Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-17T12:19:45.287Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

A. G. Di Crescenzo ,
V. Giorno ,
A. G. Nobile  and
L. M. Ricciardi
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.
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

FIRST-CROSSING TIMEABSORBING BOUNDARIESLINEAR PROCESS

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60J99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 2 , June 1995 , pp. 316 - 336
Copyright
Copyright © Applied Probability Trust 1995 

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

[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. 6–25. 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. 427–433 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, 599–611.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. 112–119. 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, 707–721.CrossRefGoogle Scholar
[7] Iyengar, S. (1985) Hitting lines with two-dimensional brownian motion. SIAM J. Appl. Math. 45, 983–989.Google Scholar
[8] Keilson, J. (1965) A review of transient behavior in regular diffusion and birth-death processes. Part II. J. Appl. Prob. 2, 405–428.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, 595–627.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