Weak Closed-loop Solvability of Linear Quadratic Stochastic Optimal Control Problems with Partial Information

TL;DR

This paper addresses the weak closed-loop solvability of linear quadratic stochastic control problems with partial information, introducing Riccati equations and BSDEs, and establishing equivalence with open-loop solvability.

Contribution

It develops a framework for weak closed-loop solvability in partial information LQSOC problems, including new Riccati equations and an equivalence result with open-loop solutions.

Findings

Established weak closed-loop solvability under partial information.

Demonstrated equivalence between open-loop and weak closed-loop solvability.

Provided an example illustrating the construction of a weak closed-loop optimal strategy.

Abstract

This paper investigates a linear quadratic stochastic optimal control (LQSOC) problem with partial information. Firstly, by introducing two Riccati equations and a backward stochastic differential equation (BSDE), we solve this LQSOC problem under standard positive semidefinite assumptions. Secondly, by means of a perturbation approach, we study open-loop solvability of this problem when the weighting matrices in the cost functional are indefinite. Thirdly, we investigate weak closed-loop solvability of this problem and prove the equivalence between open-loop and weak closed-loop solvabilities. Finally, we give an example to illustrate the way for obtaining a weak closed-loop optimal strategy.