Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T05:37:43.245Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the Volterra integral equation for the first-passage-time probability density

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

R. Gutiérrez Jáimez ,
P. Román Román  and
F. Torres Ruiz
R. Gutiérrez Jáimez*
Affiliation:
Universidad de Granada
P. Román Román*
Affiliation:
Universidad de Granada
F. Torres Ruiz*
Affiliation:
Universidad de Granada
*
∗Postal address: Departamento de Estadística e Investigación Operativa, Universidad de Granada, Avda. Fuentenueva s/n. 18071, Granada, Spain.
∗Postal address: Departamento de Estadística e Investigación Operativa, Universidad de Granada, Avda. Fuentenueva s/n. 18071, Granada, Spain.
∗Postal address: Departamento de Estadística e Investigación Operativa, Universidad de Granada, Avda. Fuentenueva s/n. 18071, Granada, Spain.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we prove the validity of the Volterra integral equation for the evaluation of first-passage-time probability densities through varying boundaries, given by Buonocore et al. [1], for the case of diffusion processes not necessarily time-homogeneous. We study, specifically those processes that can be obtained from the Wiener process in the sense of [5]. A study of the kernel of the integral equation, in the same way as that by Buonocore et al. [1], is done. We obtain the boundaries for which closed-form solutions of the integral equation, without having to solve the equation, can be obtained. Finally, a few examples are given to indicate the actual use of our method.

Keywords

TIME-NON-HOMOGENEOUS DIFFUSION PROCESSES: FIRST-PASSAGE-TIME: VARYING BOUNDARIES

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 60J65: Brownian motion
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 3 , September 1995 , pp. 635 - 648
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] Buonocore, A., Nobile, A. G. and Ricciardi, L. (1987) A new integral equation for the evaluation of first-passage-time probability densities. Adv. Appl. Prob. 19, 784–800.CrossRefGoogle Scholar
[2] Cherkasov, I. D. (1957) On the transformation of the diffusion process to a Wiener process. Theory Prob. Appl. 2, 373–377.CrossRefGoogle Scholar
[3] Giorno, V., Nobile, A. G., Ricciardi, L. and Sato, S. (1989) On the evaluation of first-passage-time probability densities via non singular integral equations. Adv. Appl. Prob. 21, 20–36.CrossRefGoogle Scholar
[4] Gutiérrez, R., De Juan, A. and Román, P. (1991) Construction of first-passage-time densities for a diffusion process which is not necessarily time-homogeneous. J. Appl. Prob. 28, 903–909.Google Scholar
[5] Ricciardi, L. (1976) On the transformation of diffusion processes into the Wiener Process. J. Math. Anal. Appl. 54, 185–199.CrossRefGoogle Scholar
[6] Ricciardi, L. and Sato, S. (1983) A note on the evaluation of first-passage-time probability densities. J. Appl. Prob. 20, 197–201.CrossRefGoogle Scholar
[7] Ricciardi, L., Sacerdote, L. and Sato, S. (1984) On an integral equation for first-passage-time probability densities. J. Appl. Prob. 21, 302–314.CrossRefGoogle Scholar
[8] Tintner, G. and Sengupta, J. K. (1972) Stochastic Economics. Academic Press, New York.CrossRefGoogle Scholar