'Ergodicity' was solved by Bellman with multi-stage sequential decision problems. There's no breakthrough here. If you leave out key parts of the problem like bankruptcy or finite capital or opportunity cost, surprise surprise, naive greedy expected value maximization gives you the wrong answer.
"Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?"
"Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?"