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Solutions for some diffusion processes with two barriers

Published online by Cambridge University Press:  14 July 2016

A. L. Sweet  and
J. C. Hardin
A. L. Sweet
Affiliation:
Purdue University, Lafayette, Indiana
J. C. Hardin
Affiliation:
NASA Langley Research Center, Hampton, Virginia
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Extract

Use is often made of the Wiener and Ornstein-Uhlenbeck (O.U.) processes in various applications of stochastic processes to problems of engineering interest. These applications frequently involve the presence of barriers. Although mathematical methods for solving Kolmogorov's forward equation for the above processes have previously been discussed ([1], [2]), many solutions for problems with two barriers do not seem to be available in the literature. Instead, one finds solutions for unrestricted processes or simulation used in place of analytical solutions in various applications ([3], [4], [5]). In this paper, solutions of Kolmogorov's forward equations in the presence of constant absorbing and/or reflecting barriers are obtained by means of separation of variables. This enables one to obtain expressions for the probability density functions for first passage times when absorbing barriers are present. The solution for the O.U. process is used to obtain a result of Breiman's [6] concerning first passage times.

Type
Research Papers
Information
Journal of Applied Probability , Volume 7 , Issue 2 , August 1970 , pp. 423 - 431
Copyright
Copyright © Applied Probability Trust 1970 

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