Optimal Control of Constrained Stochastic Linear-Quadratic Model with Applications

TL;DR

This paper derives explicit, piece-wise affine optimal control policies for constrained stochastic linear-quadratic problems, enabling efficient solutions for finite and infinite horizon cases, with applications to portfolio optimization.

Contribution

It introduces a novel explicit solution method for constrained stochastic LQ problems using Riccati equations and state separation, applicable to portfolio selection.

Findings

Optimal control is a piece-wise affine function of the state.

Solutions can be computed efficiently via two Riccati equations.

Applicable to infinite horizon problems under mild conditions.

Abstract

This paper studies a class of continuous-time scalar-state stochastic Linear-Quadratic (LQ) optimal control problem with the linear control constraints. Applying the state separation theorem induced from its special structure, we develop the explicit solution for this class of problem. The revealed optimal control policy is a piece-wise affine function of system state. This control policy can be computed efficiently by solving two Riccati equations off-line. Under some mild conditions, the stationary optimal control policy can be also derived for this class of problem with infinite horizon. This result can be used to solve the constrained dynamic mean-variance portfolio selection problem. Examples shed light on the solution procedure of implementing our method.