Path collapse for multidimensional Brownian motion with rebirth
type: Item
instance of: scholarly article
title: Path collapse for multidimensional Brownian motion with rebirth
zbMATH Open document ID: 1064.60176
author: Ilie Grigorescu
author: Min Kang
published in: Statistics \& Probability Letters
publication date: 2005-03-08T00:00:00Z
review text: Consider a bounded domain \({\mathcal R}\) in \({\mathbb R}^d\), \(d>1\), which is either a convex set with piecewise smooth boundary or a polyhedron, containing two points \(x\) and \(y\). Let \((z_x (t))_{t\in {\mathbb R}_+}\) and \((z_y (t))_{t\in {\mathbb R}_+}\) be two processess constructed from the same Brownian path starting at \(x\), resp. \(y\), and reborn in \(x\), resp. \(y\), when they hit the boundary of \({\mathcal R}\). In the main result of this paper the authors identify a grid \(G({\mathcal R})\) such that \((z_x (t))_{t\in {\mathbb R}_+}\) and \((z_y (t))_{t\in {\mathbb R}_+}\) meet with positive probability strictly less than one if \(x-y\in G({\mathcal R})\), and probability zero if \(x-y\in {\mathcal R}\setminus G({\mathcal R})\). The grid \(G({\mathcal R})\) is contructed by reunion of countably many \((d-1)\)-dimensional hypersurfaces of positive surface area. This extends a previous one-dimensional result by the same authors [J. Theor. Probab. 16, No. 1, 147--159 (2003; Zbl 1015.60071)] in which they showed that \((z_x (t))_{t\in {\mathbb R}_+}\) and \((z_y (t))_{t\in {\mathbb R}_+}\) meet with probability one or zero, depending on whether \(x-y\) belongs or does not belong to a grid constructed by integer linear combinations of the endpoints of \({\mathcal R}\). The result can be used to study the long-term behaviour of barrier options.
reviewed by: Nicolas Privault
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J50
zbMATH DE Number: 2142314
zbMATH Keywords: Absorbing Brownian motion
zbMATH Keywords: harmonic measure
zbMATH Keywords: collapsing paths
zbMATH Keywords: barrier options.
MaRDI profile type: Publication
full work available at URL: j.spl.2004.10.006
OpenAlex ID: W2058435431
cites work: Ergodic properties of multidimensional Brownian motion with rebirth
cites work: Brownian motion on the figure eight
cites work: Path collapse for an inhomogeneous random walk
cites work: The Dirichlet problem in non-smooth domains
DOI: 10.1016/J.SPL.2004.10.006
Recommended article: Simulation of Brownian motion at first-passage times
Recommended article: Pathwise differentiability for SDEs in a convex polyhedron with oblique reflection
Recommended article: Multidimensional hitting time results for Brownian bridges with moving hyperplanar boundaries
Recommended article: Cutting Brownian paths
Recommended article: The distribution of a functional of the Wiener process and its application to the Brownian sheet
Recommended article: Brownian motion on the figure eight
Recommended article: Closed form valuation of barrier options with stochastic barriers
Recommended article: Noncoalescence for the Skorohod equation in a convex domain of \({\mathbb{R}}^ 2\)
description: scientific article; zbMATH DE number 2142314