Stochastic LQ Optimal Control with Random Coefficients and a Terminal Mean-Field Cost

Abstract

This paper investigates a multidimensional non-homogeneous stochastic linear-quadratic optimal control problem featuring random coefficients and a terminal mean-field term in the cost functional, enabling its direct application to mean-variance models in financial engineering. Employing the Lagrangian duality method together with a decomposition approach for linear backward stochastic differential equations, we provide two types of sufficient conditions for solvability and derive the corresponding optimal controls. In particular, in the deterministic-coefficient case, our condition is weaker than the standard condition found in the existing literature on mean-field stochastic LQ problems. Finally, a numerical example drawn from optimal portfolio selection with multiple assets under mean-variance utility demonstrates the applicability of our results.