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Title: A note on the quantile formulation
Authors: Xu, ZQ
Keywords: Atomic
Behavioral finance
Calculus of variations
Functional optimization problem
Portfolio choice/selection
Probability weighting/distortion function
Quantile formulation
Relaxation method
Time consistency
Issue Date: 2016
Publisher: Wiley-Blackwell
Source: Mathematical finance, 2016, v. 26, no. 3, p. 589-601 How to cite?
Journal: Mathematical finance 
Abstract: 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.
ISSN: 0960-1627
DOI: 10.1111/mafi.12072
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