A note on the characterization of optimal return functions and optimal strategies for gambling problems (Q1062954)

The author considers a gambling problem with a finite set of states and with a bounded utility function. \textit{L. E. Dubins} and \textit{L. J. Savage} [Inequalities for stochastic processes. How to gamble if you must (1976; Zbl 0359.60002)] investigated this game with the lim sup payoff. They showed that the optimal, thrifty and equalizing strategies have some characterization connecting them to each other. They also gave a characterization of the optimal return function. The author shows that all the properties can be reformulated for stationary strategies. It is also pointed out that the same is true if the lim sup payoff is changed to lim inf payoff.