Some remarkable pure martingales
type: Item
instance of: scholarly article
title: Some remarkable pure martingales
zbMATH Open document ID: 1017.60054
DOI: 10.1016/S0246-0203(02)00018-3
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2003-04-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77763
review text: A continuous martingale \((M_t)\) is said to be pure if the increasing process \((\langle M\rangle_t)\) is measurable w.r.t. its DDS-Brownian motion \(\gamma\) (defined by the relation \(M_t= \gamma_{\langle M,M\rangle_t}\), \(t\geq 0\)). For some Brownian motion \(B\) and \(n\in\mathbb{N}\), set \(M_t= \int^t_0 B^n_s dB_s\). \textit{D. W. Stroock} and \textit{M. Yor} [in: Séminaire de probabilités XV. Lect. Notes Math. 850, 590-603 (1981; Zbl 0456.60048)] have shown that, if \(n\) is odd, the martingale \((M_t)\) is pure. The question about the pureness of \((M_t)\) when \(n\) is even remained open till now and is solved finally in the present paper. This is done by using the relation between the martingale \((M_t)\) and a stochastic Bessel equation \[ dX_t= dB_t+ a{dt\over X_t}\tag{1} \] of dimension \(d= 2a+ 1\in ]1,2[\). Some other properties of the solutions of (1) are investigated.
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 1901042
zbMATH Keywords: pure martingale
zbMATH Keywords: stochastic Bessel equation
zbMATH Keywords: Brownian filtration
author: Samia Beghdadi-Sakrani
reviewed by: Catherine Rainer
MaRDI profile type: Publication
OpenAlex ID: W2059362614
description: scientific article; zbMATH DE number 1901042