Likely path to extinction in simple branching models
         with large initial population

Klebaner, F. C.; Liptser, R.

doi:https://doi.org/10.1155/JAMSA/2006/60376

International Journal of Stochastic Analysis

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Volume 2006 | Article ID 060376 | https://doi.org/10.1155/JAMSA/2006/60376

Likely path to extinction in simple branching models with large initial population

F. C. Klebaner¹and R. Liptser¹

Received15 Sept 2005

Revised24 Nov 2005

Accepted04 Dec 2005

Published14 May 2006

Abstract

We give explicit formulae for most likely paths to extinction in simple branching models when initial population is large. In discrete time, we study the Galton-Watson process, and in continuous time, the branching diffusion. The most likely paths are found with the help of the large deviation principle (LDP). We also find asymptotics for the extinction probability, which gives a new expression in continuous time and recovers the known formula in discrete time. Due to the nonnegativity of the processes, the proof of LDP at the point of extinction uses a nonstandard argument of independent interest.

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Copyright

Copyright © 2006 F. C. Klebaner and R. Liptser. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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