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Approximations of boundary crossing probabilities for a Brownian motion

Part of: Stochastic processes Markov processes Numerical approximation and computational geometry

Published online by Cambridge University Press:  14 July 2016

Alex Novikov ,
Volf Frishling  and
Nino Kordzakhia
Alex Novikov*
Affiliation:
The University of Newcastle, Australia, and Steklov Mathematical Institute, Russia
Volf Frishling*
Affiliation:
Commonwealth Bank of Australia
Nino Kordzakhia*
Affiliation:
The University of Newcastle, Australia
*
∗Postal address: Department of Statistics, The University of Newcastle, University Drive, Callaghan, NSW 2308, Australia.
∗∗∗Postal address: 175 Pitt str., Treasury Dealing Department, Commonwealth Bank of Australia, Sydney, NSW 2000, Australia.
∗Postal address: Department of Statistics, The University of Newcastle, University Drive, Callaghan, NSW 2308, Australia.
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Abstract

Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.

Keywords

First passage timesWerner processGirsanov transformationnumerical integration

MSC classification

Primary: 60J65: Brownian motion
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 65D30: Numerical integration
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1019 - 1030
Copyright
Copyright © Applied Probability Trust 1999 

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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.Google Scholar
Borovkov, K. A. (1982). Rate of convergence in a boundary problem. Theory. Prob. Appl. 27, 148–149.CrossRefGoogle Scholar
Breiman, L. (1966). First exit time from a square root boundary. Proc. 5th Berkeley Symp. Math. Statist. Prob. 2, 9–16.Google Scholar
Daniels, H. (1996). Approximating the first crossing-time density for a curved boundary. Bernoulli 2, 133–143.CrossRefGoogle Scholar
DeLong, D. (1981). Crossing probabilities for a square root boundary by a Bessel process. Commun. Statist.-Theory Meth. A 10, 2197–2213.Google Scholar
Durbin, J. (1992). The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Prob. 29, 291–304.Google Scholar
Ferebee, B. (1983). An asymptotic expansion for one-sided Brownian exit densities. Z. Wahreinscheinlichkeitsch. 61, 309–326.Google Scholar
Frishling, V., Antic, A., Kuchera, A., and Rider, P. (1997). Pricing barrier options with time-dependent drift, volatility and barriers. Working paper, April 1997, Commonwealth Bank of Australia.Google Scholar
Gradshteyn, I., and Ryzhik, I. (1980). Table of Integrals, Series, and Products. Academic Press, New York.Google Scholar
Hall, W. J. (1997). The distribution of Brownian motion on linear stopping boundaries. Sequential Analysis 4, 345–352.Google Scholar
Khmaladze, E., and Shinjikashvili, E. (1998). Calculations of non-crossing probabilities for Poisson processes and its corollaries. Preprint, The University of NSW, Australia.Google Scholar
Lerche, H. R. (1986). Boundary Crossing of Brownian Motion (Lecture Notes in Statist. 40). Springer, Berlin.Google Scholar
Liptser, R. S., and Shiryaev, A. N. (1977). Statistics of Random Processes, Vol. 1. Springer, New York.CrossRefGoogle Scholar
Nagaev, S. V. (1970). The rate of convergence in a certain boundary value problem, part I. Theory Prob. Appl. 15, 179–199; part II, 15, 419–444.CrossRefGoogle Scholar
Novikov, A. A. (1971). On stopping times for a Wiener process. Theory Prob. Appl. 16, 458–465.Google Scholar
Novikov, A. A. (1979). On estimates and the asymptotic behavior of nonexit probabilities of a Wiener process to a moving boundary. Mat. Sb. 110, 539–550. (English translation in Math. USSR Sb. 38.)Google Scholar
Novikov, A. A. (1980). Estimates and asymptotic behavior of the probability of not crossing moving boundaries by sums of independent random variables. Izv. Akad. Nauk SSSR Ser. Mat. 44, 868–885.Google Scholar
Novikov, A. A., Kordzakhia, N., and Wright, I. (1998). Time-dependent barrier options and boundary crossing probabilities. Working paper, Department of Statistics, the University of Newcastle, Australia.Google Scholar
Ricciardi, L. M. (1977). Diffusion Processes and Related Topics in Biology (Lecture Notes in Biomath. 14). Springer, Berlin.Google Scholar
Roberts, G. O., and Shortland, C. F. (1997). Pricing barriers options with time-dependent coefficients. Math. Finance 7, 83–93.CrossRefGoogle Scholar
Sacerdote, L., and Tomasseti, F. (1996). On evaluations and asymptotics approximations of the first-passage-time probabilities. Adv. Appl. Prob. 28, 270–284.Google Scholar
Sahanenko, A. I. (1974). On the rate of convergence in a boundary value problem. Theory. Prob. Appl. 19, 416–421.Google Scholar
Salminen, P. (1988). On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary. J. Appl. Prob. 20, 411–426.Google Scholar
Sato, S. (1977). Evaluation of the first passage probability to a square root boundary for the Wiener process. J. Appl. Prob. 14, 53–70.CrossRefGoogle Scholar
Sen, P. K. (1981). Sequential Nonparametrics: Invariance Principles and Statistical Inference. Wiley, New York.Google Scholar
Shepp, L. A. (1967). A first passage problem for the Wiener process. Ann. Math. Statist. 38, 1912–1914.CrossRefGoogle Scholar
Siegmund, D. (1985). Sequential Analysis: Tests and Confidence Intervals. Springer, New York.CrossRefGoogle Scholar
Wang, L. and Pötzelberger, K. (1997). Boundary crossing probability for Brownian motion and general boundaries. J. Appl. Prob. 34, 54–65.Google Scholar
Wolfram, St. (1996). The Mathematica Book. Cambridge University Press.Google Scholar