Continuous-time mean-variance portfolio selection with only risky assets

We investigate in this paper a continuous-time mean-variance portfolio selection problem in a general market setting with multiple assets that all can be risky. Using the Lagrange duality method and the dynamic programming approach, we derive explicit closed-form expressions for the efficient investment strategy and the mean-variance efficient frontier. We provided a necessary and sufficient condition under which the global minimum variance is zero and there exists a risk-free wealth process. Our results reveal that, even if there is no risk-free asset in the market, there can still exist a risk-free wealth process, the global minimum variance can be zero, and the efficient frontier can be a straight line in the mean-standard derivation plane. In addition, we further prove the validity of the two-fund separation theorem.