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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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References

Billingsley, P. (1986) Probability and Measure. 2nd edn. Wiley, New York.Google Scholar
Brown, R. L., Durbin, J. and Evans, J. M. (1975) Techniques for testing the constancy of regression relationships over time. J. R. Statist. Soc. B 37, 149–192.Google Scholar
De Vore, R. A. and Lorentz, G. G. (1993) Constructive Approximation. Springer, New York.Google Scholar
Durbin, J. (1971) Boundary crossing probabilities for the Brownian motion and the Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. J. Appl. Prob. 8, 431–153.Google Scholar
Durbin, J. (1985) The first passage of a continuous Gaussian process to a general boundary. J. Appl. Prob. 22, 99–122.CrossRefGoogle Scholar
Krämer, W., Ploberger, W. and Alt, R. (1988) Testing for structural change in dynamic models. Econometrica 56, 1355–1369.CrossRefGoogle Scholar
Lerche, H. R. (1986) Boundary Crossing of Brownian Motion. (Lecture Notes in Statistics 40.) Springer, Berlin.CrossRefGoogle Scholar
Loader, C. R. and Deely, J. J. (1987) Computations of boundary crossing probabilities for the Wiener process. J. Statist. Comput. Simul. 27, 95–105.Google Scholar
Niederreiter, H. (1992) Random Number Generation and Quasi-Monte Carlo Methods. SIAM, PA.Google Scholar
Ripley, B. D. (1987) Stochastic Simulation. Wiley, New York.CrossRefGoogle Scholar
Sen, P. K. (1981) Sequential Nonparametrics: Invariance Principles and Statistical Inference. Wiley, New York.Google Scholar
Siegmund, D. (1985) Sequential Analysis: Tests and Confidence Intervals. Springer, New York.CrossRefGoogle Scholar
Siegmund, D. (1986) Boundary crossing probabilities and statistical applications. Ann. Statist. 14, 361–404.CrossRefGoogle Scholar