Constrained portfolio game with heterogeneous agents

TL;DR

This paper studies stochastic utility maximization games with heterogeneous agents in finite and infinite settings, characterizing Nash equilibria via advanced stochastic differential equations and proving convergence as the number of agents grows.

Contribution

It introduces a novel framework for analyzing constrained portfolio games with heterogeneous agents using forward-backward stochastic differential equations and graphon models.

Findings

Existence of Nash equilibrium in finite and infinite-agent settings.

Convergence of finite-agent Nash equilibria to the graphon game equilibrium.

Development of infinite-dimensional McKean-Vlasov FBSDEs for the graphon game.

Abstract

We investigate stochastic utility maximization games under relative performance concerns in both finite-agent and infinite-agent (graphon) settings. An incomplete market model is considered where agents with power (CRRA) utility functions trade in a common risk-free bond and individual stocks driven by both common and idiosyncratic noise. The Nash equilibrium for both settings is characterized by forward-backward stochastic differential equations (FBSDEs) with a quadratic growth generator, where the solution of the graphon game leads to a novel form of infinite-dimensional McKean-Vlasov FBSDEs. Under mild conditions, we prove the existence of Nash equilibrium for both the graphon game and the $n$-agent game without common noise. Furthermore, we establish a convergence result showing that, with modest assumptions on the sensitivity matrix, as the number of agents increases, the Nash equilibrium and associated equilibrium value of the finite-agent game converge to those of the graphon game.