Asymptotic results for exponential functionals of Levy processes

Li, Zenghu; Xu, Wei

doi:10.48550/arxiv.1601.02363

Cited by 2 publications

(2 citation statements)

References 18 publications

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“…A recent manuscript authored by Li and Xu [27], that appeared on Arxiv at the same time this manuscript was prepared, also studied the asymptotic behaviour of exponential functionals for Lévy processes. They obtained interesting results which are similar as the one we present below using fluctuation theory for Lévy processes and the knowledge of Lévy processes conditioned to stay positive.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic behaviour of exponential functionals of Lévy processes with applications to random processes in random environment

Palau¹,

Pardo²,

Smadi³

2016

ALEA

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Let ξ = (ξt, t ≥ 0) be a real-valued Lévy process and define its associated exponential functional as followsMotivated by applications to stochastic processes in random environment, we study the asymptotic behaviour ofwhere F = (F (x), x ≥ 0) is a function with polynomial decay at infinity and which is non increasing for large x. In particular, under some exponential moment conditions on ξ, we find five different regimes that depend on the shape of the Laplace exponent of ξ. Our proof relies on a discretization of the exponential functional It(ξ) and is closely related to the behaviour of functionals of semi-direct products of random variables. We apply our results to three questions associated to stochastic processes in random environment. We first consider the asymptotic behaviour of extinction and explosion for self-similar continuous state branching processes in a Lévy random environment. Secondly, we focus on the asymptotic behaviour of the mean population size in a model with competition or logistic growth which is affected by a Lévy random environment and finally, we study the tail behaviour of the maximum of a diffusion in a Lévy random environment.Key words and phrases: Lévy processes, exponential functional, continuous state branching processes in random environment, explosion and extinction probabilities, logistic process, diffusions in random environment.MSC 2000 subject classifications: 60G17, 60G51, 60G80.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic behaviour of exponential functionals of Lévy processes with applications to random processes in random environment

Palau¹,

Pardo²,

Smadi³

2016

ALEA

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“…Palau and Pardo (2015a) studied the same problem for a CBRE-process with stable branching in a random environment given by a Brownian motion with drift. The decay rate of the survival probability for a CBRE-process with stable branching and a general Lévy environment was studied in Li and Xu (2016). The strong Feller property of classical CBI-processes was proved in Li and Ma (2015).…”

Section: Survival and Extinction Probabilitiesmentioning

confidence: 99%

Continuous-State Branching Processes in Lévy Random Environments

2017

J Theor Probab

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A general continuous-state branching processes in random environment (CBRE-process) is defined as the strong solution of a stochastic integral equation. The environment is determined by a Lévy process with no jump less than −1. We give characterizations of the quenched and annealed transition semigroups of the process in terms of a backward stochastic integral equation driven by another Lévy process determined by the environment. The process hits zero with strictly positive probability if and only if its branching mechanism satisfies Grey's condition. In that case, a characterization of the extinction probability is given using a random differential equation with singular terminal condition. The strong Feller property of the CBRE-process is established by a coupling method. We also prove a necessary and sufficient condition for the ergodicity of the subcricital CBRE-process with immigration. (2010): Primary 60J80, 60K37; Secondary 60H20, 60G51 Mathematics Subject Classification

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Asymptotic results for exponential functionals of Levy processes

Cited by 2 publications

References 18 publications

Asymptotic behaviour of exponential functionals of Lévy processes with applications to random processes in random environment

Asymptotic behaviour of exponential functionals of Lévy processes with applications to random processes in random environment

Continuous-State Branching Processes in Lévy Random Environments

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