From wikipedia, we define a Markov kernel as:
Now, obviously, for cases where the state space is discrete, we can construct the CTMC's Rate transition matrix. In a more general setting (say, for example, where the state space is not discrete), if one is able to define a Markov kernel between the spaces of discrete times $\{n, n+1\}$ , does this Markov kernel suffice for all continuous time processes?
Minimal working example: consider the process $(X,A)_{t}$ where $X$ is a Poisson process and $A$ is a timer which displays how long the Poisson process has been in its state. Clearly the state space is of form $(\mathbb{N},[0,\infty))$
In my case, I have it (like above) such that only a finite amount amount of things change over my non-discrete state space in infinitismally small time ($[t,t+\Delta]$) – if my kernel captures this, is it sufficient?
Best Answer
I have found a sufficient answer to my problem: Piecewise Deterministic Markov Processes. For more information: https://en.wikipedia.org/wiki/Piecewise-deterministic_Markov_process.