Mean-variance portfolio selection via LQ optimal control

Concerns the problem of mean-variance portfolio selection in an incomplete market. Asset prices are solutions of stochastic differential equations and the parameters in these equations may be random. We approach this problem from the perspective of linear-quadratic (LQ) optimal control and backward stochastic differential equations (BSDEs); that is, we focus on the so-called stochastic Riccati equation (SRE) associated with the problem. Excepting certain special cases, solvability of the SRE remains an open question. Our primary theoretical contribution is a proof of existence and uniqueness of solutions of the SRE associated with the mean-variance problem. In addition, we derive closed form expressions for the optimal portfolios and efficient frontier in terms of the solution of the SRE. A generalization of the Mutual Fund Theorem and financial interpretations of the SRE are also obtained