Abstract: The cornerstones of quantitative finance are asset returns, interest rates, and volatilities. They appear in many fundamental formulas in finance.  In the classical Markowitz’s portfolio theory, the expectation and the variance of the returns of the underlying assets are assumed to be known. However, in practice, they are estimated by past values of the returns. Moreover, the successful applications of various time series models in the financial market also imply that the current returns depend on their past values. Therefore, it is natural to consider the weight vector of the portfolio as a function of past values. In this talk, I will introduce a functional optimization approach to portfolio optimization problems by treating the unknown weight vector as a function of past values instead of treating them as fixed unknown coefficients in the majority of studies. We first show that the optimal solution, in general, is not a constant function and give the optimal conditions for a vector function to be the solution. Then, we propose gradient-ascent algorithms to solve the functional optimization for mean-variance portfolio management with theorems for convergence provided. Simulations and empirical studies show that our approach can perform significantly better than the plug-in approach.