[soVeryTired]

Poisson processes are continuous time though. If you're interested in Markov chains you only need the discrete-time theory.

In discrete time and discrete space, it mostly just reduces to linear algebra.

[graycat]

No, can do continuous time discrete state space theory -- the jumps in the discrete state space are at the arrival times of the Poisson process -- that works out easily enough, especially if using Monte-Carlo. See my other post here on Red/Blue stuff.

[usgroup]

Unless you’re interested in continuous markov chains, infinite state spaces, renewal theory, excessive functions, and so on.