# How does integrating the Kolmogorov forward equation give $P = \exp (Qt)$?

I originally asked this question on the stats site and I recieved an answer and after looking through the answer, I had some pure maths bits I didn't understand. The main ones were integrating matrices, which is what this question is about.

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.

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Would you be confusing the continuous time setting and the discrete time setting? In continuous time, $X(i+1)$ is not $X(i)(I+Q)$. –  Did Jan 19 '13 at 11:49
@did Tbh I didn't really know how to solve the equation in the first place so that matrix I got was from the answer in CV. I don't get how you would integrate this in the first place.. –  Kaish Jan 19 '13 at 20:30
It is difficult to imagine a more considerate and more detailed answer than the one in CV. If you did not get it, you must be lacking some notions well before the introduction of Markov process in continuous time. To get adapted answers, you might want to describe your background in stochastics. –  Did Jan 19 '13 at 22:57
@did This module I'm doing it in is the first time I've ever done any thing to do with stochastics (as far as I'm aware) –  Kaish Jan 20 '13 at 11:02
Translation: background in stochastics = zero? –  Did Jan 20 '13 at 11:05