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Generalized deviations in risk analysis. (English) Zbl 1150.90006
The authors define deviation measures axiomatically and develop their dual counterparts which are called “risk envelops”. They are also described axiomatically. The key issues of geometry and semicontinuity are clarified. Questions are answered about how operations performed on deviation measures affect the associated risk envelops. Relations between deviation measures and “strictly expectation bounded risk measures” are studied, which in general form a class that neither includes, nor is included in, the class of coherent risk measures, although there is a major intersection. It is proved that the deviation measures that correspond to the risk measures in the intersection of the two classes are characterized by the heretofore unidentified property of being “lower range dominated”. It is shown how the deviation measures correspond to conditional value-at-risk and other such measures. Another wide array of examples are produced by introducing “basic error functionals”. For these deviation measures the risk envelopes and the exact cases in which lower range dominance is enjoyed are fully determined.
Reviewer: Yuliya Mishura

MSC:
90C25 Convex programming
90C30 Nonlinear programming
90C46 Optimality conditions and duality in mathematical programming
90C47 Minimax problems in mathematical programming
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