We present a survey of results on asymptotically optimal strategies in models of financial markets, that is, strategies whose capital grows asymptotically, with probability 1, at least as rapidly as the capital of any other strategy. Our main focus is on multi-agent models, in which the capital dynamics of agents are interconnected; classical results for the single-agent case are presented for comparison. We discuss the existence, uniqueness, and constructive description of asymptotically optimal strategies, as well as their effect on the capital dynamics of the other agents. Models with short- and long-lived assets are considered. A number of recent results are included, among them solutions of several open problems stated earlier in the literature. Bibliography: 55 titles.
