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Calculation of noncrossing probabilities for Poisson processes and its corollaries

Part of: Distribution theory Stochastic processes Statistical tables

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: estate@maths.unsw.edu.au
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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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