Dynamic Programming for Indefinite Stochastic McKean-Vlasov LQ Control Problem under Input Constraints

TL;DR

This paper develops a novel approach using infinite-dimensional HJB equations to solve indefinite stochastic McKean-Vlasov LQ control problems with input constraints, overcoming limitations of classical methods.

Contribution

It introduces an extended state space and constructs new differential equations to derive optimal controls where traditional Riccati equations fail.

Findings

Successfully characterizes optimal control under input constraints.

Applies method to mean-variance portfolio with short-selling restrictions.

Captures investment risk and market line simultaneously.

Abstract

In this note, we study a class of indefinite stochastic McKean-Vlasov linear-quadratic (LQ in short) control problem under the control taking nonnegative values. In contrast to the conventional issue, both the classical dynamic programming principle (DPP in short) and the usual Riccati equation approach fail. We tackle these difficulties by extending the state space from $\mathbb{R}$ to probability measure space, afterward derive the the corresponding the infinite dimensional Hamilton--Jacobi--Bellman (HJB in short) equation. The optimal control and value function can be obtained basing on two functions constructed via two groups of novelty ordinary differential equations satisfying the HJB equation mentioned before. As an application, we revisit the mean-variance portfolio selection problems in continuous time under the constraint that short-selling of stocks is prohibited. The investment risk and the capital market line can be captured simultaneously.