# Discretization of a Continuous Time Markov Chain

Let $$(X_t)_t$$ be a continuous time markov chain on a finite state space with initial distribution $$\alpha$$ and transition matrix $$A$$.

Suppose now we only observe $$X$$ at certain discrete time steps $$n \cdot t$$ for some $$t > 0$$, i.e. we define $$Y_n=X_{n\cdot t}$$ and $$Y_0$$ is distributed as $$\alpha$$. I would expect that $$Y_n$$ becomes a discrete time markov chain, and that its transition matrix (say $$B$$) can be derived from $$A$$.

Since the process $$X_s$$ for $$s\ge t$$ only depends on the state at $$X_t$$, not before time $$t$$, the state at discrete times $$X_{kt}$$ for integers $$k\ge n$$ only depends on the state $$X_{nt}$$ and not before, which is the definition of a discrete time Markov process.

If $$A_{ij}$$ is the transition intensity $$i\rightarrow j$$, define the vector $$a$$ by $$a_i=\sum_j A_{ij}$$ and the matrix $$Q_{ij}=A-D_{a}=a_{ij}-a_i\delta_{ij}$$.

Now, the transition probability $$P_{tij}$$ of $$i\rightarrow j$$ over a time period $$t$$ is $$P_t = e^{tQ} = \lim_{n\rightarrow\infty}\left(I+\frac{tQ}{n}\right)^n = I + tQ + \frac{t^2Q^2}{2!} + \cdots.$$