不同借贷利率下连续时间Markowitz均值-方差模型
Continuous-time Markowitz's mean-variance model under different borrowing and saving rates
AI总结：
本文在借贷利率不同的连续时间Black-Scholes市场中求解Markowitz均值-方差问题，证明价值函数具有$C^{3,2}$光滑性，并揭示最优策略由借款区、全仓股票区和储蓄区构成，且永不卖空。
摘要：
我们研究了在具有不同借贷利率的连续时间Black-Scholes市场中Markowitz的均值-方差投资组合选择问题。相应的Hamilton-Jacobi-Bellman方程是完全非线性的。通过精细的偏微分方程和验证论证，价值函数被证明具有$C^{3,2}$光滑性。研究还表明，存在一个借款边界和一个储蓄障碍，它们按升序将整个交易区域划分为借款区、全仓股票区和储蓄区。最优交易策略是连续时间策略（如大多数连续时间模型所建议的）与不连续时间策略（如带交易成本的模型所建议的）的混合：在中间的全仓股票区，应将全部财富投入股票；在另外两个区域，则以财富和时间的反馈形式连续交易。卖空股票永远不是最优的。文中还给出了数值例子，以验证理论结果并提供超越理论结果的更多金融洞见。
英文摘要：
We study Markowitz's mean-variance portfolio selection problem in a continuous-time Black-Scholes market with different borrowing and saving rates. The associated Hamilton-Jacobi-Bellman equation is fully nonlinear. Using a delicate partial differential equation and verification argument, the value function is proven to be $C^{3,2}$ smooth. It is also shown that there are a borrowing boundary and a saving barrier which divide the entire trading area into a borrowing-money region, an all-in-stock region, and a saving-money region in ascending order. The optimal trading strategy is a mixture of continuous-time strategy (as suggested by most continuous-time models) and discontinuous-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 wealth and time. It is never optimal to short sale the stock. Numerical examples are also presented to verify the theoretical results and to give more financial insights beyond them.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
