Perturbed and non-perturbed Brownian taboo processes

Abstract

In this paper we study the Brownian taboo process, which is a version of Brownian motion conditioned to stay within a finite interval, and the α-perturbed Brownian taboo process, which is an analogous version of an α-perturbed Brownian motion.We are particularly interested in the asymptotic behaviour of the supremum of the taboo process, and our main results give integral tests for upper and lower functions of the supremum as t → ∞. In the Brownian case these include extensions of recent results in Lambert [4], but are proved in a quite different way.  2001 Éditions scientifiques et médicales Elsevier SAS AMS classification: 60K05; 60J15 RÉSUMÉ. -Dans cet article, nous étudions le processus Brownien tabou qui est une version du mouvement Brownien, conditionné à rester dans un intervalle fini, et le processus Brownien tabou α-perturbé qui est une version semblable du mouvement Brownien α-perturbé. Nous sommes particulièrement intéressés par le comportement asymptotique du supremum du processus tabou et nos principaux résultats fournissent des intégrales tests pour des fonctions majorantes et minorantes du supremum lorsque t → ∞. Dans le cas Brownien, ces résultats incluent des extensions de résultats récents de Lambert [4], mais ceux-ci sont prouvés de maníère différente.  2001 Éditions scientifiques et médicales Elsevier SAS E-mail addresses: rad@ma.man.ac.uk (R.A. Doney), jasi@sci.kuniv.adu.kw (Y.B. Nakhi).

A trivariate law for certain processes related to perturbed Brownian motions

“…[18][19][20]36,39]. We became all the more motivated by our initial question when looking again at two results we obtained previously (Theorems 1.1 and 1.2 below), while seeking for a better understanding of why certain functionals of the so-called µ-perturbed Brownian motion X µ are beta-distributed.…”

A trivariate law for certain processes related to perturbed Brownian motions

“…For more details on perturbed Brownian motions, perturbed reflected Brownian motions and perturbed Bessel processes, see e.g. [18][19][20]36,39]. We became all the more motivated by our initial question when looking again at two results we obtained previously (Theorems 1.1 and 1.2 below), while seeking for a better understanding of why certain functionals of the so-called µ-perturbed Brownian motion X µ are beta-distributed.…”

A lifetime of excursions through random walks and Lévy processes

“…His publication rate went up yet another gear, with almost as many articles published during this decade as in the previous two. This was all thanks to his increased exposure to collaborative partnership as well as the inevitable depth of understanding of random walks and Lévy processes he had acquired; [51,52,55,54,56,57,53,59,60,61,62,58,63,65,64,66,72,69,68,70,67,71,74,73,76,75,77,78,79,80,82,81]. It was also during this decade that Ron began an extremely fruitful collaboration with Ross Maller, publishing 10 papers together; [53,59,58,65,70,71,76,77,78,82,94].…”