Boundary crossing of Brownian motion. Its relation to the law of the iterated logarithm and to sequential analysis
type: Item
instance of: scholarly article
title: Boundary crossing of Brownian motion. Its relation to the law of the iterated logarithm and to sequential analysis
zbMATH Open document ID: 0604.62075
author: Hans Rudolf Lerche
published in: Lecture Notes in Statistics
publication date: 1986-01-01T00:00:00Z
review text: This monograph has two parts. The first chapter is purely probabilistic; the second treats two statistical problems in sequential analysis. The first chapter studies the (stopping) time T at which a standard Brownian motion W(t) first crosses a boundary \(\psi\) (t), \(t>0\). Among the stopping bounds \(\psi\) (t) to be considered are the functions \(\psi_ a(t)\), \(a>0\), where \(\psi_ a(t)\) is the solution for x of \(h(x,t)=0\). Here \[ h(x,t)=t^{-} \phi (xt^{-})-a^{- 1}\int^{\infty}_{0}t^{-} \phi ((x-\theta)t^{-})F(d\theta) \] in which F is a sigma-finite measure on the positive half line and \(\phi\) is the standard normal density. (This is called the method of images. One can think of h(x,t) as the temperature at (x,t) after placing at \(t=0\) a unit amount of heat at \(x=0\) and a negative amount of heat with distribution \(a^{-1} F(d\theta)\) on the positive half line. Then \(\psi_ a(t)\) is the value of x at time t where the temperature is 0.) Another important function is \[ f(y,s)=\int^{\infty}_{0}\exp (\theta y-2^{-1} \theta^ 2 s)F(d\theta) \] and \(\psi_ a(t)\) is also the solution for x of \(f(xt^{-1},t^{-1})=a\). The stopping time is defined as \(T=\inf \{t>0:\) W(t)\(\geq \psi (t)\}\). It is shown that on the set \(x\leq \psi (t)\) we have \(P\{T>t\), \(W(t)\in (x,x+dx)\}=h(x,t)dx\) and \(P\{T\leq t| W(t)=x\}=a^{-1} f(xt^{-1},t^{-1})\). It is also shown how this is related to a known result of \textit{H. Robbins} and \textit{D. Siegmund} [Ann. Math. Stat. 41, 1410-1429 (1970; Zbl 0255.60058)] on first crossing by W(t) of a boundary \(\eta_ a(t)\) defined as the solution for x of \(f(x,t)=a.\) For \(\psi_ a\) defined by the method of images an explicit formula for the density \(p_ a(t)\) of T is derived. The remainder of the chapter is devoted to the tangent approximation to \(p_ a(t)\), which results if the actual bound \(\psi_ a(t)\) is replaced by the tangent to the graph of \(\psi_ a(t)\) at \((\psi_ a(t),t)\). It is shown that the ratio of approx…
reviewed by: Robert A. Wijsman
Mathematics Subject Classification ID: 62L15
Mathematics Subject Classification ID: 62-02
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 62L10
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 3976141
zbMATH Keywords: boundary crossing
zbMATH Keywords: law of the iterated logarithm
zbMATH Keywords: standard Brownian motion
zbMATH Keywords: stopping bounds
zbMATH Keywords: standard normal density
zbMATH Keywords: method of images
zbMATH Keywords: tangent approximation
zbMATH Keywords: uniform convergence
zbMATH Keywords: sequential sampling
zbMATH Keywords: normal population
zbMATH Keywords: known variance
zbMATH Keywords: unknown mean
zbMATH Keywords: Wiener process
zbMATH Keywords: normal prior
zbMATH Keywords: repeated significance test
zbMATH Keywords: upper and lower bounds
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3976141