不确定时间范围下的非凹期望效用优化
Non-concave expected utility optimization with uncertain time horizon
摘要：
我们考虑一个期望效用最大化问题，其中效用函数不一定是凹的，时间范围是不确定的。在完备金融市场中，建立了一般非凹效用函数最优的充分必要条件。我们发现，一般的凹化方法的效用函数来处理非线性，而仍然适用时，时间范围是一个停时方面的金融市场过滤，导致次优时，时间范围是独立的金融风险，因此不能直接应用。对于后者的情况下，我们建议一个递归的过程，这是基于动态规划原理。通过对随机时间范围下的凸期权补偿方案下的最优投资问题进行多期数值分析，说明了我们的研究结果。我们观察到，在确定和不确定的随机时间范围内的非凹投资组合的分布是右偏的，有一个长长的右尾，表明投资者期望频繁的小损失和一些大收益的投资。虽然（某些）平均时间范围的投资组合在提前停止日期是单峰的，随机时间范围的投资组合是多峰分布，这为投资者提供了一定的灵活性之间切换的局部最大化，这取决于市场表现。具有不同高度的多峰的多峰结构可以通过凹化过程来解释，而时域的分布对模式之间的振幅具有显著影响。
英文摘要：
We consider an expected utility maximization problem where the utility function is not necessarily concave and the time horizon is uncertain. We establish a necessary and sufficient condition for the optimality for general non-concave utility function in a complete financial market. We show that the general concavification approach of the utility function to deal with non-concavity, while being still applicable when the time horizon is a stopping time with respect to the financial market filtration, leads to sub-optimality when the time horizon is independent of the financial risk, and hence can not be directly applied. For the latter case, we suggest a recursive procedure which is based on the dynamic programming principle. We illustrate our findings by carrying out a multi-period numerical analysis for optimal investment problem under a convex option compensation scheme with random time horizon. We observe that the distribution of the non-concave portfolio in both certain and uncertain random time horizon is right-skewed with a long right tail, indicating that the investor expects frequent small losses and a few large gains from the investment. While the (certain) average time horizon portfolio at a premature stopping date is unimodal, the random time horizon portfolio is multimodal distributed which provides the investor a certain flexibility of switching between the local maximizers, depending on the market performance. The multimodal structure with multiple peaks of different heights can be explained by the concavification procedure, whereas the distribution of the time horizon has significant impact on the amplitude between the modes.
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