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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.
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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 

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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