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We justify and discuss expressions for joint lower and upper expectations inimprecise probability trees, in terms of the sub- and supermartingales that canbe associated with such trees. These imprecise probability trees can be seen asdiscrete-time stochastic processes with finite state sets and transitionprobabilities that are imprecise, in the sense that they are only known tobelong to some convex closed set of probability measures. We derive variousproperties for their joint lower and upper expectations, and in particular alaw of iterated expectations. We then focus on the special case of impreciseMarkov chains, investigate their Markov and stationarity properties, and usethese, by way of an example, to derive a system of non-linear equations forlower and upper expected transition and return times. Most importantly, weprove a game-theoretic version of the strong law of large numbers forsubmartingale differences in imprecise probability trees, and use this toderive point-wise ergodic theorems for imprecise Markov chains.

A study of imprecise Markov chains: joint lower expectations and point-wise ergodic theorems