Optimal Investment in a Large Population of Competitive and Heterogeneous Agents

Abstract

This paper studies a stochastic utility maximization game under relative performance concerns in finite agent and infinite agent settings, where a continuum of agents interact through a graphon (see definition below). We consider an incomplete market model in which agents have CARA utilities, and we obtain characterizations of Nash equilibria in both the finite agent and graphon paradigms. Under modest assumptions on the denseness of the interaction graph among the agents, we establish convergence results for the Nash equilibria and optimal utilities of the finite player problem to the infinite player problem. This result is achieved as an application of a general backward propagation of chaos type result for systems of interacting forward-backward stochastic differential equations, where the interaction is heterogeneous and through the control processes, and the generator is of quadratic growth. In addition, characterizing the graphon game gives rise to a novel form of infinite dimensional forward-backward stochastic differential equation of Mckean-Vlasov type, for which we provide well-posedness results. An interesting consequence of our result is the computation of the competition indifference capital, i.e., the capital making an investor indifferent between whether or not to compete.