Hostname: page-component-7dd5485656-hw7sx Total loading time: 0 Render date: 2025-10-31T08:31:18.813Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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