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On the inverse of the first passage time probability problem

Published online by Cambridge University Press:  14 July 2016

R. M. Capocelli  and
L. M. Ricciardi
R. M. Capocelli
Affiliation:
Laboratorio di Cibernetica del C. N. R., Arco Felice, Naples, Italy
L. M. Ricciardi
Affiliation:
University of Chicago
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Abstract

Since the pioneering work of Siegert (1951), the problem of determining the first passage time distribution for a preassigned continuous and time homogeneous Markov process described by a diffusion equation has been deeply analyzed and satisfactorily solved. Here we discuss the “inverse problem” — of applicative interest — consisting in deciding whether a given function can be considered as the first passage time probability density function for some continuous and homogeneous Markov diffusion process. A constructive criterion is proposed, and some examples are provided. One of these leads to a singular diffusion equation representing a dynamical model for the genesis of the lognormal distribution.

Keywords

DIFFUSION PROCESSESFIRST PASSAGE TIMESINGULAR DIFFUSION EQUATIONSlognormal distribution
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 2 , June 1972 , pp. 270 - 287
Copyright
Copyright © Applied Probability Trust 1972 

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