(A) sample path approach to absorbing stochastic processes

Abstract

If we assume the future status of some phenomena depends only on that of the present, not that of the past history, we can model and analyze the phenomena easily. We call this assumption the $\emph{Markov property}$ (or the $\emph {memoryless property}$) and this property plays a critical role in many fields that need stochastic modeling such as: economics, finance, manufacturing, inventory control, mathematical genetics, social mobility problem including manpower planning, telephone switches, computer and telecommunication networks. Stochastic processes can have one of transient state or recurrent state. If when the stochastic process visits a recurrent state and it remains there forever, we call it the absorbing state. About the forward process of the absorbing stochastic processes, there have been much research since the work of Kemeny and Snell, 1976 (the reprint of the 1960 original). The researchers of population genetics have studied the backward process (or the time-reversed process) of the absorbing stochastic processes partly to find the age distribution of an allele among the population. The methodologies that have been used in this area, however, are all unique ones applicable only in their own purposes. In this study, we present the unifying analyzing tool for the reversed absorbing discrete-time Markov chain. The main methodology used is ``the joint probability of the sample path``. We take a sample path approach to developing a general structure of the reversed absorbing Markov chain. The sample path approach is a fundamental method to compute probabilities: We properly define a sample space consisting of sample paths by taking relevant random variables into consideration and then compute necessary marginal and conditional probabilities using the joint probabilities of the sample paths. So the sample path approach itself is not new, but the results thus obtained are simple and straightforward. Moreover, it is a unifying method since the sam...