Mean-Variance Portfolio Choice Problem With Borrowing Rate Higher Than Deposit Rate: The Case Of Full Information And Partial Information

This paper is concerned with a continuous-time Mean-Variance optimal portfolio choice problem. The bank not only pays at an interest rate r(t) for any deposit, but also can take at a larger rate R(t) for any loan. In the first part, convex duality method linearizes convex wealth equation, and the introduction of Lagrange multiplier converts original problem to an unconstrained static optimum problem. At last, we obtain another equivalent optimum problem with convex control domain. We use maximum principle and dynamic programming principle respectively to analyze such optimum problem prelimi-naryly. Optimal portfolio is a replicating strategy for a certain contingent claim, which sums up to solve a backward stochastic differential equation. In the second part, we study the same problem with partial information. It shows that the theory of convex analysis can reduce the effect caused by partial information. Specially, when R(t) = r(t), the paper proves that the solvability of a special stochastic Riccati equation is a sufficient condition of the problem's solvability. Finally optimal portfolio and efficient frontier are obtained in an explicit form.