Robust portfolio game under relative performance and state-dependent confidence sets

This paper investigates an n-agent robust portfolio game with relative performance concerns and state-dependent confidence sets. The investors invest in a financial market with a risk-free asset and a risky asset. The ambiguity-averse investors face uncertainty over the drift of the risky asset and update posterior beliefs by Bayesian learning. The investors hold heterogeneous beliefs that the unknown drift falls within a confidence set at a certain confidence level. By maximizing the expected CARA utility of terminal wealth relative to peers under the worst-case scenario, we derive and solve a coupled 2n-dimensional Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation. The robust optimal investment strategy is obtained in semi-analytical form via a partial differential equation (PDE), for which we establish existence, uniqueness, and optimality. In the absence of relative performance concerns, the strategy consists of a myopic demand based on the worst-case drift and a hedging component, resulting in three trading regions: buying, selling, and small-trading. As learning reduces uncertainty about the drift over time, investors gradually increase their exposure to the risky asset. Ambiguity aversion leads to more conservative positions and modifies the feedback Nash equilibrium. Under relative performance, the robust optimal investment strategy is the weighted average of the peers’ strategies and the strategy without relative performance concerns, based on which we derive a feedback robust Nash equilibrium for the n-agent game. Numerical results reveal the herd effect of competition. We further illustrate the direct applicability of our results to competitive reinsurance games and competitive pension fund management, where the surplus or wealth dynamics can be mapped into our baseline model.