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Difference between non-homogeneous (time in-homogeneous) Markov and (time) homogeneous semi-Markov?
This point bugged me a big time. I worked with Markov family and got bit confused around this point. I was able to clear this confusion with the help of brilliant answer by a guy called Christian Furrer (PhD in Actuarial Mathematics). So as per mine and a general understanding, to satisfy Markov property, state holding time distribution needs to be exponential otherwise the process can’t satisfy the Markov property. The only way around this exponential distribution seemed to be semi-Markov processes (duration times can be arbitrary). In semi-Markov, state holding time modulates transition probabilities and in turn "time spent in a state affects the decision which state to enter next". Now non-homogeneous (time in-homogeneous) Markov processes have duration times that are not necessarily exponential. If that is the case that what are differences in general between time in-homogeneous simple Markov and time homogeneous semi-Markov?
Explanation by Furrer: