  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.
