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After answering [this question] (Expectation of a stopping time uniquely determined by a function) I was looking for the literature on the mean hitting/exit time for a discrete-time Markov process.

In Meyn and Tweedie, Durett and some other books I haven't found much information. Google also didn't help much for the discrete-time setting.

I guess that for the uncountable state space, mean hitting time equation shares the same properties as for the countable state space. However, I would prefer to read a book chapter on this topic.

Dynkin briefly discussed it, though for the continuous time and too briefly.

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This book was written by a professor from the university that I study:

https://www.scribd.com/document/237030386/Ferrari-e-Galves-Construction-of-Stochastic-Processes-Coupling-and-Regeneration

I'm quite sure it covers the topic you are looking for.