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A note on the evaluation of first-passage-time probability densities

Published online by Cambridge University Press:  14 July 2016

L. M. Ricciardi  and
S. Sato
L. M. Ricciardi*
Affiliation:
Università di Napoli
S. Sato*
Affiliation:
Osaka University
*
∗ Postal address: Istituto di Matematica dell ’Università, Via Mezzocannone 8, 80134 Napoli, Italy.
∗∗ Postal address: Department of Biophysical Engineering. Faculty of Engineering Science, Osaka University, Toyonaka, Osaka, Japan.
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Abstract

A procedure is indicated to estimate first-passage-time p.d.f.'s through varying boundaries for a class of diffusion processes that can be transformed into the Wiener process by rather general transformations. Although this procedure is adapted to Durbin's [4] algorithm, it could be extended to other existing computation methods.

Keywords

STANDARD WIENER PROCESSDIFFUSION PROCESS TRANSFORMATIONSDURBIN’S ALGORITHM
Type
Short Communications
Information
Journal of Applied Probability , Volume 20 , Issue 1 , March 1983 , pp. 197 - 201
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

Research carried out while this author was on leave of absence at Osaka University.

References

[1] Anderssen, K. S., De Hoog, F. R. and Weiss, R. (1973) On the numerical solution of Brownian motion processes. J. Appl. Prob. 10, 409–418.Google Scholar
[2] Blake, I. F. and Lindsey, W. C. (1973) Level-crossing problems for random processes. IEEE Trans. Information Theory IT-19, 295–315.Google Scholar
[3] De Griffi, E. M. and Favella, L. (1981) On a weakly singular Volterra integral equation. Calcolo XVIII, 153–195.Google Scholar
[4] 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.Google Scholar
[5] Favella, L., Reineri, M. T., Ricciardi, L. M. and Sacerdote, L. (1982) First passage time problems and related computational methods. Cybernetics and Systems 13, 95–128.CrossRefGoogle Scholar
[6] Holden, A. V. (1976) Models of the Stochastic Activity of Neurons. Lecture Notes in Biomathematics 16, Springer-Verlag, Berlin.Google Scholar
[7] Maruyama, T. (1977) Stochastic Problems in Population Genetics. Lecture Notes in Biomathematics 17, Springer-Verlag, Berlin.Google Scholar
[8] Park, C. and Schuurmann, F. J. (1976) Evaluations of barrier-crossing probabilities of Wiener paths. J. Appl. Prob. 13, 267–275.Google Scholar
[9] Park, C. and Schuurmann, F. J. (1980) Evaluations of absorption probabilities for the Wiener process on large intervals. J. Appl. Prob. 17, 363–372.Google Scholar
[10] Ricciardi, L. M. (1976) On the transformation of diffusion processes into the Wiener process. J. Math. Anal. Appl. 54, 185–199.Google Scholar
[11] Ricciardi, L. M. (1977) Diffusion Processes and Related Topics in Biology. Lecture Notes in Biomathematics 14, Springer-Verlag, Berlin.Google Scholar
[12] Ricciardi, L. M. and Sacerdote, L. (1979) The Ornstein-Uhlenbeck process as a model for neuronal activity, I. Mean and variance of the firing time. Biol. Cybernetics 35, 1–9.Google Scholar
[13] Ricciardi, L. M. (1982) Diffusion approximations and computation problems for single neuron's activity. In Competition and Cooperation in Neural Nets, ed. Amari, S. and Arbib, M. A., Lecture Notes in Biomathematics 45, Springer-Verlag, Berlin, 142–154.Google Scholar
[14] Sato, S. (1977) Evaluation of the first passage time probability to a square root boundary for the Wiener process. J. Appl. Prob. 14, 850–856.Google Scholar
[15] Smith, C. S. (1972) A note on boundary-crossing probabilities for the Brownian motion. J. Appl. Prob. 9, 857–861.Google Scholar