均值半方差投资组合策略的选择

风险是现代金融理论和实践的中心问题，自 Markowitz 提出均值-方差投资组合模型以来，有大量的学者对其进行了研究。但是用半方差作为风险度量比用方差作为风险度量是更为合理的。本文我们首先对近些年与均值-半方差投资组合问题相关的文献做了简单介绍，之后给出了文章中所必须的一些基本数学理论。我们构造了在部分信息下的均值-半方差投资组合选择模型，把滤波问题和投资组合选择问题分离。为解决该均值-半方差投资组合问题，我们利用分解的方法把其分成一个静态优化问题以及一个倒向随机微分方程解的问题，并且证明了这个分解的合理性。对于静态优化问题，我们首先证明了部分信息下市场具有完备性，然后考虑对半方差赋予权重的优化问题，得到了赋权后的终端财富的最优解，最后利用逼近的方法证明了对于均值-半方差投资组合问题在除去某种特殊情况下不存在最优的终端财富。最后我们证明了在部分信息情况下任意可行的终端财富都存在投资组合对其进行复制，所以尽管我们得到不存在最优的终端财富这个负面结论，我们仍然可以利用倒向随机微分方程对渐进的最优终端财富进行复制，进而可以得到均值-半方差问题的最优投资组合。

Risk is a central issue in modern financial theory and practice, and has been studied by a large number of scholars since Markowitz proposed the mean-variance portfolio model. However, it is more reasonable to use semivariance as a risk measure than variance as a risk measure. In this paper we first give a brief overview of the recent literature related to the mean semivariance portfolio problem, followed by some basic mathematical theory necessary for the discussion in the paper. We construct a model for mean semivariance portfolio selection under partial information, separating the filtering problem from the problem of portfolio selection. To solve this mean semivariance portfolio problem, we use a decomposition to split it into a static optimization problem and a problem with a backward stochastic differential equation's solution, and justify this decomposition. For the static optimisation problem, we first show that the market is complete under partial information, then consider the optimization problem with weights assigned to the semivariance to obtain the optimal solution for the terminal wealth after weighting, and finally use approximation to show that there is no optimal terminal wealth for the mean semivariance portfolio problem except a special case. Finally, we show that there are portfolios that replicate any feasible terminal wealth under partial information, so that despite the negative conclusion that there is no optimal terminal wealth, we can still replicate the asymptotically optimal terminal wealth using the backward stochastic differential equation, which in turn leads to an optimal portfolio for the mean semivariance problem.

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