Hostname: page-component-7dd5485656-gs9qr Total loading time: 0 Render date: 2025-10-31T17:13:26.922Z Has data issue: false hasContentIssue false
  • English
  • Français

On the two-boundary first-crossing-time problem for diffusion processes

Published online by Cambridge University Press:  14 July 2016

A. Buonocore ,
V. Giorno ,
A. G. Nobile  and
L. M. Ricciardi
A. Buonocore*
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, University of Naples, Via Mezzocannone 8, 80134 Naples, Italy.
∗∗ Postal address: Dipartimento di Informatica e Applicazioni, University of Salerno, 84100 Salerno, Italy.
∗∗ Postal address: Dipartimento di Informatica e Applicazioni, University of Salerno, 84100 Salerno, Italy.
∗ Postal address: Dipartimento di Matematica e Applicazioni, University of Naples, Via Mezzocannone 8, 80134 Naples, Italy.
Get access Rights & Permissions [Opens in a new window]

Abstract

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.'s from below and from above are proved to satisfy a new system of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such functions a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Finally, conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single integral equation for the first-crossing-time p.d.f.

Keywords

VARYING BOUNDARIESVOLTERRA INTEGRAL EQUATIONS

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 1 , March 1990 , pp. 102 - 114
Copyright
Copyright © Applied Probability Trust 1990 

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.)

Article purchase

Temporarily unavailable

References

[1] Blake, I. F. and Lindsey, W. C. (1973) Level crossing problems for random processes. IEEE Trans. Inf. Theory 19, 295–315.CrossRefGoogle 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] 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
[4] Darling, D. A. and Siegert, A. J. F. (1953) The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639.Google Scholar
[5] Durbin, J. (1971) Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. J. Appl. Prob. 8, 431–453.Google Scholar
[6] Fortet, R. (1943) Les fonctions aléatoires du type de Markoff associées à certaines équations linéaires aux dérivées partielles du type parabolique. J. Math. Pures Appl. 22, 177–243.Google Scholar
[7] Giorno, V., Nobile, A. G., Ricciardi, L. M. and Sato, S. (1989) On the evaluation of first-passage-time probability densities via non-singular integral equations. Adv. Appl. Prob. 21, 20–36.Google Scholar
[8] 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. 26, 707–721.Google Scholar
[9] Tricomi, F. G. (1970) Integral Equations. Interscience, New York.Google Scholar