On integral equations arising in the first-passage problem for Brownian motion
type: Item
instance of: scholarly article
title: On integral equations arising in the first-passage problem for Brownian motion
zbMATH Open document ID: 1044.60076
published in: Journal of Integral Equations and Applications
publication date: 2004-03-17T00:00:00Z
review text: Let \(B_t, t \geq 0\), be a standard Brownian motion started at zero and \(g(t)\) be a continuous non-negative function defined on \((0, \infty)\). The first passage problem is considered. Namely, let \(\tau = \inf \{t>0: B_t \geq g(t)\}\) and \(F\) be the distribution function of \(\tau\). Then the following system of integral equations holds: \[ t^{n/2}H_n\left(\frac{g(t)}{\sqrt{t}}\right)= \int_x^{\infty} (t-s)^{n/2}H_n\left(\frac{g(t)-g(s)}{\sqrt{t-s}}\right)F(ds), \] where \(H_n(x)= \int_x^{\infty} H_{n-1}(z)dz, n \geq 0,\) and \(H_{-1}(x)= (1/\sqrt{2 \pi})e^{{-x^2}/2}\).
reviewed by: Sabir Umarov
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 2059977
zbMATH Keywords: first passage problem
zbMATH Keywords: Brownian motion
zbMATH Keywords: Markov process
zbMATH Keywords: Chapman-Kolmogorov equation
zbMATH Keywords: system of integral equations
author: Goran Peskir
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/jiea/1181074930
OpenAlex ID: W2028801808
cites work: A new integral equation for the evaluation of first-passage-time probability densities
cites work: Heuristic Approach to the Kolmogorov-Smirnov Theorems
cites work: The first-passage density of a continuous gaussian process to a general boundary
cites work: The first-passage density of the Brownian motion process to a curved boundary
cites work: The tangent approximation to one-sided Brownian exit densities
cites work: Les fonctions aléatoires du type de Markoff associees à certaines équations linéaires aux dérivées partielles du type parabolique
cites work: Q5674478
cites work: Q5332526
cites work: Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung
cites work: Zur Theorie der stetigen zufälligen Prozesse
cites work: Q5775184
cites work: On Certain Confidence Contours for Distribution Functions
cites work: Probabilities of Wiener paths crossing differentiable curves
cites work: Evaluations of barrier-crossing probabilities of Wiener paths
cites work: Limit at zero of the Brownian first-passage density
cites work: On an integral equation for first-passage-time probability densities
cites work: On the First Passage Time Probability Problem
cites work: Q5600531
DOI: 10.1216/JIEA/1181074930
Recommended article: On an integral equation for first-passage-time probability densities
Recommended article: On the first-passage time of integrated Brownian motion
Recommended article: Moments of first-passage places and related results for the integrated Brownian motion
Recommended article: On a first passage problem for branching Brownian motions
Recommended article: AN INTEGRAL EQUATION FOR THE DISTRIBUTION OF THE FIRST EXIT TIME OF A REFLECTED BROWNIAN MOTION
Recommended article: On the first passage problem for correlated Brownian motion
Recommended article: Brownian motion and exposed solutions of differential inclusions
Recommended article: On Large Deviations of Stochastic Integrodifferential Equations with Brownian Motion
Recommended article: The First Passage Time Density of Brownian Motion and the Heat Equation with Dirichlet Boundary Condition in Time Dependent Domains
description: scientific article; zbMATH DE number 2059977