Asymptotic distribution of capital in a market model with competition

We consider a stochastic game-theoretic model of an investment market with continuous time, where investors compete for the distribution of income from several assets. The main results are related to the question in what proportions the total capital will be distributed among the investors, and what will be the prices of the assets asymptotically on the infinite time horizon depending on the strategies of the investors. We prove that there exists a strategy with the following properties:
1) the proportion of its capital on the entire time horizon is separated from zero with probability 1 regardless of the strategies of competitors;
2) the relative asset prices are asymptotically determined by it;
3) the proportion of capital of investors who follow essentially different strategies tends to zero.
This strategy is analogous to the Kelly (growth-optimal) strategy in stock market models without competition.