Set-valued Bellman's principle: evidence from the mean-risk problem
Abstract
Selecting a portfolio of risky assets which maximizes the expected terminal values at the same time as it minimizes portfolio risk is known as the mean-risk problem. The usual approach in the literature is to combine the mean and the risk to obtain a problem with a single objective. In a dynamic setting this scalarization, however, comes at the cost of time inconsistency. We show that these difficulties disappear by considering the problem in its natural form, that is, as a bi-objective optimization problem. As such the mean-risk problem satisfies an appropriate notion of time consistency, closely related to existence of a moving scalarization. Moreover, the mean-risk problem satisfies a Bellman's principle appropriate for a bi-objective optimization problem: a set-valued Bellman's principle. Based on joint work with Birgit Rudloff.
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