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On an integral equation for first-passage-time probability densities

Published online by Cambridge University Press:  14 July 2016

L. M. Ricciardi ,
L. Sacerdote  and
S. Sato
L. M. Ricciardi*
Affiliation:
Unioersitd di Napoli
L. Sacerdote*
Affiliation:
Unioersitd di Salerno
S. Sato*
Affiliation:
Uniuersitd di Napoli
*
∗ Postal address: Istituto di Matematica ‘Renato Caccioppoli’, Universita di Napoli, Via Mezzocannone 8, 1-80134 Napoli, Italy.
∗∗ Postal address: Istituto di Scienze dell ‘Inforrnazione, Universita de Salerno, Salerno, Italy.
∗ Postal address: Istituto di Matematica ‘Renato Caccioppoli’, Universita di Napoli, Via Mezzocannone 8, 1-80134 Napoli, Italy.
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Abstract

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ∼ 2, a, b E ∼) for which no explicit analytical results have previously been available.

Keywords

DIFFUSION PROCESSESWIENER PROCESSVARYING BOUNDARIES
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 2 , June 1984 , pp. 302 - 314
Copyright
Copyright © Applied Probability Trust 1984 

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Footnotes

†

On leave of absence from the Faculty of Engineering Science, Osaka University.

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