[Paper Review] Mean-Field Stochastic Linear Quadratic Optimal Control Problems: Closed-Loop Solvability

Xun Li, Jingrui Sun+1 more|arXiv (Cornell University)|Feb 25, 2016

Stochastic processes and financial applications13 references4 citations

TL;DR

This paper establishes the closed-loop solvability of mean-field stochastic linear quadratic (LQ) optimal control problems by characterizing the existence of an optimal closed-loop strategy through the solvability of coupled generalized Riccati equations, along with constraints on solutions to a linear backward SDE and a terminal ODE. The key contribution is a necessary and sufficient condition for closed-loop solvability in terms of a regular solution to these equations and auxiliary constraints.

ABSTRACT

An optimal control problem is studied for a linear mean-field stochastic differential equation with a quadratic cost functional. The coefficients and the weighting matrices in the cost functional are all assumed to be deterministic. Closed-loop strategies are introduced, which require to be independent of initial states; and such a nature makes it very useful and convenient in applications. In this paper, the existence of an optimal closed-loop strategy for the system (also called the closed-loop solvability of the problem) is characterized by the existence of a regular solution to the coupled two (generalized) Riccati equations, together with some constraints on the adapted solution to a linear backward stochastic differential equation and a linear terminal value problem of an ordinary differential equation.

Motivation & Objective

• To establish necessary and sufficient conditions for closed-loop solvability in mean-field stochastic linear quadratic optimal control problems.
• To characterize optimal closed-loop strategies that are independent of initial states, enhancing applicability in real-world systems.
• To extend classical LQ theory to mean-field SDEs by incorporating expectations of state and control processes.
• To resolve fundamental challenges in the solvability of MF-SDE-based LQ problems with deterministic coefficients.
• To provide a theoretical foundation for applications in finance, risk management, and portfolio optimization.

Proposed method

• Introduces closed-loop strategies that are independent of initial states, ensuring practical implementability.
• Derives a coupled system of two generalized Riccati equations (GREs) as the core condition for closed-loop solvability.
• Imposes constraints on the adapted solution of a linear backward stochastic differential equation (BSDE) and a terminal value ODE.
• Uses a verification theorem to show that the optimal control is given by a feedback form involving the solution of the GREs.
• Applies the stochastic maximum principle and duality arguments to derive the necessary conditions for optimality.
• Establishes equivalence between closed-loop solvability and the existence of a regular solution to the coupled GREs with auxiliary constraints.

Experimental results

Research questions

• RQ1 Under what conditions does a mean-field stochastic LQ optimal control problem admit a closed-loop optimal strategy?
• RQ2 How can the optimal control be expressed in feedback form independent of the initial state?
• RQ3 What is the role of the generalized Riccati equations in ensuring closed-loop solvability?
• RQ4 How do the constraints on the BSDE and ODE solutions affect the solvability of the problem?
• RQ5 What is the relationship between the closed-loop solvability and the regular solution of the coupled GREs?

Key findings

• Closed-loop solvability of the MF-LQ problem is equivalent to the existence of a regular solution to the coupled generalized Riccati equations.
• The optimal closed-loop strategy is given by a feedback law involving the solution of the GREs, ensuring state and expectation dependence.
• The auxiliary constraints on the BSDE and ODE solutions are necessary for the optimality of the closed-loop strategy.
• In the absence of deterministic drift and noise terms, closed-loop solvability reduces to the regular solvability of the GREs alone.
• The value function is explicitly represented using the solution of the GREs and the auxiliary processes.
• The proof relies on a verification argument that shows the optimality of the feedback strategy through an identity involving the cost functional and the solution of the GREs.