The importance of strictly local martingales; applications to radial Ornstein-Uhlenbeck processes
type: Item
instance of: scholarly article
title: The importance of strictly local martingales; applications to radial Ornstein-Uhlenbeck processes
zbMATH Open document ID: 0960.60046
author: Xue-Mei Li
author: Marc Yor
author: K. D. Elworthy
publication date: 2001-05-13T00:00:00Z
review text: The authors study continuous local martingales. First they introduce a default function \(\gamma_M(t)\) of a local martingale \(M\) by \(\gamma_M (t)= EM_0-EM_t\) and note that a local martingale is a martingale if and only if \(\gamma_M(T)=0\) for every bounded stopping time \(T\). Assume now that the local martingale \(M\) is continuous and \(T\) is a finite stopping time such that the process \((M^T)^-\) belongs to class \(D\). Then they show that \[ E\bigl[ M_T\mathbf{1}_{\{\sup_{t\leq T}M_t< x\}}\bigr] +xP\left(\sup_{t\leq T}M_t\geq x\right)+ E(M_0-x)^+ =EM_0, \] and the following default formula holds \[ \lim_{x\to \infty} xP \left(\sup_{t\leq T}M_t\geq x\right)= \gamma_M(T). \] The authors give analogous results for continuous semi-martingales and apply these results to study the integrability of functionals of local martingales. They also describe the default functions of Bessel processes and radial Ornstein-Uhlenbeck processes in relation to their first hitting and last exit times.
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 1389147
zbMATH Keywords: local martingales
zbMATH Keywords: Bessel processes
zbMATH Keywords: Ornstein-Uhlenbeck processes
reviewed by: Esko Valkeila
MaRDI profile type: Publication
full work available at URL: s004400050240
OpenAlex ID: W2149286090
DOI: 10.1007/S004400050240
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1389147