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The first-passage density of a continuous gaussian process to a general boundary

Published online by Cambridge University Press:  14 July 2016

J. Durbin
J. Durbin*
Affiliation:
London School of Economics and Political Science
*
∗Postal address: The London School of Economics and Political Science, Department of Statistical and Mathematical Sciences, Houghton St, London WC2A 2AE, UK.
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Abstract

Under mild conditions an explicit expression is obtained for the first-passage density of sample paths of a continuous Gaussian process to a general boundary. Since this expression will usually be hard to compute, an approximation is given which is computationally simple and which is exact in the limit as the boundary becomes increasingly remote. The integral of this approximating density is itself approximated by a simple formula and this also is exact in the limit. A new integral equation is derived for the first-passage density of a continuous Gaussian Markov process. This is used to obtain further approximations.

Keywords

BROWNIAN MOTIONCONTINUOUS MARKOV PROCESSBOUNDARY-CROSSING PROBABILITYEXIT DENSITYVOLTERRA INTEGRAL EQUATION OF THE SECOND KIND
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 1 , March 1985 , pp. 99 - 122
Copyright
Copyright © Applied Probability Trust 1985 

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References

Bachelier, L. (1900) Théorie de la spéculation. Ann. Sci. Ecole. Norm. Sup. 3, 17–21.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Borovkov, A. A. (1965) On the first passage time for one class of processes with independent increments. Theory Prob. Appl. 10, 331–334.Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
Cuzick, J. (1981) Boundary crossing probabilities for stationary Gaussian processes and Brownian motion. Trans. Amer. Math. Soc. 263, 469–492.CrossRefGoogle Scholar
Daniels, H. E. (1974) The maximum size of a closed epidemic. Adv. Appl. Prob. 6, 607–621.Google Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
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.CrossRefGoogle Scholar
Durbin, J. (1973a) Weak convergence of the sample distribution function when parameters are estimated. Ann. Statist. 1, 279–290.CrossRefGoogle Scholar
Durbin, J. (1973b) Distribution Theory for Tests Based on the Sample Distribution Function. SIAM, Philadelphia.Google Scholar
Durbin, J. (1983) Boundary crossing probabilities for Gaussian processes and applications to Kolmogorov–Smirnov statistics when parameters are estimated. In Proc. Colloq. Limit Theorems in Probability and Statistics, Veszprém, Hungary, 1982.Google Scholar
Ferebee, B. (1983) An asymptotic expansion for one-sided Brownian exit densities. Z. Wahrscheinlichkeitsth. 63, 1–15.Google Scholar
Jennen, C. and Lerche, H. R. (1981) First-exit densities of Brownian motion through one-side moving boundaries. Z. Wahrscheinlichkeitsth. 55, 133–148.Google Scholar
Keilson, J. (1963) The first passage time density for homogeneous skip-free walks on the continuum. Ann. Math. Statist. 34, 1003–1011.CrossRefGoogle Scholar
Tyurin, Y. N. and Neshenko, G. A. (1978) The asymptotics of the Kolmogorov statistic for a parametric family. Dokl. Akad. Nauk SSSR 239, 1292–1294. (English translation: Soviet Math. Dokl. 19, 516–519. Reviewed in Math. Rev. 57 (1979), #17935.) Google Scholar