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A new integral equation for the evaluation of first-passage-time probability densities

Published online by Cambridge University Press:  01 July 2016

A. Buonocore ,
A. G. Nobile  and
L. M. Ricciardi
A. Buonocore*
Affiliation:
Università di Napoli
A. G. Nobile*
Affiliation:
Università di Salerno
L. M. Ricciardi*
Affiliation:
Università di Napoli
*
∗Dipartimento di Matematica e Applicazioni, Università di Napoli, Via Mezzocannone 8, 80134 Napoli, Italy.
∗∗ Dipartimento di Informatica e Applicazioni, Facoltà di Scienze, Università di Salerno, 84100 Salerno, Italy.
∗Dipartimento di Matematica e Applicazioni, Università di Napoli, Via Mezzocannone 8, 80134 Napoli, Italy.
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Abstract

The first-passage-time p.d.f. through a time-dependent boundary for one-dimensional diffusion processes is proved to satisfy a new Volterra integral equation of the second kind involving two arbitrary continuous functions. Use of this equation is made to prove that for the Wiener and the Ornstein–Uhlenbeck processes the singularity of the kernel can be removed by a suitable choice of these functions. A simple and efficient numerical procedure for the solution of the integral equation is provided and its convergence is briefly discussed. Use of this equation is finally made to obtain closed-form expressions for first-passage-time p.d.f.'s in the case of various time-dependent boundaries.

Keywords

DIFFUSIONWIENER PROCESSORNSTEIN–UHLENBECK PROCESSVARYING BOUNDARIESDANIELS BOUNDARYVOLTERRA INTEGRAL EQUATIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 4 , December 1987 , pp. 784 - 800
Copyright
Copyright © Applied Probability Trust 1987 

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Footnotes

Research carried out under CNR-JSPS Scientific Cooperation Programme, Contract No. 84.00227.01, CNR Contract No. 85.00002.01 and under MPI support.

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