Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes
type: Item
instance of: scholarly article
title: Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes
zbMATH Open document ID: 1055.60017
author: Alain Rouault
author: Marc Yor
author: Marguerite Zani
author: Catherine Donati-Martin
publication date: 2004-10-05T00:00:00Z
review text: This paper is concerned with the Large Deviations (LD) behaviour of squared Bessel processes (as well as a few related processes) considered over a finite time-horizon, when the dimensional parameter of the Bessel process tends to infinity. \newline First of all, fixing a space \(C^{\alpha}[0,T]\) of \(\alpha\)-Hölder continuous functions (where \(0<\alpha<1/2\)), the exponential tightness of the corresponding probability distributions is established. Several proofs are then given for the LD upper and lower bounds: a first method of proof consists in using the properties of an auxiliary exponential martingale, a second method of proof relies on an additivity property for squared Bessel processes and uses Cramér's theorem; the advantage of this second approach lies in the fact that it should be applicable also for other families of Markov processes having a similar additivity property, e.g. continuous state branching processes. The authors also show that the Freidlin-Wentzell approach may be extended to the present context, even though the diffusion coefficient in the stochastic differential equation of squared Bessel processes fails to be Lipschitz continuous. These different approaches lead to several variational expressions for the rate function of the LD principle, and these variational expressions are compared in a separate section. LD results are also derived for other related processes (squared radial Ornstein-Uhlenbeck processes, squared Bessel bridges), using essentially the contraction principle.
reviewed by: Michel Sortais
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 2106563
zbMATH Keywords: Bessel processes
zbMATH Keywords: Ornstein-Uhlenbeck processes
zbMATH Keywords: additivity property
zbMATH Keywords: Freidlin-Wentzell method
MaRDI profile type: Publication
full work available at URL: s00440-004-0338-y
OpenAlex ID: W2002133168
DOI: 10.1007/S00440-004-0338-Y
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 2106563