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On evaluations and asymptotic approximations of first-passage-time probabilities

Part of: Markov processes Distribution theory Stochastic processes

Published online by Cambridge University Press:  01 July 2016

L. Sacerdote  and
F. Tomassetti
L. Sacerdote*
Affiliation:
Università di Torino
F. Tomassetti*
Affiliation:
Universita di Napoli
*
* Postal address: Dipartimento di Matematica, Universita di Torino, Via Carlo Alberto 10, 10123 Torino, Italy.
** Postal address: Dipartimento di Matematica e Applicazioni, Universita di Napoli, Via Cintia, 80126 Napoli, Italy.
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Abstract

The series expansion for the solution of the integral equation for the first-passage-time probability density function, obtained by resorting to the fixed point theorem, is used to achieve approximate evaluations for which error bounds are indicated. A different use of the fixed point theorem is then made to determine lower and upper bounds for asymptotic approximations, and to examine their range of validity.

Keywords

FIRST-PASSAGE-TIMEDIFFUSION PROCESSESORSTEIN-UHLENBECK PROCESSESFELLER PROCESSES

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60J60: Diffusion processes 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 62E17: Approximations to distributions (nonasymptotic)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 1 , March 1996 , pp. 270 - 284
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

Research carried out under C.N.R. contracts and under M.U.R.S.T. financial support.

References

[1] Banach, S. (1922) Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Mat. 3, 133–181.Google Scholar
[2] Buonocore, A., Nobile, A. G. and Ricciardi, L. M. (1987) A new integral equation for the evaluation of the first-passage-time probability densities. Adv. Appl. Prob. 19, 784–800.Google Scholar
[3] Daniels, H. E. (1969) The minimum of stationary Markov process superimposed on a U-shaped trend. J. Appl. Prob. 6, 399–408.Google Scholar
[4] Darling, D. A. and Siegert, A. J. F. (1953) The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639.CrossRefGoogle Scholar
[5] Durbin, J. (1971) Boundary crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov theorem. J. Appl. Prob. 8, 431–453.Google Scholar
[6] Durbin, J. (1985) The first-passage density of a Gaussian process to a general boundary. J. Appl. Prob. 22, 99–122.Google Scholar
[7] Durbin, J. (1992) The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Prob. 29, 291–304.Google Scholar
[8] Fortet, R. (1943) Les fonctions aleatoires du type de Markoff associées a certaines equations linéaires aux derivées partielles du type parabolique. Math. Pures Appl. 21, 177–243.Google Scholar
[9] Giorno, V., Lansky, P., Nobile, A. G. and Ricciardi, L. M. (1988) Diffusion approximation and first-passage-time problem for a model neuron. III. A birth-and-death process approach. Biol. Cybern. 58, 387–404.Google Scholar
[10] Giorno, V., Nobile, A. G., Ricciardi, L. M. and Sato, S. (1989) On the evaluation of the first-passage-time probability densities via nonsingular integral equations. Adv. Appl. Prob. 21, 20–36.Google Scholar
[11] Hochstadt, H. (1989) Integral Equations. Wiley, New York.Google Scholar
[12] Karlin, S. and Taylor, H. M. (1981) A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
[13] Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985) A note on first-passage-time and some related problems. J. Appl. Prob. 22, 346–359.Google Scholar
[14] Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985) Exponential trends of Ornstein-Uhlenbeck first-passage-time densities. J. Appl. Prob. 22, 360–369.Google Scholar
[15] 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
[16] 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. Cybern. 35, 1–9.Google Scholar
[17] Ricciardi, L. M., Sacerdote, L. and Sato, S. (1983) Diffusion approximation and the first-passage-time problem for a model neuron. II Outline of computation method. Math. Biosci. 64, 29–44.Google Scholar
[18] 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