Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 60376, 23 pages
doi:10.1155/JAMSA/2006/60376
Likely path to extinction in simple branching models
with large initial population
School of Mathematical Sciences, Monash
University, University of Sciences and Technology
Houary Boumediene, Victoria 3800, Australia
Received 15 September 2005; Revised 24 November 2005; Accepted 4 December 2005
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.
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.