Mean Field Portfolio Games with Consumption

TL;DR

This paper analyzes mean field portfolio games with consumption, establishing a correspondence between Nash equilibria and solutions to FBSDEs, providing explicit solutions in certain market conditions, and addressing uniqueness of equilibria.

Contribution

It introduces a general framework linking Nash equilibria to FBSDE solutions for portfolio games with consumption, covering various utility functions and proving uniqueness in specific cases.

Findings

Established a one-to-one correspondence between Nash equilibria and FBSDE solutions.

Derived explicit Nash equilibria in time-independent market parameters.

Proved the uniqueness of strong equilibrium in bounded spaces.

Abstract

We study mean field portfolio games with consumption. For general market parameters, we establish a one-to-one correspondence between Nash equilibria of the game and solutions to some FBSDE, which is proved to be equivalent to some BSDE. Our approach, which is general enough to cover power, exponential and log utilities, relies on martingale optimality principle in [3,9] and dynamic programming principle in [6,7]. When the market parameters do not depend on the Brownian paths, we get the unique Nash equilibrium in closed form. As a byproduct, when all market parameters are time-independent, we answer the question proposed in [12]: the strong equilibrium obtained in [12] is unique in the essentially bounded space.