Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-28T00:57:22.278Z Has data issue: false hasContentIssue false
  • English
  • Français

Some remarks on the Rayleigh process

Published online by Cambridge University Press:  14 July 2016

V. Giorno ,
A. G. Nobile ,
L. M. Ricciardi  and
L. Sacerdote
V. Giorno*
Affiliation:
University of Salerno
A. G. Nobile*
Affiliation:
University of Salerno
L. M. Ricciardi*
Affiliation:
University of Naples
L. Sacerdote*
Affiliation:
University of Turin
*
∗Postal address: Dipartimento di Informatica e Applicazioni, Facoltà de Scienze, Università, 84100 Salerno, Italy.
∗Postal address: Dipartimento di Informatica e Applicazioni, Facoltà de Scienze, Università, 84100 Salerno, Italy.
∗∗Postal address: Dipartimento di Matematica e Applicazioni, Università di Napoli, Via Mezzocannone 8, 80134 Napoli, Italy.
∗∗∗Postal address: Dipartimento di Matematica, Università di Torino, Via Principe Amadeo 8, 10123 Torino, Italy.
Get access Rights & Permissions [Opens in a new window]

Abstract

The transition p.d.f. for a one-dimensional Rayleigh process in the presence of an absorption condition or a zero-flux condition in the origin is obtained in closed form. The first-passage-time problem through an arbitrary constant boundary is then considered and the moment-generating function is determined. In some particular cases the first-passage-time p.d.f. is explicitly derived. Use of some of these results is finally made to obtain the transition p.d.f. of the affine drift-linear infinitesimal-variance diffusion process when the origin is an entrance or a regular boundary in the presence of a reflection condition.

Keywords

BESSEL PROCESSFIRST-PASSAGE TIMECONTINUOUS-STATE BRANCHINGDIFFUSION
Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 2 , June 1986 , pp. 398 - 408
Copyright
Copyright © Applied Probability Trust 1986 

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–JNPS Scientific Cooperation Programme, Contracts 83.00032.01 and 84.00227.01, and under MPI support.

References

[1] Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1953) Higher Transcendental Functions, Vol. 1. McGraw-Hill, New York.Google Scholar
[2] Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1953) Higher Transcendental Functions, Vol. 2. McGraw-Hill, New York.Google Scholar
[3] Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1954) Tables of Integral Transforms, Vol. 1. McGraw-Hill, New York.Google Scholar
[4] Feller, W. (1951) Two singular diffusion processes. Ann. Math. 54, 173–182.Google Scholar
[5] Gradshteyn, I. S. and Ryzhik, I. M. (1965) Tables of Integrals, Series and Products. Academic Press, New York.Google Scholar
[6] Ito, K. and Mckean, H. P. (1965) Diffusion Processes and their Sample Paths. Academic Press, New York.Google Scholar
[7] Karlin, S. and Taylor, H. W. (1981) A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
[8] Kent, J. (1978) Some probabilistic properties of Bessel functions. Ann. Prob. 6, 760–770.CrossRefGoogle Scholar
[9] Stratonovich, R. L. (1963) Topics in the Theory of Random Noise, Vol. 1. Gordon and Breach, New York.Google Scholar
[10] Tricomi, F. G. (1954) Funzioni Ipergeometriche Confluenti. Cremonese, Roma.Google Scholar