Dynamic mean-variance portfolio with a benchmark process and no-shorting constraints

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