Asymptotically optimal strategies in a diffusion approximation of a repeated betting game

Abstract: 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.