Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion
type: Item
instance of: scholarly article
title: Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion
zbMATH Open document ID: 0945.60082
publication date: 2000-09-24T00:00:00Z
review text: Let \(B\) be a Brownian motion starting from 0 and \(-\infty <\alpha ,\beta <1\); two stochastic functional equations are studied in this paper. They are \[ (1) \quad W_t=B_t+\alpha M_t^W+\frac 12L_t^W \quad \text{and}\quad (2)\;X_t=B_t+\alpha M_ t^X+\beta I_t^X, \] where \(M_t^Y\), \(I_t^Y\) and \(L_t^Y\) are the maximal process, minimal process of the process \(Y\) before time \(t\) and the local time of \(Y\) at 0, respectively. It is proven that for these two equations there exist pathwise unique solutions \(W\) and \(X\), respectively, and are adapted to the filtration of \(B\). Furthermore, \(W\) behaves like reflected Brownian motion (RBM) except when it attains a new maximum and it is then called \(\alpha\)-perturbed RBM. Similarly, \(X\) behaves like Brownian motion (BM) except when it attains a new maximum or minimum and it is then called \(\alpha,\beta\)-doubly perturbed BM. The time change of the positive (respectively, negative) part of an \(\alpha,\beta\)-doubly perturbed BM is an \(\alpha\)-perturbed RBM (respectively, \(\beta\)-perturbed RBM). And these two processes are independent.
reviewed by: Gong Guanglu
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J99
zbMATH DE Number: 1300444
zbMATH Keywords: \(\alpha \)-perturbed reflected Brownian motion
zbMATH Keywords: pathwise uniqueness
zbMATH Keywords: \(\alpha, \beta \)-doubly perturbed Brownian motion
author: Loïc Chaumont
author: Ronald A. Doney
MaRDI profile type: Publication
full work available at URL: s004400050216
OpenAlex ID: W1984474027
DOI: 10.1007/S004400050216
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1300444