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Approximation to the exit probability of a continuous Gaussian process over a U-shaped boundary of increasing curvature

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

A. N. Balabushkin
A. N. Balabushkin*
Affiliation:
Institute of Control Sciences, Moscow
*
∗Postal address: 1–267 Proizvodstvennaya Street, Moscow 119619, Russia.
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Abstract

A simple approximation to the probability of crossing a U-shaped boundary by a Brownian motion is given. The larger the second derivative of the curve at a minimum point, the higher the accuracy of the approximation. The result is also extended to a class of continuous Gaussian processes with definite properties. Numerical examples are given.

Keywords

BROWNIAN MOTIONCONTINUOUS GAUSSIAN PROCESSBOUNDARY CROSSING

MSC classification

Primary: 60J65: Brownian motion
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 2 , June 1995 , pp. 429 - 442
Copyright
Copyright © Applied Probability Trust 1995 

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References

[1] Balabushkin, A. N. (1991) Forecasting of the state of a dynamic object at the time of reaching the boundary under small perturbations. Automation and Remote Control 52, 1533–1538.Google Scholar
[2] Balabushkin, A. N. and Gul'Ko, F. B. (1988) Prediction of extremal values of phase variables of stochastic systems. Automation and Remote Control 49, 743–748.Google Scholar
[3] Durbin, J. (1985) The first-passage density of a continuous Gaussian process to a general boundary. J. Appl. Prob. 22, 99–122.Google Scholar
[4] Durbin, J. (1992) The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Prob. 29, 291–304.Google Scholar
[5] Ferebee, B. (1982) The tangent approximation to one-sided Brownian exit densities. Z. Wahrscheinlichkeitsth. 61, 309–326.Google Scholar
[6] Ferebee, B. (1983) An asymptotic expansion for one-sided Brownian exit densities. Z. Wahrscheinlichkeitsth. 63, 1–15.Google Scholar
[7] Fernique, X. (1975) Regularité des Trajectoires des Fonctions Aléatoires Gaussiennes. Lecture Notes in Mathematics 480, Springer-Verlag, Berlin.Google Scholar
[8] Jennen, C. and Lerche, H. R. (1981) First-exit densities of Brownian motion through one-sided moving boundaries. Z. Wahrscheinlichkeitsth. 55, 133–148.Google Scholar
[9] Slepian, D. (1962) The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41, 463–501.Google Scholar