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Linear and Nonlinear Boundary Crossing Probabilities for Brownian Motion and Related Processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

James C. Fu  and
Tung-Lung Wu
James C. Fu*
Affiliation:
University of Manitoba
Tung-Lung Wu*
Affiliation:
University of Manitoba
*
∗Postal address: Department of Statistics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada.
∗Postal address: Department of Statistics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada.
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Abstract

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We propose a new method to obtain the boundary crossing probabilities or the first passage time distribution for linear and nonlinear boundaries for Brownian motion. The method also covers certain classes of stochastic processes associated with Brownian motion. The basic idea of the method is based on being able to construct a finite Markov chain, and the boundary crossing probability of Brownian motion is cast as the limiting probability of the finite Markov chain entering a set of absorbing states induced by the boundaries. Error bounds are obtained. Numerical results for various types of boundary studied in the literature are provided in order to illustrate our method.

Keywords

Boundary crossing probabilityfirst passage timeBrownian motiondiffusion processfinite Markov chain imbeddingdiscretizationrandom walk

MSC classification

Secondary: 60J65: Brownian motion 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60J60: Diffusion processes 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1058 - 1071
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] 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
[2] Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
[3] Bluman, G. W. (1980). On the transformation of diffusion processes into the Wiener process. SIAM J. Appl. Math. 39, 238–247.CrossRefGoogle Scholar
[4] Borovkov, K. and Novikov, A. (2005). Explicit bounds for approximation rates of boundary crossing probabilities for the Wiener process. J. Appl. Prob. 42, 82–92.CrossRefGoogle Scholar
[5] Cherkasov, I. D. (1957). On the transformation of the diffusion process to a Wiener process. Theory Prob. Appl. 2, 373–377.CrossRefGoogle Scholar
[6] Daniels, H. E. (1969). The minimum of stationary Markov process superimposed on a U-shaped trend. J. Appl. Prob. 6, 399–408.CrossRefGoogle Scholar
[7] Daniels, H. E. (1996). Approximating the first crossing-time density for a curved boundary. Bernoulli 2, 133–143.CrossRefGoogle Scholar
[8] Durbin, J. (1971). Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov–Smirnov test. J. Appl. Prob. 8, 431–453.Google Scholar
[9] Durbin, J. (1992). The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Prob. 29, 291–304.CrossRefGoogle Scholar
[10] Fu, J. C. and Lou, W. Y. W. (2003). Distribution Theory of Runs and Patterns and Its Applications. World Scientific, River Edge, NJ.Google Scholar
[11] Kac, M. (1947). Random walk and the theory of Brownian motion. Amer. Math. Monthly 54, 369–391.CrossRefGoogle Scholar
[12] Klebaner, F. C. (2005). Introduction to Stochastic Calculus with Applications. Imperial College, London.CrossRefGoogle Scholar
[13] Kolmogorov, A. N. (1931). Über die analytischen methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104, 415–458.CrossRefGoogle Scholar
[14] Nagaev, S. V. (1970). On the speed of convergence in a boundary value problem. I. Theory Prob. Appl. 15, 163–186.CrossRefGoogle Scholar
[15] Nagaev, S. V. (1970). On the speed of convergence in a boundary value problem. II. Theory Prob. Appl. 15, 403–429.CrossRefGoogle Scholar
[16] Novikov, A., Frishling, V. and Kordzakhia, N. (1999). Approximations of boundary crossing probabilities for a Brownian motion. J. Appl. Prob. 36, 1019–1030.Google Scholar
[17] Pötzelberger, K. and Wang, L. (2001). Boundary crossing probability for Brownian Motion. J. Appl. Prob. 38, 152–164.CrossRefGoogle Scholar
[18] Robbins, H. and Siegmund, D. (1970). Boundary crossing probabilities for Wiener process and sample sums. Ann. Math. Statist. 41, 1410–1429.CrossRefGoogle Scholar
[19] Roberts, G. O. and Shortland, C. F. (1997). Pricing barrier options with time-dependent coefficients. Math. Finance 7, 83–93.Google Scholar
[20] Ricciardi, L. M. and Sacerdote, L. (1979). The Ornstein-Uhlenbeck process as a model of neuronal activity. Biol. Cybernet. 35, 1–9.CrossRefGoogle Scholar
[21] Ricciardi, L. M., Di Crescenzo, A., Giorno, V. and Nobile, A. G. (1999). An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling. Math. Japon. 50, 247–322.Google Scholar
[22] Sacerdote, L. and Tomassetti, F. (1996). On the evaluations and asymptotic approximations of first-passage-time probabilities. Adv. Appl. Prob. 28, 270–284.Google Scholar
[23] Scheike, T. H. (1992). A boundary-crossing result for Brownian motion. J. Appl. Prob. 29, 448–453.CrossRefGoogle Scholar
[24] Siegmund, D. (1986). Boundary crossing probabilities and statistical applications. Ann. Statist. 14, 361–404.Google Scholar
[25] Strassen, V. (1967). Almost sure behavior of sums of independent random variables and martingales. In Proc. 5th Berkeley Symp. Math. Statist. and Prob., Vol. II, University of California Press, Berkeley, CA, pp. 315–343.Google Scholar
[26] Wang, L. and Pötzelberger, K. (1997). Boundary crossing probability for Brownian Motion and general boundaries. J. Appl. Prob. 34, 54–65.CrossRefGoogle Scholar
[27] Wang, L. and Pötzelberger, K. (2007). Crossing probabilities for diffusion processes with piecewise continuous boundaries. Methodology Comput. Appl. Prob. 9, 21–40.CrossRefGoogle Scholar