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Calculation of noncrossing probabilities for Poisson processes and its corollaries

Part of: Distribution theory Statistical tables Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Estate Khmaladze  and
Eka Shinjikashvili
Estate Khmaladze*
Affiliation:
University of New South Wales and A. Razmadze Mathematical Institute, Tbilisi
Eka Shinjikashvili*
Affiliation:
University of New South Wales
*
∗ Postal address: Department of Statistics, School of Mathematics, University of New South Wales, Sydney 2052, Australia.
∗∗ Email address: [email protected]
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Abstract

The paper describes a new numerical method for the calculation of noncrossing probabilities for arbitrary boundaries by a Poisson process. We find the method to be simple in implementation, quick and efficient - it works reliably for Poisson processes of very high intensity n, up to several thousand. Hence, it can be used to detect unusual features in the finite-sample behaviour of empirical process and trace it down to very high sample sizes. It also can be used as a good approximation for noncrossing probabilities for Brownian motion and Brownian bridge, in particular when the boundaries are not regular. As a numerical example we demonstrate the divergence of normalized Kolmogorov-Smirnov statistics from their prescribed limiting distributions (Eicker (1979), Jaeshke (1979)) for quite large n in contrast to very regular behaviour of statistics of Mason (1983). For the Brownian motion case we considered square-root, Daniels' (1969) and Grooneboom's (1989) boundaries.

Keywords

Barrier optionscurved boundariesextreme value distributionsempirical processweighted Kolmogorov-Smirnov statisticsBrownian motion and Brownian bridge

MSC classification

Primary: 62Q05: Statistical tables 62E15: Exact distribution theory
Secondary: 62E20: Asymptotic distribution theory 60G55: Point processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 3 , September 2001 , pp. 702 - 716
Copyright
Copyright © Applied Probability Trust 2001 

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References

Antic, A., Frishling, V., Kucera, A. and Rider, P. (1998). Pricing barrier options with time dependent drift, volatility and barriers. Working Paper, Commonwealth Bank of Australia.Google Scholar
Borovkov, A. A. and Syčeva, N. M. (1968). Certain asymptotically optimal nonparametric tests. Theory Prob. Appl. 13, 359–393.Google Scholar
Brémaud, P., (1981). Point Processes and Queues. Martingale Dynamics. Springer, New York.CrossRefGoogle Scholar
Chesney, M. et al. (1979). Parisian pricing. Risk 10, 77–80.Google Scholar
Daniels, H. (1969). The minimum of a stationary Markov process superimposed on a U-shaped trend. J. Appl. Prob. 6, 399–408.CrossRefGoogle Scholar
Daniels, H. (1996). Approximating the first crossing time density for a curved boundary. Bernoulli 2, 133–143.Google Scholar
Durbin, J. (1971). Boundary-crossing probabilities for the Brownian motion and Poisson process and techniques for computing the power of the Kolmogorov–Smirnov test. J. Appl. Prob. 8, 431–453.Google Scholar
Eicker, F. (1979). The asymptotic distribution of the suprema of the standardized empirical process. Ann. Statist. 7, 116–138.Google Scholar
Einmahl, J. H. J. (1996). Extension to higher dimensions of the Jaeschke–Eicker result on the standardized empirical process. Commun. Statist. Theory Meth. 25, 813–822.CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 2. John Wiley, New York.Google Scholar
Galambos, J. (1994). The development of the mathematical theory of extremes in the past half century. Theory Prob. Appl. 39, 234–248.Google Scholar
Grooneboom, P. (1989). Brownian motion with a Parabolic drift and airy functions. Prob. Theory Rel. Fields 81, 79–109.CrossRefGoogle Scholar
Jaeschke, D. (1979). The asymptotic distribution of the supremum of the standardized empirical distribution function on subintervals. Ann. Statist. 7, 108–115.CrossRefGoogle Scholar
Karr, A. F. (1991). Point Processes and Their Statistical Inference. Marcel Dekker, New York.Google Scholar
Khmaladze, E. and Shinjikashvili, E. (1998). Calculation of noncrossing probabilities for Poisson process and its corollaries. Res. Rept S98-8 Department of Statistics, University of New South Wales.Google Scholar
Kotelnikova, V. F. and Khmaladze, E. V. (1983). Calculation of the probability of an empirical process not crossing a curvilinear boundary. Theory Prob. Appl. 27, 640–648.CrossRefGoogle Scholar
Lerche, H. R. (1986). Boundary Crossing of Brownian Motion (Lecture Notes Statist. 40). Springer, Berlin.Google Scholar
Liptser, R. S. and Shiryayev, A. N. (1978). Statistics of Random Processes II. Applications. Springer, New York.Google Scholar
Loader, C. R. and Deely, J. J. (1987). Computations of boundary crossing probabilities for the Wiener process. J. Statist. Comput. Simulation 27, 95–105.Google Scholar
Mason, D. M. (1983). The asymptotic distribution of weighted empirical distribution functions. Stoch. Proc. Appl. 15, 99–109.CrossRefGoogle Scholar
Musiela, M. and Rutkowski, M. (1997). Martingale Methods in Financial Modelling. Springer, Berlin.CrossRefGoogle Scholar
Nesenenko, G. A. and Tjurin, Ju. N. (1978). Asymptotic behavior of the Kolmogorov statistic for a parametric family. Dokl. Akad. Nauk SSSR 239, 1292–1294 (in Russian).Google Scholar
Niederhausen, H. (1981). Scheffe polynomials for computing exact Kolmogorov–Smirnov and Renyi type distributions. Ann. Statist. 9, 528–531.Google Scholar
Noe, M. (1972). The calculation of distributions of two-sided Kolmogorov–Smirnov type statistics. Ann. Math. Statist. 43, 58–64.Google Scholar
Noe, M. and Vandewiele, G. (1968). The calculation of distributions of two-sided Kolmogorov–Smirnov type statistics including a table of significance points for a particular case. Ann. Math. Statist. 39, 233–241.Google Scholar
Novikov, A., Frishling, V. and Kordzakhia, N. (1999). Approximations of boundary crossing probabilities for the Brownian motion. J. Appl. Prob. 36, 1019–1030.Google Scholar
Owen, A. B. (1995). Nonparametric likelihood confidence bands for a distribution function. J. Amer. Statist. Assoc. 90, 516–521.Google Scholar
Ricciardi, L. M., Sacerdote, L. and Sato, S. (1984). On an integral equation for first-passage-time probability densities. J. Appl. Prob. 21, 302–314.Google Scholar
Sacerdote, L. and Tomassetti, F. (1996). On evaluations and asymptotic approximations of first-passage-time probabilities. Adv. Appl. Prob. 28, 270–284.Google Scholar
Shiryaev, A. N. (1999). Essentials of Stochastic Finance: Facts, Models, Theory. Fazis, Moscow (in Russian). English translation: World Scientific, River Edge, NJ.Google Scholar
Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. John Wiley, New York.Google Scholar
Steck, G. P. (1968). The Smirnov two-sample tests as rank tests. Ann. Math. Statist. 40, 1449–1466.CrossRefGoogle Scholar
Stephens, M. A. (1974). EDF statistics for goodness of fit and some comparisons. J. Amer. Statist. Assoc. 69, 730–737.Google Scholar