Optimal portfolio selection of many players under relative performance criteria in the market model with random coefficients

TL;DR

This paper investigates optimal portfolio strategies in a multi-agent market with random coefficients, considering relative performance criteria and different risk preferences, establishing equilibrium existence and extending results to mean field models.

Contribution

It introduces a framework for portfolio optimization with random market parameters and relative performance, providing equilibrium existence results and extending to McKean-Vlasov control.

Findings

Existence of forward Nash and mean field equilibria under random coefficients.

Derivation of explicit portfolio formulas extending constant coefficient models.

Application of measure-dependent forward relative performance processes.

Abstract

We study the optimal portfolio selection problem under relative performance criteria in the market model with random coefficients from the perspective of many players game theory. We consider five random coefficients which consist of three market parameters which are used in the risky asset price modeling and two preference parameters which are related to risk attitude and impact of relative performance. We focus on two cases; either all agents have Constant Absolute Risk Aversion (CARA) risk preferences or all agents have Constant Relative Risk Aversion (CRRA) risk preferences for their investment optimization problem. For each case, we show that the forward Nash equilibrium and the mean field equilibrium exist for the n-agent game and the corresponding mean field stochastic optimal control problem, respectively. To extend the n-agent game to the continuum of players game, we introduce a measure dependent forward relative performance process and apply an optimization over controlled dynamics of McKean-Vlasov type. We conclude that our optimal portfolio formulas extend the corresponding results of the market model with constant coefficients.