Explicit Solution for Constrained Stochastic Linear-Quadratic Control with Multiplicative Noise

TL;DR

This paper derives an explicit, piece-wise affine control policy for constrained stochastic LQ problems with multiplicative noise, applicable in financial risk management, by solving coupled Riccati equations.

Contribution

It introduces a novel analytical solution for constrained stochastic LQ control problems with multiplicative noise, overcoming previous structural challenges.

Findings

Control policy is a piece-wise affine function of the state.

Optimal control can be computed by solving two coupled Riccati equations.

Method applies to infinite horizon problems under mild conditions.

Abstract

We study in this paper a class of constrained linear-quadratic (LQ) optimal control problem formulations for the scalar-state stochastic system with multiplicative noise, which has various applications, especially in the financial risk management. The linear constraint on both the control and state variables considered in our model destroys the elegant structure of the conventional LQ formulation and has blocked the derivation of an explicit control policy so far in the literature. We successfully derive in this paper the analytical control policy for such a class of problems by utilizing the state separation property induced from its structure. We reveal that the optimal control policy is a piece-wise affine function of the state and can be computed off-line efficiently by solving two coupled Riccati equations. Under some mild conditions, we also obtain the stationary control policy for infinite time horizon. We demonstrate the implementation of our method via some illustrative examples and show how to calibrate our model to solve dynamic constrained portfolio optimization problems.