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

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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.