I have a question about the optimal stop. Is it possible to give an example of the problem of optimal stopping on a Markov chain with a countable number of states and discrete time in which the optimal stopping time does not exist? It seems to me that such an example should exist, but I cannot invent it.
2026-03-31 01:08:39.1774919319
Example where optimal stopping does not exist
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Let $X_n$ be any Markov chain. Let the reward for stopping after $n$ steps be $1 - 1/n$.