Estimation of first crossing time distribution for N-parameter Brownian motion processes relative to upper class boundaries
type: Item
instance of: scholarly article
title: Estimation of first crossing time distribution for N-parameter Brownian motion processes relative to upper class boundaries
zbMATH Open document ID: 0577.60077
DOI: 10.1016/0047-259X(84)90026-5
author: Pradip Kumar Sen
author: Michael Wichura
published in: Journal of Multivariate Analysis
publication date: 1984-01-01T00:00:00Z
review text: Let \(X_ t\), \(t=(t_ 1,...,t_ N)\), be an N-parameter Brownian sheet, \(\phi\) (u) a positive function with suitable growth conditions, \(g(u)=u^{1/2}\phi (u)\). Denote \(\delta (t)=t_ 1...t_ N\), \(I=[0,1]^ N\), \(T_ g=\inf \{\partial (t):\) \(t\in I\), \(X_ t\geq g(\delta (t))\}\), \(P_{\tau}=P(T_ g\leq \tau)\). The main result is the following: For each \(\phi\) (\(\cdot)\) from the upper class for \(X_ t\) at the origin \[ 0\leq a_ N\leq \underline{\lim}_{\tau \to 0}P_{\tau}/I_ g(\tau,N)\leq \overline{\lim}_{\tau \to 0}P_{\tau}/I_ g(\tau,N)\leq 2^ Ne^ N(1-N^{-1})^{N-1}/N!, \] where \[ I_ g(\tau,N)=(2\pi)^{-1/2}\int_{0}^{\tau}H_ g(u,N)du,\quad H_ g(u,N)=\exp \{-\phi^ 2(u)\}\phi^{2N-1}(u)| \log u^{- 1}|^{N-1}u^{\quad -1}. \] To prove this fact the authors used methods analogous to their earlier work for standard Brownian motion [Estimation of first crossing time distributions for Brownian motion processes relative to upper class boundaries. to appear]. Some numerical results of evaluating \(a_ N\) are given.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 3923816
zbMATH Keywords: upper and lower functions
zbMATH Keywords: first crossing time
zbMATH Keywords: Kolmogorov-type
zbMATH Keywords: integral test
zbMATH Keywords: Brownian sheet
zbMATH Keywords: numerical results
MaRDI profile type: Publication
cites work: Q5538132
cites work: Sample functions of the \(N\)-parameter Wiener process
cites work: On the Continuity of Brownian Motion with a Multidimensional Parameter
cites work: On Some Asymptotic Properties Concerning Brownian Motion
description: scientific article; zbMATH DE number 3923816