Multi-attribute preference robust utility based shortfall risk optimization and distributionally robust reward risk optimization

Abstract

The study of decision making under uncertainty is important in many areas (e.g. portfolio theory, control theory and utility theory). The exogenous and endogenous uncertainties, such as variations in stock prices, changes in consumer demand and ambiguity about investor’s risk attitude, are beyond deciders’ control and knowledge and significantly influence the effectiveness of any decision. In this thesis, we concentrate on this issue and propose some efficient models to deal with the uncertainties. Specifically, (a) we introduce a utility-based reward-risk ratio (URR) optimization model and consider a situation where an investor does not have complete information on the probability distribution of the underlying random variables, and we propose a distributionally robust URR optimization model to mitigate the risk arising from ambiguity of the true probability distribution; (b) we introduce a multivariate utility-based shortfall risk measure (MSR) and focus on a case that a decision maker’s true loss function in the definition of MSR is unknown but it is possible to elicit a set of plausible loss functions with partial information, and consequently propose a robust formulation of MSR based on the worst case loss function; (c) we investigate an issue that whether a statistical estimator such as the optimal value of a preference robust optimization model based on empirical data is reliable when the empirical data contain some noise, and we derive moderate sufficient conditions under which the optimal value of the model is robust against perturbation of the exogenous uncertainty data.

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Identifiers

ORCID for Huifu Xu: orcid.org/0000-0001-8307-2920

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