We generalize the idea of semi-self-financing strategies, originally discussed in Ehrbar (1990), and later formalized in Cui et al (2012), for the pre-commitment mean-variance (MV) optimal portfolio allocation problem. The proposed semi-self-financing strategies are built upon a numerical solution framework for Hamilton–Jacobi–Bellman equations, and can be readily employed in a very general setting, namely continuous or discrete re-balancing, jump-diffusions with finite activity, and realistic portfolio constraints. We show that if the portfolio wealth exceeds a threshold, an MV optimal strategy is to withdraw cash. These semi-self-financing strategies are generally non-unique. Numerical results confirming the superiority of the efficient frontiers produced by the strategies with positive cash withdrawals are presented. Tests based on estimation of parameters from historical time series show that the semi-self-financing strategy is robust to estimation ambiguities. 
Better than pre-commitment mean-variance portfolio allocation strategies: A semi-self-financing Hamilton–Jacobi–Bellman equation approach