On positively divisible non-Markovian processes

Abstract

There are some positively divisible non-Markovian processes whose transition matrices satisfy the Chapman-Kolmogorov equation. These processes should also satisfy the Kolmogorov consistency conditions, an essential requirement for a process to be classified as a stochastic process. Combining the Kolmogorov consistency conditions with the Chapman-Kolmogorov equation, we derive a necessary condition for positively divisible stochastic processes on a finite sample space. This necessary condition enables a systematic approach to the manipulation of certain Markov processes in order to obtain a positively divisible non-Markovian process. We illustrate this idea by an example and, in addition, analyze a classic example given by Feller in the light of our approach.

Figures (1)

• Figure 1: Realizations of the Markov process $X(T)$ with the joint probability distribution in equation (\ref{['Two state Markov process']}), and the non-Markovian process $\tilde{X}(T)$ with the joint probability distribution in equation (\ref{['Two state non-Markovian process']}) for different values of the parameter $\varepsilon$. The probability vectors $\mathbf{p}(t_{3k-2})$ have been chosen as $(0.25,0.75)^T$ for all $k=1,2,3,\ldots$. The first two time steps of the realizations have been generated using the same random numbers since there is no difference between the Markovian and the non-Markovian process for the first two time steps. From these realizations, one finds the following numerical values: $\mu = 0.575, \tilde{\mu}=0.585$ (for $\varepsilon =0.75$), $\tilde{\mu}=0.580$ (for $\varepsilon = 1$), $\rm{Var}(X)=0.022, \rm{Var}(\tilde{X})=0.039$ (for $\varepsilon =0.75$), $\rm{Var}(\tilde{X})=0.050$ (for $\varepsilon =1$), which are in accordance with the analytical predictions within statistical error.

Theorems & Definitions (3)

• Definition 1
• Corollary 1
• Remark 1