Explicit Solution for Constrained Scalar-State Stochastic Linear-Quadratic Control With Multiplicative Noise

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 piecewise affine function of the state and can be computed offline efficiently by solving two coupled Riccati equations. Under some mild conditions, we also obtain the stationary control policy for an 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.