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Calculation of noncrossing probabilities for Poisson processes and its corollaries

  • Estate Khmaladze (a1) and Eka Shinjikashvili (a2)

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.

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Corresponding author

Postal address: Department of Statistics, School of Mathematics, University of New South Wales, Sydney 2052, Australia.
∗∗ Email address: estate@maths.unsw.edu.au

References

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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.
Borovkov, A. A. and Syčeva, N. M. (1968). Certain asymptotically optimal nonparametric tests. Theory Prob. Appl. 13, 359393.
Brémaud, P., (1981). Point Processes and Queues. Martingale Dynamics. Springer, New York.
Chesney, M. et al. (1979). Parisian pricing. Risk 10, 7780.
Daniels, H. (1969). The minimum of a stationary Markov process superimposed on a U-shaped trend. J. Appl. Prob. 6, 399408.
Daniels, H. (1996). Approximating the first crossing time density for a curved boundary. Bernoulli 2, 133143.
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, 431453.
Eicker, F. (1979). The asymptotic distribution of the suprema of the standardized empirical process. Ann. Statist. 7, 116138.
Einmahl, J. H. J. (1996). Extension to higher dimensions of the Jaeschke–Eicker result on the standardized empirical process. Commun. Statist. Theory Meth. 25, 813822.
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 2. John Wiley, New York.
Galambos, J. (1994). The development of the mathematical theory of extremes in the past half century. Theory Prob. Appl. 39, 234248.
Grooneboom, P. (1989). Brownian motion with a Parabolic drift and airy functions. Prob. Theory Rel. Fields 81, 79109.
Jaeschke, D. (1979). The asymptotic distribution of the supremum of the standardized empirical distribution function on subintervals. Ann. Statist. 7, 108115.
Karr, A. F. (1991). Point Processes and Their Statistical Inference. Marcel Dekker, New York.
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.
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, 640648.
Lerche, H. R. (1986). Boundary Crossing of Brownian Motion (Lecture Notes Statist. 40). Springer, Berlin.
Liptser, R. S. and Shiryayev, A. N. (1978). Statistics of Random Processes II. Applications. Springer, New York.
Loader, C. R. and Deely, J. J. (1987). Computations of boundary crossing probabilities for the Wiener process. J. Statist. Comput. Simulation 27, 95105.
Mason, D. M. (1983). The asymptotic distribution of weighted empirical distribution functions. Stoch. Proc. Appl. 15, 99109.
Musiela, M. and Rutkowski, M. (1997). Martingale Methods in Financial Modelling. Springer, Berlin.
Nesenenko, G. A. and Tjurin, Ju. N. (1978). Asymptotic behavior of the Kolmogorov statistic for a parametric family. Dokl. Akad. Nauk SSSR 239, 12921294 (in Russian).
Niederhausen, H. (1981). Scheffe polynomials for computing exact Kolmogorov–Smirnov and Renyi type distributions. Ann. Statist. 9, 528531.
Noe, M. (1972). The calculation of distributions of two-sided Kolmogorov–Smirnov type statistics. Ann. Math. Statist. 43, 5864.
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, 233241.
Novikov, A., Frishling, V. and Kordzakhia, N. (1999). Approximations of boundary crossing probabilities for the Brownian motion. J. Appl. Prob. 36, 10191030.
Owen, A. B. (1995). Nonparametric likelihood confidence bands for a distribution function. J. Amer. Statist. Assoc. 90, 516521.
Ricciardi, L. M., Sacerdote, L. and Sato, S. (1984). On an integral equation for first-passage-time probability densities. J. Appl. Prob. 21, 302314.
Sacerdote, L. and Tomassetti, F. (1996). On evaluations and asymptotic approximations of first-passage-time probabilities. Adv. Appl. Prob. 28, 270284.
Shiryaev, A. N. (1999). Essentials of Stochastic Finance: Facts, Models, Theory. Fazis, Moscow (in Russian). English translation: World Scientific, River Edge, NJ.
Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. John Wiley, New York.
Steck, G. P. (1968). The Smirnov two-sample tests as rank tests. Ann. Math. Statist. 40, 14491466.
Stephens, M. A. (1974). EDF statistics for goodness of fit and some comparisons. J. Amer. Statist. Assoc. 69, 730737.

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Calculation of noncrossing probabilities for Poisson processes and its corollaries

  • Estate Khmaladze (a1) and Eka Shinjikashvili (a2)

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