Mean–variance portfolio selection under no-shorting rules : a BSDE approach

Abstract

This paper revisits the mean–variance portfolio selection problem in continuous-time within the framework of short-selling of stocks is prohibited via backward stochastic differential equation approach. To relax the strong condition in Li et al. (Li et al. 2002), the above issue is formulated as a stochastic recursive optimal linear–quadratic control problem. Due to no-shorting rules (namely, the portfolio taking non-negative values), the well-known “completion of squares” no longer applies directly. To overcome this difficulty, we study the corresponding Hamilton–Jacobi–Bellman (HJB, for short) equation inherently and derive the two groups of Riccati equations. On one hand, the value function constructed via Riccati equations is shown to be a viscosity solution of the HJB equation mentioned before; On the other hand, by means of these Riccati equations and backward semigroup, we are able to get explicitly the efficient frontier and efficient investment strategies for the recursive utility mean–variance portfolio optimization problem.