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Calculating extinction probabilities for the birth and death chain in a random environment

Published online by Cambridge University Press:  14 July 2016

William C. Torrez
William C. Torrez*
Affiliation:
New Mexico State University
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Abstract

In a previous investigation (Torrez (1978)) conditions were given for extinction and instability of a stochastic process (Zn) evolving in a random environment controlled by an irreducible Markov chain (Yn) with state space 𝒴 The process (Yn, Zn) is Markovian with state space 𝒴 × {0,1, ···, N} where 𝒴 = {1,· ··,m} and the marginal process (Zn) is a birth and death chain on {0,1,· ··,N}, with 0 and N made absorbing, when conditioned on a fixed sequence of environmental states of (Yn). This paper provides bivariate finite difference methods for calculating (i) P(Zn → 0) when this probability is not one; and (ii) the expected duration of the process Zn. For (i), the cases when the transition probabilities of the (Yn)-conditioned process (Zn) are non-homogeneous and homogeneous are considered separately. Examples are given to illustrate these methods.

Keywords

BIRTH AND DEATH CHAINEXTINCTION PROBABILITYRANDOM ENVIRONMENTBIVARIATE FINITE DIFFERENCE EQUATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 4 , December 1979 , pp. 709 - 720
Copyright
Copyright © Applied Probability Trust 1979 

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References

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