On independent times and positions for Brownian motions.
type: Item
instance of: scholarly article
title: On independent times and positions for Brownian motions.
zbMATH Open document ID: 1055.60078
DOI: 10.4171/RMI/328
author: Bernard Roynette
author: Pierre Vallois
author: Marc Yor
author: Bernard De Meyer
published in: Revista Matemática Iberoamericana
publication date: 2002-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/39694
review text: Let \((B_t\); \(t\geq 0)\) (resp. \(((X_t,Y_t)\); \(t\geq 0)\))) be a one- (resp. two-) dimensional Brownian motion started at 0. Let \(T\) be a stopping time such that \((B_{t\wedge T}\); \(t\geq 0)\) (resp. \((X_{t\wedge T}\); \(t\geq 0)\); \((Y_{t\wedge T}\); \(t\geq 0)\)) is uniformly integrable. The authors look for some additional assumptions on the joint law of \((B_T,T)\) under which one hopes that the law of \(T\) is determined from the law \(\mu\) of \(B_T\). The main result of the paper states that: (1) if \(T\) and \(B_T\) are independent and \(T\) has all exponential moments, then \(T\) is constant; (2) if \(X_T\) and \(Y_T\) are independent and have all exponential moments, then \(X_T\) and \(Y_T\) are Gaussian. The proposed approach is rather general and allows also to prove similar results, when the Brownian motion \((B_t\); \(t\geq 0)\) is replaced by Brownian motion with drift, Ornstein-Uhlenbeck or Bessel processes. A number of examples of stopping times \(T\) (obtained by iteration or from intertwinings) with only some exponential moments, such that \(T\) and \(B_T\) are independent (and similarly for \(X_T\) and \(Y_T\)), are presented. The authors also obtain a negative answer on Tortrat's conjecture and exhibit a class of non-constant stopping times such that \(X_T\) and \(Y_T\) are independent and Gaussian. In the Appendix the relation of the obtained results with Skorokhod embedding problem for space-time Brownian motion \(((t,B_t)\); \(t\geq 0)\) is discussed.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 60E10
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60J25
zbMATH DE Number: 1931491
zbMATH Keywords: space-time Brownian motion
zbMATH Keywords: Bessel process
zbMATH Keywords: Ornstein-Uhlenbeck process
zbMATH Keywords: exponential moment
zbMATH Keywords: stopping time
zbMATH Keywords: intertwinings
zbMATH Keywords: holomorphic function
MaRDI profile type: Publication
cites work: Q3975569
cites work: Q4747344
cites work: Representation of measures by balayage from a regular recurrent point
cites work: Q5833361
cites work: Q5563135
OpenAlex ID: W4299806671
description: scientific article; zbMATH DE number 1931491