Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T15:20:12.815Z Has data issue: false hasContentIssue false
  • English
  • Français

On the inverse of the first passage time probability problem

Published online by Cambridge University Press:  14 July 2016

R. M. Capocelli  and
L. M. Ricciardi
R. M. Capocelli
Affiliation:
Laboratorio di Cibernetica del C. N. R., Arco Felice, Naples, Italy
L. M. Ricciardi
Affiliation:
University of Chicago
Get access Rights & Permissions [Opens in a new window]

Abstract

Since the pioneering work of Siegert (1951), the problem of determining the first passage time distribution for a preassigned continuous and time homogeneous Markov process described by a diffusion equation has been deeply analyzed and satisfactorily solved. Here we discuss the “inverse problem” — of applicative interest — consisting in deciding whether a given function can be considered as the first passage time probability density function for some continuous and homogeneous Markov diffusion process. A constructive criterion is proposed, and some examples are provided. One of these leads to a singular diffusion equation representing a dynamical model for the genesis of the lognormal distribution.

Keywords

DIFFUSION PROCESSESFIRST PASSAGE TIMESINGULAR DIFFUSION EQUATIONSlognormal distribution
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 2 , June 1972 , pp. 270 - 287
Copyright
Copyright © Applied Probability Trust 1972 

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.)

References

Aitchinson, J. and Brown, J. A. C. (1966) The Lognormal Distribution. Cambridge University Press, Cambridge.Google Scholar
Bharucha-Reid, A. T. (1960) Elements of the Theory of Markov Processes and their Applications. McGraw-Hill, New York.Google Scholar
Capocelli, R. M. and Ricciardi, L. M. (1971) Diffusion approximation and first passage time problem for a model neuron. Kybernetik 8, 214–223.Google Scholar
Cowan, J. D. (1971) The first passage time problem for the Ornstein-Uhlenbeck process and its generalization. (Preprint).Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Wiley, New York.Google Scholar
Darling, D. A. and Siegert, A. J. F. (1953) The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639.Google Scholar
Feller, W. (1952) The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 55, 468–519.Google Scholar
Feller, W. (1954) Diffusion processes in one dimension. Trans. Amer. Math. Soc. 77, 1–31.Google Scholar
Johannesma, P. I. M. (1968) Diffusion models for the stochastic activity of neurons. In Neural Networks (Caianiello, E. R., ed.). Springer-Verlag, Heidelberg.Google Scholar
Roy, B. K. and Smith, D. R. (1969) Analysis of the exponential decay model of the neuron showing frequency threshold effects. Bull. Math. Biophys. 31, 341–357.CrossRefGoogle ScholarPubMed
Siegert, A. J. F. (1951) On the first passage time probability problem. Phys. Rev. 81, 617–623.Google Scholar
Stratonovich, R. L. (1963) Topics in the Theory of Random Noise. Vol. I. Gordon and Breach, New York.Google Scholar
Sugiyama, H., Moore, G. P. and Perkel, D. H. (1970) Solutions for a stochastic model of neuronal spike production. Math. Biosciences 8, 323–341.CrossRefGoogle Scholar
Tricomi, F. G. (1954) Funzioni Ipergeometriche Confluenti. Cremonese, Rome.Google Scholar
Uhlenbeck, G. E. and Ornstein, L. S. (1930) On the theory of the Brownian motion. Phys. Rev. 36, 823–841.CrossRefGoogle Scholar
Wang, M. C. and Uhlenbeck, G. E. (1945) On the theory of the Brownian motion II. Rev. Modern Phys. 17, 323–342.CrossRefGoogle Scholar