Let
$$T = \left(\begin{matrix} -2 & 1 & 1&0 \\
2 & -3 & 1&0 \\
1 & 2 & -4 & 1\\
1 & 3 & 1 & -5\end{matrix} \right) $$
be a transition rate matrix of an imbedded discrete time Markov chain with state space $S=\{0,1,2,3\}$ of a continuous time Markov chain $X$ . Then how do we get the transition probability matrix $P$ of $X$ by using $T$?
Best Answer
Before answering the question, it seems necessary to dispel some misconceptions:
The matrix $T$ is the generator of a continuous time Markov chain hence, by definition, $$P_t=\mathrm e^{tT}=\sum_{n\geqslant0}\frac{t^n}{n!}T^n.$$ A general approach to compute each $P_t$ is to diagonalize the symmetric matrix $T$ as $T=P^{-1}DP$ for some diagonal matrix $D$ with entries $\lambda_i$ the eigenvalues of $T$, then $\mathrm e^{tD}$ is diagonal with entries $\mathrm e^{t\lambda_i}$ and $$P_t=\mathrm e^{tT}=P^{-1}\mathrm e^{tD}P.$$