If $Q$ is a generator matrix of a continuous time Markov chain (CTMC), and I need to use this matrix to solve the Kolmogorov forward equation, I would need to start by integrating it. But I haven't got a clue how to do it. Can someone show me please?
I know something like, let's assume $i$ represents the current state of a CTMC. Then, the forward equation basically tells us that we can work out $X(i + 1)$ by doing
$$X(i + 1) = X(i) \cdot (Id_2 + Q)$$
To look at the difference in time, we can subtract $X(i)$ from both sides and get
$$X(i + 1) - X(i) = X(i) Q$$
Thinking of this in terms of functions instead of matrices, we can say that this can be written as
$$ P'(t) = X(i) Q dt$$
But I don't get how you get to this bit and how you can integrate from here.
I would really appreciate any help. Thank you.