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Stochastic Boundary Crossing Probabilities for the Brownian Motion

Part of: Stochastic processes Applications

Published online by Cambridge University Press:  30 January 2018

Xiaonan Che  and
Angelos Dassios
Xiaonan Che*
Affiliation:
London School of Economics and Political Science
Angelos Dassios*
Affiliation:
London School of Economics and Political Science
*
∗ Postal address: Department of Statistics, London School of Economics and Political Science, Houghton Street, London, WC2A 2AE, UK.
∗ Postal address: Department of Statistics, London School of Economics and Political Science, Houghton Street, London, WC2A 2AE, UK.
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Abstract

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Using martingale methods, we derive a set of theorems of boundary crossing probabilities for a Brownian motion with different kinds of stochastic boundaries, in particular compound Poisson process boundaries. We present both the numerical results and simulation experiments. The paper is motivated by limits on exposure of UK banks set by CHAPS. The central and participating banks are interested in the probability that the limits are exceeded. The problem can be reduced to the calculation of the boundary crossing probability from a Brownian motion with stochastic boundaries. Boundary crossing problems are also very popular in many fields of statistics.

Keywords

Boundary crossing probabilityfirst passage time probabilityBrownian motion

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 62P05: Applications to actuarial sciences and financial mathematics 60G51: Processes with independent increments; L'evy processes
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 2 , June 2013 , pp. 419 - 429
Copyright
© Applied Probability Trust 

References

Abundo, M. (2002). Some conditional crossing results of Brownian motion over a piecewise-linear boundary. Statist. Prob. Lett. 58, 131–145.CrossRefGoogle Scholar
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
Bech, M., Preisig, C. and Soramäki, K. (2008). Global trends in large-value payments. Econom. Policy Rev. 14, 59–81.Google Scholar
Breiman, L. (1968). Probability. Addison Welsley, Reading, MA.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
Che, X. (2012). Markov typed models for large-valued interbank payment systems. Doctoral Thesis, London School of Economics.Google Scholar
Che, X., and Dassios, A. (2011). Asymetric linear and stochastic boundary crossing probabilities of Brownian paths. Working paper, London School of Economics.Google Scholar
Doob, J. L. (1949). Heuristic approach to Kolmogorov-Smirnov theorems. Ann. Math. Statist. 20, 393–403.CrossRefGoogle Scholar
Escribá, L. B. (1987). A stopped Brownian motion formula with two sloping line boundaries. Ann. Prob. 15, 1524–1526.Google Scholar
Frishling, V., Antic, A., Kuchera, A. and Rider, P. (1997). Pricing barrier options with time-dependent drift, volatility and barriers. Working paper, Commonwealth Bank of Australia.Google Scholar
Hall, W. J. (1997). The distribution of Brownian motion on linear stopping boundaries. Sequent. 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
Novikov, A., Frishling, V. and Kordzakhia, N. (1999). Approximations of boundary crossing probabilities for a Brownian motion. J. Appl. Prob. 36, 1019–1030.CrossRefGoogle Scholar
Novikov, A., Frishling, V. and Kordzakhia, N. (2003). Time-dependent barrier options and boundary crossing probabilities. Georgian Math. J. 2, 325–334.Google Scholar
Ricciardi, L. M. (1977). Diffusion Processes and Related Topics in Biology (Lecture Notes Biomath. 14). Springer, Berlin.CrossRefGoogle Scholar
Robbins, H. (1970). Statistical methods related to the law of the iterated logarithm. Ann. Math. Statist. 41, 1397–1409.Google Scholar
Robbins, H. and Siegmund, D. (1973). Statistical tests of power one and the integral representation of solutions of certain partial differential equations. Bull. Inst. Math. 1, 93–120.Google Scholar
Roberts, G. O. and Shortland, C. F. (1997). Pricing barriers options with time-dependent coefficients. Math. Finance 7, 83–93.CrossRefGoogle Scholar
Sen, P. K. (1981). Sequential Nonparametrics: Invariance Principles and Statistical Inference. John Wiley, New York.Google Scholar
Scheike, T. H. (1992). A boundary-crossing result for Brownian motion. J. Appl. Prob. 29, 448–453.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.Google Scholar
Shiryaev, A. N. (2007). On martingale methods in the boundary crossing problems of Brownian motion. Sovrem. Probl. Mat. 8, 3–78 (in Russian).Google Scholar
Wang, L. and Pötzelberger, K. (1997). Boundary crossing probability for Brownian motion and general boundaries. J. Appl. Prob. 34, 54–65.Google Scholar