Mean Field Game of Optimal Relative Investment with Jump Risk

TL;DR

This paper analyzes a mean field game of optimal relative investment considering jump risks and peer competition, providing explicit equilibrium characterizations and approximate Nash equilibria for large agent populations.

Contribution

It introduces a mean field game model with jump risks and derives explicit equilibrium solutions, linking the n-player game to the mean field approximation.

Findings

Explicit mean field equilibrium under jump risk

Approximate Nash equilibrium for large n

Quantitative error bounds for approximation

Abstract

This paper studies the n-player game and the mean field game under the CRRA relative performance on terminal wealth, in which the interaction occurs by peer competition. In the model with n agents, the price dynamics of underlying risky assets depend on a common noise and contagious jump risk modelled by a multi-dimensional nonlinear Hawkes process. With a continuum of agents, we formulate the MFG problem and characterize a deterministic mean field equilibrium in an analytical form under some conditions, allowing us to investigate some impacts of model parameters in the limiting model and discuss some financial implications. Moreover, based on the mean field equilibrium, we construct an approximate Nash equilibrium for the n-player game when n is sufficiently large. The explicit order of the approximation error is also derived.