Let's say we have some Markov process $(X(t): t \ge 0)$ with state space $S = \{ 1, 2, 3 \}$. Furthermore, let's say we're also given a generator matrix
$$Q = \begin{bmatrix} -1 & 0 & 1 \\ 4 & -5 & 1 \\ 1 & 8 & -9 \end{bmatrix}$$
Now define the one-step transition matrix to be $\mathcal{P}$. I was told that we can calculate an approximate value for $\mathcal{P}(t)$ for a given value of $t$. However, I don't understand exactly what this means, let alone how it is done. What does it mean for one to calculate an approximate value for $\mathcal{P}(t)$ for a given value of $t$, and how is it done?
The generator $Q$ is for a continuus-time process. The Matrix $P(t)$ is for a single discrete step of size $t$. They are related by the differential equation $$ P'(t)=P(t)Q$$ with starting value $P(0)=I$ (the identity matrix). In order to see why this equation makes sense you should imagine a small time-increment $\delta t$ where you can assume that at most one transition takes place in the time $\delta t$.
Anyway, the solution of the differential equation is exactly $$ P(t) = \exp(tQ)=I+tQ+\frac{1}{2}t^2Q^2+...$$ take as many terms as you like (or even compute it exactly. There are many methods of computing matrix exponentials)
edit: sry @E-A, your comment makes this answer unneccessary^^, I was too slow