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We study Markowitz's mean-variance portfolio selection problem in acontinuous-time Black-Scholes market with different borrowing and saving rates.The associated Hamilton-Jacobi-Bellman equation is fully nonlinear. Using adelicate partial differential equation and verification argument, the valuefunction is proven to be $C^{3,2}$ smooth. It is also shown that there are aborrowing boundary and a saving barrier which divide the entire trading areainto a borrowing-money region, an all-in-stock region, and a saving-moneyregion in ascending order. The optimal trading strategy is a mixture ofcontinuous-time strategy (as suggested by most continuous-time models) anddiscontinuous-time strategy (as suggested by models with transaction costs):one should put all her wealth in the stock in the middle all-in-stock region,and continuously trade it in the other two regions in a feedback form of wealthand time. It is never optimal to short sale the stock. Numerical examples arealso presented to verify the theoretical results and to give more financialinsights beyond them.

Continuous-time Markowitz's mean-variance model under different borrowing and saving rates