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Exponential trends of Ornstein–Uhlenbeck first-passage-time densities

Published online by Cambridge University Press:  14 July 2016

A. G. Nobile ,
L. M. Ricciardi  and
L. Sacerdote
A. G. Nobile*
Affiliation:
University of Salerno
L. M. Ricciardi*
Affiliation:
University of Naples
L. Sacerdote*
Affiliation:
University of Salerno
*
∗Postal address: Dipartimento di Informatica e Applicazioni, Università di Salerno, 84100 Salerno, Italy.
∗∗Postal address: Dipartimento di Matematica e Applicazioni, Università di Napoli, Via Mezzocannone 8, 80134 Napoli, Italy.
∗Postal address: Dipartimento di Informatica e Applicazioni, Università di Salerno, 84100 Salerno, Italy.
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Abstract

The asymptotic behaviour of the first-passage-time p.d.f. through a constant boundary for an Ornstein–Uhlenbeck process is investigated for large boundaries. It is shown that an exponential p.d.f. arises, whose mean is the average first-passage time from 0 to the boundary. The proof relies on a new recursive expression of the moments of the first-passage-time p.d.f. The excellent agreement of theoretical and computational results is pointed out. It is also remarked that in many cases the exponential behaviour actually occurs even for small values of time and boundary.

Keywords

ASYMPTOTIC BEHAVIOURFIRST-PASSAGE-TIME MOMENTS
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 2 , June 1985 , pp. 360 - 369
Copyright
Copyright © Applied Probability Trust 1985 

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Footnotes

Research carried out under CNR-JSPS Scientific Cooperation Programme, Contracts No. 83.0032.01 and No. 84.00227.01, and with MPI financial support.

References

[1] Anderssen, R. S., De Hoog, F. R. and Weiss, R. (1973) On the numerical solution of Brownian motion processes. J. Appl. Prob. 10, 409–418.CrossRefGoogle Scholar
[2] Balossino, N., Ricciardi, L. M. and Sacerdote, L. (1985) On the evaluation of first-passage-time densities for diffusion processes. Cybernetics and Systems. To appear.Google Scholar
[3] Cerbone, G., Ricciardi, L. M. and Sacerdote, L. (1981) Mean variance and skewness of the first-passage time for the Ornstein–Uhlenbeck process. Cybernetics and Systems 12, 395–429.Google Scholar
[4] Daniels, H. E. (1969) The minimum of a stationary Markov process superimposed on a U-shaped trend. J. Appl. Prob. 6, 399–408.CrossRefGoogle Scholar
[5] De Griffi, E. M. and Favella, L. (1981) On a weakly singular Volterra integral equation. Calcolo XVIII, 153–195.Google Scholar
[6] Durbin, J. (1971) Boundary crossing probabilities for Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov–Smirnov test. J. Appl. Prob. 8, 431–453.CrossRefGoogle Scholar
[7] Favella, L., Reineri, M. T., Ricciardi, L. M. and Sacerdote, L. (1982) First passage time problems and some related computational methods. Cybernetics and Systems 13, 95–128.Google Scholar
[8] Keilson, J. and Ross, H. (1975) Passage time distribution for the Ornstein–Uhlenbeck process. Sel. Tables Math. Statist. 3, 233–328.Google Scholar
[9] Ricciardi, L. M. (1977) Diffusion Processes and Related Topics in Biology. Lecture Notes in Biomathematics, Springer-Verlag, Berlin.Google Scholar
[10] 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
[11] Ricciardi, L. M. and Sato, S. (1983) A note on the evaluation of the first-passage-time probability densities. J. Appl. Prob. 20, 197–201.Google Scholar
[12] Ricciardi, L. M., Sacerdote, L. and Sato, S. (1983) Diffusion approximation and first passage time problem for a model neuron. II. Outline of a computational method. Math. Biosci. 64, 29–44.Google Scholar
[13] Ricciardi, L. M., Sacerdote, L. and Sato, S. (1984) On an integral equation for first-passage time probability densities. J. Appl. Prob. 21, 302–314.Google Scholar
[14] Roy, B. and Smith, D. R. (1969) Analysis of the exponential decay of the neuron showing frequency threshold effects. Bull. Math. Biophys. 31, 341–357.Google Scholar
[15] 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
[16] Sato, S. (1978) On the moments of the firing interval of the diffusion approximated model neuron. Math. Biosci. 39, 53–70.Google Scholar
[17] Siebert, W. M. (1969) On stochastic neural models of the diffusion type Q. Progr. Rept., Res. Lab. Electron., MIT 94, 281–287.Google Scholar
[18] Siegert, A. J. F. (1951) On the first passage time probability problem. Phys. Rev. 81, 617–623.Google Scholar
[19] Sugiyama, H., Moore, G. and Perkel, D. M. (1970) Solutions for a stochastic model of neuronal spike production. Math. Biosci. 8, 323–341.Google Scholar