When is it necessary to solve Kolmogorov forward equations (KFE) for a Markov Chain?

Say I have a continuous time markov chain, time homogeneous $$X$$ with a few states (say, 2). I want to know the distribution of where $$X$$ is at time $$t$$, call it $$\mu_t$$, which will be a vector of 2 elements since there are two states. From the book Grimmet and Stirzaker, we have the Chapman Kolmogorov equations:

$$\mu_{t+s}=\mu_t*P_s, \forall s,t \ge 0$$

where $$P_s$$ is the infinitesimal probability transition matrix at time $$s$$.

I know we also have the Kolmogorov forward and Kolmogorov backward equations which tell us how $$P_t$$ evolves with time.

I'd like to when it is necessary to use the KFE equations. In every Markov Chain problem, we are given either:

• The probability transition matrix, $$P_t$$ explicitly, OR
• The generator matrix $$G$$

IF I have the generator matrix $$G$$ then like it shows on page 258, I can easily get the $$P_t$$ matrix by $$p_ii=1+g_{ii}*h+o(h)$$ and $$p_ij=g_{ij}*h+o(h)$$ (as shown on page 258)

and if I have $$P_t$$ matrix, then I can always use the Chapman-Kolmogorov equation in order to get $$\mu_t$$ at an instant in time $$t$$: $$\mu_{t}=\mu_0*P_t$$ (as I showed above). Is that correct?

Then, given $$\mu_0$$, the initial state distribution, I don’t see any reason to solve the KFE equations. When would I need to solve them?

• You have tagged this as stochastic-pde. Is there a stochastic-pde somewhere in your question? – user608030 Dec 30 '18 at 1:49