Dynamic Mean-Variance Portfolio Selection with Liability and No-Shorting Constraints

In this paper, we formulate a mean-variance portfolio selection model with liability under the constraint that short-selling is prohibited. Due to the introduction of the liability and no-shorting constraints, our problem is not a conventional stochastic optimal linear-quadratic(LQ) control problem, and the corresponding HJB equation has no continuous solution. we construct a lower-semicontinuous function through two Ricttati equations, and show that this function is a viscosity solution of the HJB equation. we get explicitly the optimal dynamic strategy and the mean-variance efficient frontier in closed forms.