Mean-variance portfolio selection with random parameters in a complete market

The paper concerns the continuous-time, mean-variance portfolio selection problems in a complete market with random interest rate, appreciation rates, and volatility coefficients. The problem is tackled using the results of stochastic linear-quadratic (LQ) optimal control and backward stochastic differential equations (BSDEs), two theories that have been extensively studied and developed in recent years. Specifically, the mean-variance problem is formulated as a linearly constrained stochastic LQ control problem, Solvability of this LQ problem is reduced, in turn, to proving global solvability of a stochastic Riccati equation. The proof of existence and uniqueness of this Riccati equation, which is a fully nonlinear and singular BSDE with random coefficients, is interesting in its own right and relies heavily on the structural properties of the equation. Efficient investment strategies as well as the mean-variance efficient frontier are then analytically derived in terms of the solution of this equation. In particular, it is demonstrated that the efficient frontier in the mean-standard deviation diagram is still a straight line or, equivalently, risk-free investment is still possible, even when the interest rate is random. Finally, a version of the Mutual Fund Theorem is presented.