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Boundary crossing probability for Brownian motion and general boundaries

Part of: Probabilistic methods, simulation and stochastic differential equations Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Liqun Wang  and
Klaus Pötzelberger
Liqun Wang*
Affiliation:
University of Basel
Klaus Pötzelberger*
Affiliation:
University of Economics and Business Administration Vienna
*
∗Postal address: Institute for Statistics and Econometrics, University of Basel, Holbeinstrasse 12, CH-4051 Basel, Switzerland.
∗∗Postal address: Institute of Statistics, University of Economics and Business Administration Vienna, Augasse 2–6, A-1090 Vienna, Austria.
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Abstract

An explicit formula for the probability that a Brownian motion crosses a piecewise linear boundary in a finite time interval is derived. This formula is used to obtain approximations to the crossing probabilities for general boundaries which are the uniform limits of piecewise linear functions. The rules for assessing the accuracies of the approximations are given. The calculations of the crossing probabilities are easily carried out through Monte Carlo methods. Some numerical examples are provided.

Keywords

WIENER PROCESSFIRST PASSAGE TIMEOPTIONAL STOPPING TIMESTRUCTURAL CHANGELOCAL POWERNUMERICAL COMPUTATIONMONTE CARLO SIMULATION

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 65C05: Monte Carlo methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 1 , March 1997 , pp. 54 - 65
Copyright
Copyright © Applied Probability Trust 1997 

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