Continuous Time Evolutionary Market Dynamics : The Case of Fix-Mix Strategies

We develop a continuous-time evolutionary market model where prices are endogenously generated by supply and demand. Investment strategies are assumed to be fix-mix, which means that the relative budget shares are constant in time. The model is therefore a hybrid. While given portfolio rules remain constant over time, assets, market-clearing and in particular market shares of the individual portfolio strategies evolve in continuous time. Our main goal is to understand the wealth dynamics which describes the evolution of market shares. We study its asymptotic properties and identify evolutionary stable investment strategies. These strategies prevent entrants to the financial market from gaining wealth in the long run and furthermore, in the existence of a small diversified number of mutant strategies, drive the invading strategy out of the market. Our definition of evolutionary stability is therefore a close adaptation of Maynard-Smith and Price's (1973) original definition of an ESS [8]