Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T07:15:04.904Z Has data issue: false hasContentIssue false
  • English
  • Français

On some first-crossing-time probabilities for a two-dimensional random walk with correlated components

Part of: Stochastic processes Combinatorics

Published online by Cambridge University Press:  01 July 2016

A. Di Crescenzo ,
V. Giorno  and
A. G. Nobile
A. Di Crescenzo*
Affiliation:
University of Naples
V. Giorno*
Affiliation:
University of Salerno
A. G. Nobile*
Affiliation:
University of Udine
*
∗ Postal address: Dipartimento di Matematica e Applicazioni, University of Naples, via Cintia, 80126 Naples, Italy.
∗∗ Postal address: Dipartimento di Informatica e Applicazioni, University of Salerno, via S. Allende, 84081 Baronissi (SA), Italy. Correspondence should be addressed to this author.
∗∗∗ Postal address: Dipartimento di Matematica e Informatica, University of Udine, via Zanon 6, 33100 Udine, Italy.
Get access Rights & Permissions [Opens in a new window]

Abstract

For a two-dimensional random walk {X (n) = (X(n)1, X(n)2)T, n ∈ ℕ0} with correlated components the first-crossing-time probability problem through unit-slope straight lines x2 = x1 - r(r = 0,1) is analysed. The p.g.f.'s for the first-crossing-time probabilities are expressed as solutions of a fourth-degree algebraic equation and are then exploited to obtain the first-crossing-time probabilities. Several additional results, including the mean first-crossing time and the probability of ultimate crossing, are also given.

Keywords

UNIT-SLOPE LINESULTIMATE CROSSINGMEAN CROSSING TIME

MSC classification

Secondary: 05A15: Exact enumeration problems, generating functions 92D25: Population dynamics (general) 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 2 , June 1992 , pp. 441 - 454
Copyright
Copyright © Applied Probability Trust 1992 

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] Buonocore, A., Giorno, V., Nobile, A. G. and Ricciardi, L. M. (1990) On the two-boundary first-crossing-time problem for diffusion processes. J. Appl. Prob. 27, 102–114.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] Csàki, E., Mohanty, S. G. and Saran, J. (1990) On random walks in a plane. Ars Combinatoria 29, 309–318.Google Scholar
[4] Di Crescenzo, 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
[5] Giorno, V. and Nobile, A. G. (1988) On the distribution of the range of an asymmetric random walk. Ricerche di Matematica XXXVII, 315–324.Google Scholar
[6] Giorno, V., Nobile, A. G. and Ricciardi, L. ?. (1989) A symmetry-based constructive approach to probability densities for one-dimensional diffusion processes. J. Appl. Prob. 27, 707–721.Google Scholar
[7] Giorno, V., Nobile, A. G. and Ricciardi, L. ?. (1990) On the asymptotic behaviour of first-passage-time densities for one-dimensional diffusion processes and varying boundaries. Adv Appl. Prob. 22, 883–914.CrossRefGoogle Scholar
[8] Giorno, V., Nobile, A. G., Ricciardi, L. M. and Sato, S. (1989) On the evaluation of first-passage-time probability densities via non-singular equations. Adv. Appl. Prob. 21, 20–36.CrossRefGoogle Scholar
[9] Notari, V. (1987) Le equazioni di quarto grado ed i sistemi di due equazioni di secondo grado in due incognite. In Questioni riguardanti le matematiche elementari, raccolte e coordinate da F. Enriques. Zanichelli, Bologna.Google Scholar
[10] Ricciardi, L. M. (1977) Diffusion Processes and Related Topics in Biology. Lecture Notes in Biomathematics, 14. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[11] Vassiliou, P.-C. G. (1981) A random walk between a reflecting and a random absorbing barrier. Rev. Roum. Math. Pures Appl. XXVI, 1391–1400.Google Scholar