Mean-field game of mean-variance portfolio management with peer-based relative risk aversion

This paper investigates a mean-field game (MFG) problem for mean-variance (MV) portfolio management, highlighting a new type of relative performance encoded by the peer-based risk aversion. Specifically, the risk aversion is formulated as a piecewise form that depends on whether the individual's wealth is above or below the population average. Due to the inherent time-inconsistency in the MV criterion, together with the piecewise risk aversion, we encounter a class of time-inconsistent MFG, new to the literature. Our goal is to seek a mean-field equilibrium, characterized by a forward-backward stochastic differential equation (FBSDE) system and a mean-field consistency condition. The new challenge stems from the discontinuous coefficients induced by the piecewise risk aversion. In response, we first propose a smooth regularization technique and obtain the existence of the equilibrium in the intra-personal game for the representative agent by establishing the solution to the discontinuous multi-dimensional FBSDE. Next, by invoking fixed-point arguments and convergence analysis as smoothing regularization vanishes, we conclude the existence of the mean-field equilibrium in the time-inconsistent MFG.