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

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