Assume that you want to create a finite state continuos time markov chain. You have intensities $\mu_{ij}(t)$ which are continuous.
Then you use the Kolmogorov backwards or forwards equations to find the probabilities $P_{ij}(t)$.
My question is. Does it then exist a probability space $(\Omega,\mathcal{F},P)$ where the process exist? And can it be assumed to be cádlág? One problem is that we don't have the probabilities at $t=0$ we only have the transtions probabilities. But can this be overcome?
I have not seen in any books that they actually prove that the process exist, they just assume it. But in practice, we start with the intensities, then get the probabilities, so we must prove that it exist? Can you please help?