On a best response problem arising in mean field stochastic growth games with common noise

This paper analyzes the best response control problem arising in mean field stochastic growth. We consider mean field games in the setting of stochastic growth where each player's capital stock is described by Cobb-Douglas production dynamics subject to stochastic depreciation and common noise. Each individual's utility functional consists of one's own utility and relative utility. Due to random mean field dynamics, the analysis of the best response relies on a stochastic Hamilton-Jacobi-Bellman (SHJB) equation, which in turn derives a special nonlinear backward stochastic differential equation (BSDE). We analyze the BSDE and use it to determine the solution equation system of the mean field game. Further, we extend the analysis to an AK model for the growth dynamics.