Optimal Regulators for Mean‐Field Stochastic System With State‐Dependent Weight Costs

ABSTRACT This article investigates optimal control problems for mean‐field type stochastic systems with state‐dependent weight costs, where the state system is linear. It generalizes linear‐quadratic problems by allowing the coefficients of the quadratic terms to depend on a power function of the state process. By employing the method of completing the square, the optimal cost and the linear state feedback form of the optimal control are derived. Additionally, we extend this analysis to a risk‐sensitive setting. By introducing an auxiliary state process and control function, we obtain the optimal cost and the nonlinear state feedback form of the optimal control pair. The article also addresses the corresponding zero‐sum stochastic differential game problem, and the explicit form of the saddle point is provided.