Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T00:39:39.518Z Has data issue: false hasContentIssue false
  • English
  • Français

A new integral equation for the evaluation of first-passage-time probability densities

Published online by Cambridge University Press:  01 July 2016

A. Buonocore ,
A. G. Nobile  and
L. M. Ricciardi
A. Buonocore*
Affiliation:
Università di Napoli
A. G. Nobile*
Affiliation:
Università di Salerno
L. M. Ricciardi*
Affiliation:
Università di Napoli
*
∗Dipartimento di Matematica e Applicazioni, Università di Napoli, Via Mezzocannone 8, 80134 Napoli, Italy.
∗∗ Dipartimento di Informatica e Applicazioni, Facoltà di Scienze, Università di Salerno, 84100 Salerno, Italy.
∗Dipartimento di Matematica e Applicazioni, Università di Napoli, Via Mezzocannone 8, 80134 Napoli, Italy.
Get access Rights & Permissions [Opens in a new window]

Abstract

The first-passage-time p.d.f. through a time-dependent boundary for one-dimensional diffusion processes is proved to satisfy a new Volterra integral equation of the second kind involving two arbitrary continuous functions. Use of this equation is made to prove that for the Wiener and the Ornstein–Uhlenbeck processes the singularity of the kernel can be removed by a suitable choice of these functions. A simple and efficient numerical procedure for the solution of the integral equation is provided and its convergence is briefly discussed. Use of this equation is finally made to obtain closed-form expressions for first-passage-time p.d.f.'s in the case of various time-dependent boundaries.

Keywords

DIFFUSIONWIENER PROCESSORNSTEIN–UHLENBECK PROCESSVARYING BOUNDARIESDANIELS BOUNDARYVOLTERRA INTEGRAL EQUATIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 4 , December 1987 , pp. 784 - 800
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.)

Footnotes

Research carried out under CNR-JSPS Scientific Cooperation Programme, Contract No. 84.00227.01, CNR Contract No. 85.00002.01 and under MPI support.

References

[1] Abrahams, J. (1986) A survey of recent progress on level crossing problems for random processes. In Communications and Networks. A Survey of Recent Advances, ed. Blake, I. F. and Poor, H. V., Springer-Verlag, New York, 6–25.Google Scholar
[2] 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.Google Scholar
[3] Baker, C. T. H. (1978) The Numerical Treatment of Integral Equations. Oxford University Press, Oxford.Google Scholar
[4] Balossino, N., Ricciardi, L. M. and Sacerdote, L. (1985). On the evaluation of first-passage-time densities for diffusion processes. Cybernet. Syst. 16, 325–339.Google Scholar
[5] Blake, I. F. and Lindsey, W. C. (1973) Level crossing problems for random processes. IEEE Trans. Inf. Theory 19, 295–315.Google Scholar
[6] 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
[7] 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
[8] Favella, L. F. and De Griffi, R. M. (1981) On a weakly singular Volterra integral equation. Calcolo XVIII, 153–195.Google Scholar
[9] Favella, L., Reineri, M. T., Ricciardi, L. M. and Sacerdote, L. (1982) First-passage-time problems and some related computational methods. Cybernet. Syst. 13, 95–128.CrossRefGoogle Scholar
[10] Feller, W. (1952) The parabolic differential equations and the associated semigroup-transformations. Ann. Math. 55, 468–518.Google Scholar
[11] Fortet, R. (1943) Les fonctions aléatoires du type de Markoff associées à certaines équations lineáires aux dérivées partielles du type parabolique. J. Math. Pures Appl. 22, 177–243.Google Scholar
[12] Heath, R. A. (1981) A tandem random walk model for psychological discrimination. Br. J. Math. Statist. Psychol. 34, 76–92.CrossRefGoogle ScholarPubMed
[13] Holden, A. V. (1976) Models of the Stochastic Activity of Neurons. Lecture Notes in Biomathematics, 16. Springer-Verlag, Berlin.Google Scholar
[14] Maruyama, T. (1977) Stochastic Problems in Population Genetics. Lecture Notes in Biomathematics, 17. Springer-Verlag, Berlin.Google Scholar
[15] Nobile, A. G. and Ricciardi, L. M. (1984a) Growth with regulation in fluctuating environments. I. Alternative logistic-like diffusion models. Biol. Cybernet. 49, 179–188.CrossRefGoogle Scholar
[16] Nobile, A. G. and Ricciardi, L. M. (1984b) Growth with regulation in fluctuating environments. II. Intrinsic lower bounds to population size. Biol. Cybernet. 50, 285–299.CrossRefGoogle ScholarPubMed
[17] Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985). A note on first-passage-time and related problems. J. Appl. Prob. 22, 346–359.Google Scholar
[18] Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985). Exponential trends of Omstein–Uhlenbeck first-passage-time-densities. J. Appl. Prob. 22, 346–359.Google Scholar
[19] Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985) Exponential trends of first-passage-time-densities for a class of diffusion processes with steady-state distribution. J. Appl. Prob. 22, 611–618.Google Scholar
[20] Park, C. and Paranjape, S. R. (1974) Probabilities of Wiener paths crossing differentiable curves. Pacific J. Math. 53, 579–583.Google Scholar
[21] Park, C. and Shuurmann, F. J. (1976) Evaluations of barrier-crossing probabilities of Wiener paths. J. Appl. Prob. 13, 267–275.Google Scholar
[22] Park, C. and Shuurmann, F. J. (1980) Evaluations of absorption probabilities for the Wiener process on large intervals. J. Appl. Prob. 17, 363–372.Google Scholar
[23] Ratcliff, R. (1980) A note on modelling accumulation of information when the rate of accumulation changes with time. J. Math. Psychol. 21, 178–184.Google Scholar
[24] Ricciardi, L. M. (1977) Diffusion Processes and Related Topics in Biology. Lecture Notes in Biomathematics, 14. Springer-Verlag, Berlin.Google Scholar
[25] Ricciardi, L. M. and Sato, S. (1983) A note on the evaluation of first-passage-time probability densities. J. Appl. Prob. 20, 197–201.Google Scholar
[26] 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
[27] 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
[28] Smithies, F. (1958) Integral Equations. Cambridge University Press, Cambridge.Google Scholar
[29] Stratonovich, R. L. (1963) Topics in the Theory of Random Noise, Vol. I. Gordon and Breach, New York.Google Scholar