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摘要翻译：
研究了一类连续时间Markowitz均值-方差证券组合选择问题。这是一个奇异随机控制问题，内在地处于有限时间范围内。通过一系列的变换，将该问题转化为一个所谓的双障碍问题，这是一个在物理和偏微分方程文献中研究较多的问题，具有两个时变自由边界。这两个边界，定义了购买，销售，和无贸易区域，被证明是平滑的时间。这反过来又通过Skorokhod问题描述了最优策略，即试图在无贸易区域内保持一定的调整后的债券-股票头寸。揭示了最优策略的几个显著不同于无交易成本策略的特征。研究表明，在投资目标和股票波动率之间存在一个与股票超额收益和交易费用有关的临界时间长度，因此，如果计划期限短于该临界时间长度，则可能无法实现预期的最终收益（而在没有交易费用的情况下，任何预期收益都可以在任意时间内达到）。它进一步证明，任何遵循最优策略的人都不应该超过到期时间短于上述临界长度的点购买股票。此外，当到期日越来越近时，投资者购买股票的可能性更小，而出售股票的可能性更大。这些特征，虽然符合广泛接受的投资智慧，表明规划的地平线是投资机会的一个组成部分。
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英文标题：
《Continuous-Time Markowitz's Model with Transaction Costs》
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作者：
Min Dai, Zuo Quan Xu, Xun Yu Zhou
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最新提交年份：
2009
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分类信息：
一级分类：Quantitative Finance 数量金融学
二级分类：Portfolio Management 项目组合管理
分类描述：Security selection and optimization, capital allocation, investment strategies and performance measurement
证券选择与优化、资本配置、投资策略与绩效评价
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英文摘要：
A continuous-time Markowitz's mean-variance portfolio selection problem is studied in a market with one stock, one bond, and proportional transaction costs. This is a singular stochastic control problem,inherently in a finite time horizon. With a series of transformations, the problem is turned into a so-called double obstacle problem, a well studied problem in physics and partial differential equation literature, featuring two time-varying free boundaries. The two boundaries, which define the buy, sell, and no-trade regions, are proved to be smooth in time. This in turn characterizes the optimal strategy, via a Skorokhod problem, as one that tries to keep a certain adjusted bond-stock position within the no-trade region. Several features of the optimal strategy are revealed that are remarkably different from its no-transaction-cost counterpart. It is shown that there exists a critical length in time, which is dependent on the stock excess return as well as the transaction fees but independent of the investment target and the stock volatility, so that an expected terminal return may not be achievable if the planning horizon is shorter than that critical length (while in the absence of transaction costs any expected return can be reached in an arbitrary period of time). It is further demonstrated that anyone following the optimal strategy should not buy the stock beyond the point when the time to maturity is shorter than the aforementioned critical length. Moreover, the investor would be less likely to buy the stock and more likely to sell the stock when the maturity date is getting closer. These features, while consistent with the widely accepted investment wisdom, suggest that the planning horizon is an integral part of the investment opportunities.
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PDF链接：
https://arxiv.org/pdf/0906.0678
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