关于分位数公式的注记
A Note on the Quantile Formulation
摘要：
许多离散或连续时间环境下的投资模型归结为决策变量分位数函数的目标最大化。这个分位数优化问题被称为原始投资问题的分位数公式化。在一定的单调性假设下，文献中已经提出了几种解决这种分位数优化问题的方案。在本文中，我们提出了一个变量的变化和松弛的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。该方法是证明通过一个投资组合选择问题下的秩相关效用理论（RDUT）。我们发现，这个问题是等价于一个经典的默顿的投资组合选择问题下的期望效用理论具有相同的效用函数，但不同的定价核明确确定给定的定价核和概率加权函数。利用这一结果，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。它还表明，解决功能优化问题可能会减少到解决概率优化问题。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下的最优停时模型。
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.
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。
