RandomFunction on Markov or Transformed processes

Questions
1

I am unable to grasp the behaviour of RandomFunction on nontrivial processes.

Attached is a screenshot of my attempts of using it with two invocations (providing an explicit dt, and not providing it) on several different processes: first a Poisson process (on which it works fine), then a transformed Poisson, a continuous and a discrete Markov (one of the invocations fails on each of those), and finally on combinations of Poisson with both of the Markov (discrete and continuous), on which I am simply unable to make it work.

Are these calls not implemented yet? Or am I doing something wrong?

2

This is a bit strange. But treat the TransformedProcess as a continuous process. The documentation for RandomFunction says that a continuous process needs a step size:

RandomFunction[Ppoisson2, {0, 10, 0.1}]

The problem is, once a process has been transformed, it’s not exactly clear if it is a continuous process or not. It’s basically treated as a continuous process.

Also, when posting, please try to post a small code example instead of a picture. It makes it easier to try to look and play around with the problem.