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Sufficient Conditions for a First Passage Time Process to be that of Brownian Motion

Published online by Cambridge University Press:  14 July 2016

M.T. Wasan
M.T. Wasan*
Affiliation:
Queen's University, Kingston, Ontario
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Extract

In this paper we assign a set of conditions to a strong Markov process and arrive at a differential equation analogous to the Kolmogorov equation. However, in this case the duration variable is the net distance travelled and the state variable is a time, a situation precisely opposite to that of Brownian motion. Solving this differential equation under certain boundary conditions produces the density function of the first passage times of Brownian motion with positive drift (see [1]), with the aid of which we define a new stochastic process.

Type
Short Communications
Information
Journal of Applied Probability , Volume 6 , Issue 1 , April 1969 , pp. 218 - 223
Copyright
Copyright © Applied Probability Trust 

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References

[1] Bartlett, M. S. (1966) An Introduction to Stochastic Processes. Cambridge University Press.Google Scholar
[2] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. John Wiley and Sons, New York.Google Scholar
[3] Tucker, H. G. (1967) A Graduate Course in Probability. Academic Press, New York.Google Scholar
[4] Tweedie, M. C. K. (1957) Statistical properties of inverse Gaussian distributions. Ann. Math. Statist. 28, 362–377.Google Scholar
[5] Watson, G. N. (1944) A Treatise on the Theory of Bessel Functions. Cambridge University Press.Google Scholar