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The accuracy of the diffusion approximation to the expected time to extinction for some discrete stochastic processes

Published online by Cambridge University Press:  14 July 2016

J. Grasman  and
D. Ludwig
J. Grasman*
Affiliation:
Mathematisch Centrum, Amsterdam
D. Ludwig*
Affiliation:
University of British Columbia
*
∗ Postal address: Mathematisch Centrum, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands.
∗∗ Postal address: Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1W5.
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Abstract

Asymptotic approximations and numerical computations are used to estimate the accuracy of the diffusion approximation for the expected time to extinction for some stochastic processes. The results differ for processes with a continuant transition matrix (e.g. a birth and death process), and those with a noncontinuant transition matrix (e.g. a non-linear branching process). In the latter case, the diffusion equation does not hold near the point of exit. Consequently, high-order corrections do not result in substantial improvement over the diffusion approximation.

Keywords

BIRTH AND DEATH PROCESSDIFFUSION APPROXIMATIONLOGISTIC EQUATIONPOPULATION DYNAMICSDISCRETE MARKOV PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 2 , June 1983 , pp. 305 - 321
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

Research supported in part by NSERC, Canada, under Grant No. A-9239 and performed in the academic year 1981/82 when the first author was visiting the Department of Mathematics, University of British Columbia.

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