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On a symmetry-based constructive approach to probability densities for two-dimensional diffusion processes

Published online by Cambridge University Press:  14 July 2016

A. G. Di Crescenzo*
Affiliation:
University of Naples
V. Giorno*
Affiliation:
University of Salerno
A. G. Nobile*
Affiliation:
University of Salerno
L. M. Ricciardi*
Affiliation:
University of Naples
*
Postal address: Dipartimento di Matematica e Applicazioni, Università di Napoli ‘Federico II′, Via Cintia, 80126 Naples, Italy.
∗∗Postal address: Dipartimento di Informatica e Applicazioni, Università di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy.
∗∗Postal address: Dipartimento di Informatica e Applicazioni, Università di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy.
Postal address: Dipartimento di Matematica e Applicazioni, Università di Napoli ‘Federico II′, Via Cintia, 80126 Naples, Italy.

Abstract

The method earlier introduced for one-dimensional diffusion processes [6] is extended to obtain closed form expressions for the transition p.d.f.'s of two-dimensional diffusion processes in the presence of absorbing boundaries and for the first-crossing time p.d.f.'s through such boundaries. Use of such a method is finally made to analyse a two-dimensional linear process.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1995 

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References

[1] Abrahams, J. (1986) A survey of recent progress on level-crossing problems for random processes. In Comunications and Networks. A Survey of Recent Advances, ed. Blake, I. F. and Poor, H. V., pp. 625. Springer-Verlag, New York.Google Scholar
[2] Di Snescenzo, A., Giorno, V., Nobile, A. G. and Ricciardi, L. M. (1990) Some preliminary results on first crossing time densities for two-dimensional diffusion processes. In Cybernetics and Systems '90, ed. Trappl, R., pp. 427433 World Scientific, Singapore.Google Scholar
[3] Di Crescenzo, A., Giorno, V., Nobile, A. G. and Ricciardi, L. M. (1991) On the reduction to one dimension of first-passage-time problems for diffusion processes. J. Math. Phys. Sci. 25, 599611.Google Scholar
[4] Gardiner, C. W. (1985) Handbook of Stochastic Methods. Springer-Verlag, Berlin.Google Scholar
[5] Giorno, V Di Crescenzo, A Nobile, A G and Ricciardi, L. M. (1989) First-crossing time problem diffusion processes in ℝ through particular closed curves. In 6th European Young Statisticians Meeting, ed. Hála, M. and Malí, M., pp. 112119. Charles University, Prague.Google Scholar
[6] Giorno, V., Nobile, A. G. and Ricciardi, L. M. (1989) A symmetry-based constructive approach to probability densities for one-dimensional diffusion processes. J. Appl. Prob. 27, 707721.CrossRefGoogle Scholar
[7] Iyengar, S. (1985) Hitting lines with two-dimensional brownian motion. SIAM J. Appl. Math. 45, 983989.Google Scholar
[8] Keilson, J. (1965) A review of transient behavior in regular diffusion and birth-death processes. Part II. J. Appl. Prob. 2, 405428.Google Scholar
[9] Naeh, T., Klosek, M. M., Matkowsky, B. J. and Schuss, Z. (1990) A direct approach to the exit problem. SIAM J. Appl. Math. 50, 595627.Google Scholar
[10] Soong, T. T. (1973) Random Differential Equations in Science and Engineering. Academic Press, New York.Google Scholar
[11] Wong, E. (1971) Stochastic Processes in Information and Dynamical Systems. McGraw-Hill, New York.Google Scholar