Utility Maximization in a Market with Competitive Heterogeneous Agents: backward propagation of chaos and learning

Abstract

This thesis is dedicated to the theory and numerics of expected utility maximization game under relative performance concerns, in a large population of competitive, heterogeneous agents.First, we consider an incomplete market model in which agents have CARA utilities, and we obtain characterizations of Nash equilibria in both the finite and infinite agent settings. The infinite agent setting leads to a stochastic graphon game, where a continuum of agents interact through a graphon. Under modest assumptions on the denseness of the interaction graph, we establish convergence results for the Nash equilibria of the finite agent problem to the infinite agent 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 (FBSDEs), where the interaction is heterogeneous and through the control processes, and the generator is of quadratic growth. Characterizing the graphon game gives rise to a novel form of infinite-dimensional FBSDEs of McKean-Vlasov type, for which we provide well-posedness results. Next, we consider a class of McKean-Vlasov type BSDEs, with the interaction term dependent more generally on the laws of both the value processes and the diagonals of the control processes. We prove a convergence result from the finite particle system to the infinite particle system. Under certain assumptions on the Malliavin differentiabilities of the terminal value and the generator, we improve upon the usual Lipschitz conditions imposed on the generator, by allowing the generator to be locally Lipschitz. Under stronger conditions, we prove a point-wise convergence result for the control processes. Lastly, we adopt a neural network algorithm to learning Mckean-Vlasov type graphon FBSDEs, in order to approximate numerically the equilibrium of the heterogeneous utility maximization game. We analyze the accuracy and efficiency of the algorithm in a complete market when stock prices are lognormal. We analyze the quality of the learnt control by computing exploitability of the algorithm for different graphons. Lastly, We provide inferences of the risk aversion levels and graphons on equilibrium wealth, benchmarked wealth, and equilibrium utilities, in a complete market when the stock price coefficients are Markovian.