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