Study of a functional linked to Bessel bridge
type: Item
instance of: scholarly article
title: Study of a functional linked to Bessel bridge
zbMATH Open document ID: 0842.60076
author: Bernard Roynette
author: Gérard Lorang
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1996-03-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77528
review text: The authors investigate the behaviour of \(I_t (x,y):= E[\exp \int^t_0 l( X^{x,y}_{s,t} )ds]\), where (i) \((X^{x,y}_{s,t}\); \(0\leq s\leq t)\) denotes the \(n\)-dimensional Bessel bridge from \(x\) to \(y\) over time \(t\), \(n> 2\), (ii) \(l: \mathbb{R}_+\to \mathbb{R}_+\) is a bounded function verifying: \(\gamma_-< l(x)< \gamma_+\) (resp. \(c_1\leq (l(x)- {{k(p)} \over {x^2}}) x^{2+ q-p}\leq c_2)\) for any \(x\in [0,1 ]\) (resp. \([1, +\infty[)\), where \(\gamma_-\), \(\gamma_+\), \(c_1\) and \(c_2\) are explicit and fixed constants, \(k(p)= p(n-p- 2)/2\), \(0<p< (n-2)/ 2\), \(q>p\). The authors show that for any \(y_0>0\), there exists a positive constant \(C\) (depending only on \(l\), \(y_0\), \(n\), \(p)\) such that \(\forall t>0\), \(\forall x\geq 0\), \(0\leq y\leq y_0\), \[ C^{-1} \Biggl( {{1+x+t} \over {1+x}} \Biggr)^p\leq I_t (x,y)\leq C \Biggl( {{1+x+t} \over {1+x}} \Biggr)^p. \] This result allows the authors to give explicit lower and upper bounds of the density of hyperbolic diffusions on \(\mathbb{R}_+\), i.e. the generator is the differential operator \[ H_{\alpha, \beta} (f) (r)= (f'' (r)+ \gamma_{\alpha, \beta} (r) f' (r))/2, \qquad \gamma_{\alpha, \beta} (r)= \alpha \coth r+ 2\beta \coth 2r. \]
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 847541
zbMATH Keywords: \(n\)-dimensional Bessel bridge
zbMATH Keywords: density of hyperbolic diffusions
zbMATH Keywords: differential operator
reviewed by: Pierre Vallois
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 847541