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Boundary crossing probability for Brownian motion

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

Published online by Cambridge University Press:  14 July 2016

Klaus Pötzelberger  and
Liqun Wang
Klaus Pötzelberger*
Affiliation:
Vienna University of Economics and Business Administration
Liqun Wang*
Affiliation:
University of Manitoba
*
∗ Postal address: Institute of Statistics, Vienna University of Economics and Business Administration, Augasse 2–6, A-1090 Vienna, Austria.
∗∗ Postal address: Department of Statistics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2. Email address: [email protected]
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Abstract

Wang and Pötzelberger (1997) derived an explicit formula for the probability that a Brownian motion crosses a one-sided piecewise linear boundary and used this formula to approximate the boundary crossing probability for general nonlinear boundaries. The present paper gives a sharper asymptotic upper bound of the approximation error for the formula, and generalizes the results to two-sided boundaries. Numerical computations are easily carried out using the Monte Carlo simulation method. A rule is proposed for choosing optimal nodes for the approximating piecewise linear boundaries, so that the corresponding approximation errors of boundary crossing probabilities converge to zero at a rate of O(1/n2).

Keywords

first hitting timefirst passage timeoptimal stoppingcurved boundariesWiener processrandom walkbarrier optionscumulative sumssequential analysisMonte Carlo simulation

MSC classification

Primary: 60J65: Brownian motion
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 65C05: Monte Carlo methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 152 - 164
Copyright
Copyright © by the Applied Probability Trust 2001 

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References

Anderson, T. W. (1960). A modification of the sequential probability ratio test to reduce the sample size. Ann. Math. Statist. 31, 165–197.CrossRefGoogle 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
Daniels, H. E. (1996). Approximating the first crossing-time density for a curved boundary. Bernoulli 2, 133–143.CrossRefGoogle 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, 431–453.CrossRefGoogle Scholar
Durbin, J. (1992). The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Prob. 29, 291–304.CrossRefGoogle Scholar
Hall, W. J. (1997). The distribution of Brownian motion on linear stopping boundaries. Sequential Anal. 16, 345–352.CrossRefGoogle Scholar
Krämer, W., Ploberger, W., and Alt, R. (1988). Testing for structural change in dynamic models. Econometrica 56, 1355–1369.CrossRefGoogle Scholar
Lin, X. S. (1998). Double barrier hitting time distributions with applications to exotic options. Insurance: Math. Econ. 23, 45–58.Google Scholar
Martin-Löf, A. (1998). The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier. J. Appl. Prob. 35, 671–682.CrossRefGoogle Scholar
Novikov, A., Frishling, V., and Kordzakhia, N. (1999). Approximations of boundary crossing probabilities for a Brownian motion. J. Appl. Prob. 99, 1019–1030.CrossRefGoogle Scholar
Roberts, G. O., and Shortland, C. F. (1997). Pricing barrier options with time-dependent coefficients. Math. Finance 7, 83–93.CrossRefGoogle Scholar
Sacerdote, L., and Tomassetti, F. (1996). On evaluations and asymptotic approximations of first-passage-time probabilities. Adv. Appl. Prob. 28, 270–284.CrossRefGoogle Scholar
Sen, P. K. (1981). Sequential Nonparametrics: Invariance Principles and Statistical Inference. John 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
Wang, L. and Pötzelberger, K. (1997). Boundary crossing probability for Brownian motion and general boundaries. J. Appl. Prob. 34, 54–65.CrossRefGoogle Scholar