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On the probability densities of an Ornstein–Uhlenbeck process with a reflecting boundary

Published online by Cambridge University Press:  14 July 2016

L. M. Ricciardi  and
L. Sacerdote
L. M. Ricciardi*
Affiliation:
University of Naples
L. Sacerdote*
Affiliation:
University of Turin
*
∗Postal address: Dipartimento di Matematica e Applicazioni, University of Naples, Via Mezzocannone 8, 80134 Naples, Italy.
∗∗Postal address: Dipartimento di Matematica, University of Turin, Via Principe Amedeo 8, 10123 Turin, Italy.
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Abstract

We show that the transition p.d.f. of the Ornstein–Uhlenbeck process with a reflection condition at an assigned state S is related by integral-type equations to the free transition p.d.f., to the transition p.d.f. in the presence of an absorption condition at S, to the first-passage-time p.d.f. to S and to the probability current. Such equations, which are also useful for computational purposes, yield as an immediate consequence all known closed-form results for Wiener and Ornstein–Uhlenbeck processes.

Keywords

DIFFUSION PROCESSESWIENER PROCESSFIRST-PASSAGE TIME
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 2 , June 1987 , pp. 355 - 369
Copyright
Copyright © Applied Probability Trust 1987 

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References

[1] Abrahams, J. (1983) A survey of recent progress on level crossing problems for random processes.Google 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. CrossRefGoogle Scholar
[3] Blake, I. F. and Lindsey, W. C. (1973) Level crossing problems for random processes. I.E.E.E. Trans. Inf. Theory IT-19, 295–315.Google Scholar
[4] Cox, J. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Wiley, New York.Google Scholar
[5] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1953) Higher Transcendental Functions, Vol. II. McGraw-Hill, New York.Google Scholar
[6] 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
[7] Giorno, V., Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1986) Some remarks on the Rayleigh process. J. Appl. Prob. 23, 398–408.CrossRefGoogle Scholar
[8] Giorno, V., Nobile, A. G. and Ricciardi, L. ?. (1986) On some diffusion approximations to queueing systems. Adv. Appl. Prob. 18, 991–1014.Google Scholar
[9] Heath, R. A. (1981) A tandem random walk model for psychological discrimination. Br. J. Math. Stat. Psychol. 34, 76–92.Google Scholar
[10] Holden, A. V. (1976) Models of the Stochastic Activity of Neurons. Lecture Notes in Biomathematics, Springer-Verlag, Berlin.Google Scholar
[11] Maruyama, T. (1977) Stochastic Problems in Population Genetics. Lecture Notes in Biomathematics, Springer-Verlag, Berlin.Google Scholar
[12] Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985) A note on first-passage time and some related problems. J. Appl. Prob. 22, 346–360.Google Scholar
[13] Nobile, A. G., Ricciardi, L. ?. and Sacerdote, L. (1985) Exponential trends of Ornstein–Uhlenbeck first-passage-time densities. J. Appl. Prob. 22, 360–369.Google Scholar
[14] Ratcliff, R. (1980) A note on modelling accumulation of information when the rate of accumulation changes with time. J. Math. Psych. 21, 178–184.Google Scholar
[15] Ricciardi, L. M. (1977) Diffusion Processes and Some Related Topics in Biology. Lecture Notes in Biomathematics, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[16] Ricciardi, L. M. (1985) Stochastic population models. II. Diffusion models. Lecture Notes at the International School on Mathematical Ecology.Google Scholar
[17] 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. Cybernet. 35, 1–9.Google Scholar
[18] Ricciardi, L. M., Sacerdote, L. and Sato, S. (1983) Diffusion approximation and first passage time problem for a model neuron. II. Outline of a computation method. Math. Biosci. 64, 29–44.CrossRefGoogle Scholar
[19] 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
[20] 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
[21] Siegert, A. J. F. (1951) On the first passage time probability problem. Phys. Rev. 81, 617–623.Google Scholar