(\xi_s, s\geq 0)$$ denotes a Lévy process. When $$A_\infty <\infty$$, we show that in most cases, the law of $$A_\infty$$ is a solution of an integro-differential equation; moreover, this law is characterized by its integral moments. When the process $$\xi$$ is asymptotically $$\alpha$$-stable, we prove that $$t^{-1/ \alpha} \log A_t$$ converges in law, as $$t\to \infty$$, to the supremum of an $$\alpha$$-stable Lévy process; in particular, if $$\mathbb{E} [\xi_1]>0$$, then $$\alpha=1$$ and $$(1/t) \log A_t$$ converges almost surely to $$\mathbb{E} [\xi_1]$$. Eventually, we use Girsanov’s transform to give the explicit behavior of $$\mathbb{E} [(a+A_t (\xi))^{-1}]$$ as $$t\to \infty$$, where $$a$$ is a constant, and deduce from this the rate of decay of the tail of the distribution of the maximum of a diffusion process in a random Lévy environment.
For the entire collection see [Zbl 0889.00015].

##### MSC:
 60J60 Diffusion processes 60J99 Markov processes