英文标题：
《A Note on the Quantile Formulation》
---
作者：
Zuo Quan Xu
---
最新提交年份：
2014
---
英文摘要：
  Many investment models in discrete or continuous-time settings boil down to maximizing an objective of the quantile function of the decision variable. This quantile optimization problem is known as the quantile formulation of the original investment problem. Under certain monotonicity assumptions, several schemes to solve such quantile optimization problems have been proposed in the literature. In this paper, we propose a change-of-variable and relaxation method to solve the quantile optimization problems without using the calculus of variations or making any monotonicity assumptions. The method is demonstrated through a portfolio choice problem under rank-dependent utility theory (RDUT). We show that this problem is equivalent to a classical Merton\'s portfolio choice problem under expected utility theory with the same utility function but a different pricing kernel explicitly determined by the given pricing kernel and probability weighting function. With this result, the feasibility, well-posedness, attainability and uniqueness issues for the portfolio choice problem under RDUT are solved. It is also shown that solving functional optimization problems may reduce to solving probabilistic optimization problems. The method is applicable to general models with law-invariant preference measures including portfolio choice models under cumulative prospect theory (CPT) or RDUT, Yaari\'s dual model, Lopes\' SP/A model, and optimal stopping models under CPT or RDUT.
---
中文摘要：
许多离散或连续时间的投资模型归结为最大化决策变量分位数函数的目标。这个分位数优化问题被称为原始投资问题的分位数公式。在某些单调性假设下，文献中已经提出了几种解决此类分位数优化问题的方案。在本文中，我们提出了一种变量变化和松弛方法来解决分位数优化问题，而不使用变分法或任何单调性假设。该方法通过秩相关效用理论（RDUT）下的投资组合问题进行了验证。我们证明了该问题等价于期望效用理论下的经典Merton投资组合问题，其效用函数相同，但定价核不同，由给定的定价核和概率加权函数明确确定。利用这个结果，解决了RDUT下投资组合选择问题的可行性、适定性、可达性和唯一性问题。研究还表明，求解函数优化问题可以简化为求解概率优化问题。该方法适用于具有规律不变偏好测度的一般模型，包括累积前景理论（CPT）或RDUT下的投资组合选择模型、Yaari的对偶模型、Lopes的SP/A模型以及CPT或RDUT下的最优停止模型。
---
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
--

---
PDF下载：
-->