  We construct a diffusion approximation of a repeated game in which agents
make bets on outcomes of i.i.d. random vectors and their strategies are close
to an asymptotically optimal strategy. This model can be interpreted as trading
in an asset market with short-lived assets. We obtain sufficient conditions for
a strategy to maintain a strictly positive share of total wealth over the
infinite time horizon. For the game with two players, we find necessary and
sufficient conditions for the wealth share process to be transient or recurrent
in this model, and also in its generalization with Markovian regime switching.
