Equilibrium Strategies for the N-agent Mean-Variance Investment Problem over a Random Horizon

TL;DR

This paper derives explicit equilibrium strategies for a large group of competing agents in a mean-variance investment model with a random horizon, considering both finite and mean-field games.

Contribution

It extends stochastic control methods to derive explicit equilibrium feedback strategies in a multi-agent, random horizon setting with dynamic risk aversion.

Findings

Explicit equilibrium strategies are obtained for both finite and mean-field games.

Strategies depend on current wealth and competitors' wealth, influenced by the random horizon.

Special cases recover known results when risk aversion is constant or competition is absent.

Abstract

We study equilibrium feedback strategies for a family of dynamic mean-variance problems with competition among a large group of agents. We assume that the time horizon is random and each agent's risk aversion depends dynamically on the current wealth. We consider both the finite population game and the corresponding mean-field one. Each agent can invest in a risk-free asset and a specific individual stock, which is correlated with other stocks by a common noise. By applying stochastic control theory, we derive the extended Hamilton-Jacobi-Bellman (HJB) system of equations for both $n$-agent and mean-field games. Under an exponentially distributed random horizon, in each case, we explicitly obtain the equilibrium feedback strategies and the value function. Our results show that the agent's equilibrium feedback strategy depends not only on his/her current wealth but also on the wealth of other competitors. Moreover, when the risk aversion is state-independent and the risk-free interest rate is zero, the equilibrium strategies degenerate to constants, which is identical to the unique equilibrium obtained in \citet{lacker2019mean} with exponential risk preferences; when the competition parameter goes to zero and the risk aversion equals some specific value, the equilibrium strategies coincide with the ones derived in \citet{landriault2018equilibrium}.