文章基本信息

标题：Asymptotic behaviour of exponential functionals of Lévy processes with applications to random processes in random environment
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作者：Sandra Palau ; Juan Carlos Pardo ; Charline Smadi 等
期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2016
卷号：XIII
期号：2
页码：1235-1258
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：Let ξ = (ξt, t ≥ 0) be a real-valued L´evy process and define its associatedexponential functional as followsIt(ξ) :=Z t0exp{−ξs}ds, t ≥ 0.Motivated by applications to stochastic processes in random environment, we studythe asymptotic behaviour ofEhF􀀀It(ξ)ias t → ∞,where F = (F(x), x ≥ 0) is a function with polynomial decay at infinity andwhich is non increasing for large x. In particular, under some exponential momentconditions on ξ, we find five different regimes that depend on the shape of theLaplace exponent of ξ. Our proof relies on a discretization of the exponentialfunctional It(ξ) and is closely related to the behaviour of functionals of semi-directproducts of random variables.We apply our results to three questions associated to stochastic processes in randomenvironment. We first consider the asymptotic behaviour of extinction andexplosion for self-similar continuous state branching processes in a L´evy randomenvironment. Secondly, we focus on the asymptotic behaviour of the mean populationsize in a model with competition or logistic growth which is affected by a L´evyrandom environment and finally, we study the tail behaviour of the maximum of adiffusion in a L´evy random environment.
关键词：L´evy processes; exponential functional; continuous state branching;processes in random environment; explosion and extinction probabilities; logistic process; diffu-;sions in random environment.