Hi

I have a stochastic process $\displaystyle \{X_t, t \geq 1\}$ representing a non homogeneous Markov chain. The transition matrix between $\displaystyle X_1$ and $\displaystyle X_2$ is given by

$\displaystyle \begin{pmatrix}A &B& C\cr D&E&F\cr 0&0&1

\end{pmatrix}$

From the time $\displaystyle t=2$, the transition matrix between $\displaystyle X_t$ and $\displaystyle X_{t+1}$ becomes time independent and is given by

$\displaystyle \begin{pmatrix}G&H&K\cr

L&M&N\cr 0&0&1\end{pmatrix}$

For homgeneous Markov chains, it is well know that

$\displaystyle n_{ij}=(I-Q)^{-1},$

where

$\displaystyle n_{ij}$ is the number of times the process visits the transient state $\displaystyle j$ before it eventually enters a recurrent state, having initially started from the transient state $\displaystyle i$.

$\displaystyle I$ is the identity matrix.

$\displaystyle Q$ is the transition matrix between transient states.

How can I obtain the number of visits for my case?

Can I transform the non homogeneous model to a homogeneous one?