Discrete-Time Backward Stochastic LQ Control Problem

TL;DR

This paper develops a comprehensive analytical framework for discrete-time backward stochastic LQ control problems, addressing unique terminal constraints and cross-term costs with novel transformation and decoupling techniques.

Contribution

It introduces a new approach to solve backward stochastic LQ problems with cross-term costs, including conditions for solvability, a maximum principle, and explicit feedback control.

Findings

Derived necessary and sufficient conditions for problem solvability.

Established a backward stochastic maximum principle and Riccati equation.

Provided explicit state feedback control and verified with numerical examples.

Abstract

This paper focuses on the discrete-time backward stochastic linear quadratic (BSLQ) optimal control problem with nonhomogeneous system terms and cost function cross terms. The terminal constraint of such systems distinguishes it from forward stochastic systems, posing unique challenges for analysis and solution. Within the Hilbert space framework, we first clarify the necessary and sufficient conditions for problem solvability, then introduce the backward stochastic system maximum principle to derive the Hamiltonian system characterizing the optimal control. After equivalent transformation of the original problem, we use the decoupling method to obtain the corresponding Riccati equation, and present the explicit state feedback expression of the optimal control and the analytical form of the value function. Finally, numerical examples verify the effectiveness and feasibility of the proposed method. The innovation lies in expanding the model generality: addressing the structural asymmetry issue in the Riccati equation with cross-term cost functions, we propose system equivalent transformation and decoupling techniques. Our theoretical results provide a new analytical framework for dynamic optimization problems such as financial portfolio optimization and risk management.