Relatively growth optimal investment strategies in a market model with competition
We consider a game-theoretic model of a market where investors compete for payoffs yielded by several assets. The main result consists in a proof of existence and uniqueness of a strategy, called relatively growth optimal, such that the logarithm of the share of its wealth in the total wealth of the market is a submartingale for any strategies of the other investors. It is also shown that this strategy is asymptotically optimal in the sense that it achieves the maximal capital growth rate when compared to competing strategies. Based on the obtained results, we study the asymptotic structure of the market when all the investors use the relatively growth optimal strategy.