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

TL;DR

This paper develops a diffusion approximation for a repeated betting game, analyzing strategies that sustain positive wealth share over time and characterizing conditions for wealth dynamics in both two-player and regime-switching scenarios.

Contribution

It introduces a diffusion model for repeated betting with asymptotically optimal strategies and provides conditions for wealth persistence and recurrence in complex market settings.

Findings

Strategies can maintain positive wealth share indefinitely.

Necessary and sufficient conditions for wealth process transience or recurrence.

Extension of results to Markovian regime switching models.

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.