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Well-timed diffusion approximations

Published online by Cambridge University Press:  01 July 2016

John B. Walsh
John B. Walsh*
Affiliation:
The University of British Columbia
*
∗ Postal address: Department of Mathematics. The University of British Columbia, #121–1984 Mathematics Road, University Campus, Vancouver, B.C., Canada V6T 1Y4.
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Abstract

Let X be a Markov process on the line. Under certain conditions it is possible to find a diffusion process which is an approximation to X in the following sense:

(1) X can be embedded in that is there are stopping times (Tt ) such that {Xt , t ≥ 0} and have the same distribution;

(2) for each t, E{Tt } = t.

We call the well-timed diffusion approximation to X, and suggest that it is useful for approximating quantities like the first-exit probabilities and expected first-exit times of X.

We determine the well-timed approximation in several special cases and give an asymptotic approximation for use in cases in which cannot be exactly determined.

Keywords

DIFFUSION APPROXIMATIONTIME-CHANGEMARTINGALE EMBEDDINGS

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 2 , June 1981 , pp. 352 - 368
Copyright
Copyright © Applied Probability Trust 1981 

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