Indefinite Linear-Quadratic Optimal Control Problems of Backward Stochastic Differential Equations with Partial Information

TL;DR

This paper addresses indefinite linear-quadratic optimal control problems for backward stochastic differential equations with partial information, deriving conditions for optimality and explicit control formulas using stochastic filtering and variational methods.

Contribution

It introduces a novel approach combining stochastic filtering and variational techniques to solve indefinite LQ control problems for BSDEs with partial information.

Findings

Derived necessary and sufficient optimality conditions.

Established solvability of associated matrix differential equations and BSDEs.

Provided explicit formulas for optimal control and value function.

Abstract

This paper is concerned with a kind of linear-quadratic (LQ) optimal control problem of backward stochastic differential equation (BSDE) with partial information. The cost functional includes cross terms between the state and control, and the weighting matrices are allowed to be indefinite. Through variational methods and stochastic filtering techniques, we derive the necessary and sufficient conditions for the optimal control, where a Hamiltonian system plays a crucial role. Moreover, to construct the optimal control, we introduce a matrix-valued differential equation and a BSDE with filtering, and establish their solvability under the assumption that the cost functional is uniformly convex. Finally, we present explicit forms of the optimal control and value function.