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A symmetry-based constructive approach to probability densities for one-dimensional diffusion processes

Published online by Cambridge University Press:  14 July 2016

V. Giorno ,
A. G. Nobile  and
L. M. Ricciardi
V. Giorno*
Affiliation:
University of Salerno
A. G. Nobile*
Affiliation:
University of Salerno
L. M. Ricciardi*
Affiliation:
University of Naples
*
∗ Postal address: Dipartimento di Informatica e Applicazioni, Università di Salerno, Salerno, Italy.
∗ Postal address: Dipartimento di Informatica e Applicazioni, Università di Salerno, Salerno, Italy.
∗∗ Postal address: Dipartimento di Matematica e Applicazioni, Università di Napoli, Via Mezzocannone 8, 80134 Napoli, Italy.
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Abstract

Special symmetry conditions on the transition p.d.f. of one-dimensional time-homogeneous diffusion processes with natural boundaries are investigated and exploited to derive closed-form results concerning the transition p.d.f.'s in the presence of absorbing and reflecting boundaries and the first-passage-time p.d.f. through time-dependent boundaries.

Keywords

TIME-HOMOGENEOUS DIFFUSIONSVARYING BOUNDARIESREFLECTIONABSORBTIONFIRST-PASSAGE TIME
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 4 , December 1989 , pp. 707 - 721
Copyright
Copyright © Applied Probability Trust 1989 

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References

[1] Blake, I. F. and Lindsey, W. C. (1973) Level crossing problems for random processes. IEEE Trans. Inf. Theory 19, 295–315.Google Scholar
[2] Buonocore, A., Nobile, A. G. and Ricciardi, L. M. (1987) A new integral equation for the evaluation of first-passage-time probability densities. Adv. Appl. Prob. 19, 784–800.Google Scholar
[3] Cox, D. R. and Miller, H. D. (1970) The Theory of Stochastic Processes. Methuen, London.Google Scholar
[4] Daniels, H. E. (1969) The minimum of a stationary Markov process superimposed on a U-shaped trend. J. Appl. Prob. 6, 399–408.Google Scholar
[5] Feller, W. (1952) The parabolic differential equations and the associated semigroup-transformations. Ann. Math. 55, 468–518.CrossRefGoogle Scholar
[6] Giorno, V., Lánský, 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. Cybernetics 58, 387–404.CrossRefGoogle Scholar
[7] Giorno, V., Nobile, A. G. and Ricciardi, L. M. (1986) On some diffusion approximations to queueing systems. Adv. Appl. Prob. 18, 991–1014.Google Scholar
[8] Giorno, V., Nobile, A. G. and Ricciardi, L. M. (1987) On some time-non-homogeneous diffusion approximations to queueing systems. Adv. Appl. Prob. 19, 974–994.Google Scholar
[9] Giorno, V., Nobile, A. G. and Ricciardi, L. M. (1988) A new approach to the construction of first-passage-time densities. In Cybernetics and Systems 1988, ed. Trappl, R., Kluwer Academic Publishers, Dordrecht, 375–381.Google Scholar
[10] Giorno, V., Nobile, A. G., Ricciardi, L. M. and Sato, S. (1989) On the evaluation of firstpassage-time probability densities via nonsingular integral equations. Adv. Appl. Prob. 21, 20–36.Google Scholar
[11] Heath, R. A. (1981) A tandem random walk model for psychological discrimination. Br. J. Math. Statist. Psychol. 34, 76–92.Google Scholar
[12] Hongler, M. O. (1979) Exact solutions of a class of non-linear Fokker–Planck equations. Phys. Lett. 75A, 3–4.CrossRefGoogle Scholar
[13] Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
[14] Lánský, P. and Lánská, V. (1987) Diffusion approximations of the neuronal model with synaptic reversal potentials. Biol. Cybernetics 56, 19–26.Google Scholar
[15] Maruyama, T. (1977) Stochastic Problems in Population Genetics. Lecture Notes in Biomathematics 17, Springer-Verlag, Berlin.Google Scholar
[16] Nobile, A. G. and Ricciardi, L. M. (1984) Growth with regulation in fluctuating environments. I. Alternative logistic-like diffusion models. Biol. Cybernetics 49, 179–188.Google Scholar
[17] Nobile, A. G. and Ricciardi, L. M. (1984) Growth with regulation in fluctuating environments. II. Intrinsic lower bounds to population size. Biol. Cybernetics 50, 285–299.CrossRefGoogle ScholarPubMed
[18] Ricciardi, L. M. (1986) Stochastic population theory: Diffusion processes. In Mathematical Ecology, ed. Hallam, T. G. and Levin, S. Springer-Verlag, Berlin, 191–238.Google Scholar
[19] Tuckwell, H. C. (1988) Nonlinear random reaction-diffusion systems. In Biomathematics and Related Computational Problems, ed. Ricciardi, L. M., Kluwer Academic Publishers, Dordrecht, 581–590.Google Scholar