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What are the difference and relation between a Markov jump process and a continuous-time discrete-state Markov process?
By "a continuous-time discrete-state Markov process", I understand it same as a continuous-time Markov chain.
Is a "jump process" here same concept as a discrete-state process?
Thanks and regards!
Presumably because not all jump processes are not Markov jump processes.
Also jump processes do not have discrete space. Take a compound Poisson process, for example, that is a process for which jumps happen at a fixed rate $\lambda$, but the jump distribution is not a constant 1, but instead can be a distribution (which may be continuous), therefore the space is not discrete.
Also there are also jump processes, which has independent increment property (and markov property) but are not compound Poisson process. These processes, with Brownian motions are the only processes with independent increment properties. they are called Levy processes.
Notice jump continuous-time discrete space and compound Poisson process has finite variation, but this is not necessarily the case with general Levy processes (even without the Brownian motion component), which are pure jump processes.
Of course, this is only a small class of Markovian jump processes. I am sure there are plenty of others which I have not mentioned here. Also there are markov processes for which the transition probability is time dependent, but these won't have the Q matrices as generator which you might have had in mind.
