Evolutionary Finance: Models with Long-Lived Assets

Evolutionary Finance explores the "survival and extinction" questions of investment strategies (portfolio rules) in the market selection process. It models the stochastic dynamics of financial markets based on behavioral and evolutionary principles, where asset prices are determined endogenously by short-run equilibrium between supply and demand, arising from the interaction of competing portfolio rules. This paper presents a survey of developments in Evolutionary Finance with a focus on long-lived, dividend-paying risky securities, where the budget of each investor comes from asset dividends and capital gains. We review several key models in this area addressing the following problems in order: 1) the most general results under the most general assumptions; 2) global evolutionary stability under restrictive assumptions; 3) viewing the model from a different, game-theoretic, perspective and examining almost sure Nash equilibrium strategies under restrictive assumptions. A central goal of the study is to identify an investment strategy that allows an investor to survive in the market selection process, i.e., to keep with probability one, a strictly positive, bounded away from zero share of market wealth over an infinite time horizon, irrespective of the strategies used by other investors. The main results are under general assumptions, such a survival strategy -- an analogue of the famous Kelly rule of "betting one's beliefs" exists -- and is asymptotically unique (within a specific class of strategies called basic). Moreover, under the required stronger assumptions, the Kelly rule is globally evolutionarily stable and is the unique investment strategy that forms a symmetric Nash equilibrium almost surely.