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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].

60J60 Diffusion processes
60J99 Markov processes