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By Georg Ch Pflug

This ebook is the 1st available in the market to regard unmarried- and multi-period threat measures (risk functionals) in an intensive, finished demeanour. It combines the remedy of houses of the chance measures with the similar features of selection making less than probability. The publication introduces the speculation of hazard measures in a mathematically sound means. It comprises houses, characterizations and representations of chance functionals for single-period and multi-period actions, and likewise indicates the embedding of such functionals in selection versions and the homes of those types.

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S. s. June 3, 2007 11:37 World Scientific Book - 9in x 6in MERI˙WS Modeling uncertain outcomes 19 The construction for (i) is to set (Y (1) , Y (2) ) = for a U uniform in [0,1]. For the constructions (ii) and (iii) see [Strassen (1965)]. Proof. 15 Definition. 17)). For a distribution with only non-negative values, the value LG (p) of the Lorenz curve indicates the percentage of the total outcome, which is attributed to the p percent of worst scenarios. 8. 9. 2 0 0 Fig. 25, 0) Right: the pertaining Lorenz Lorenz curves are related to the convex ordering by the following relationship.

Call the function U supergradientbounded from below, if c := inf{a ∈ ∂U (v) : v ∈ R} is not equal to −∞. If c ≥ 0, then U is nondecreasing. If U is supergradient-bounded from below, then U (w) − cw is nondecreasing and concave. Thus according to the assumption, E[U (Y (1) )] − cE[Y (1) ] ≤ E[U (Y (2) )] − cE[Y (2) ], whence E[U (Y (1) )] ≤ E[U (Y (2) )]. If U is not supergradient-bounded from below, it may be approximated by a sequence of supergradient-bounded concave functions U (n) such that U (n) ↓ U and E(U (n) (Y (i) )) ↓ E(U (Y (i) )) for i = 1, 2.

For an agreed set U of utility functions, it is desirable that the probability functional respects the order induced by U. Otherwise contradiction would appear between decision makers choosing a utility function U from U and those basing their decision on the probability functional: There would exist two outcome distributions Y (1) and Y (2) , such that utility maximizers would prefer Y (1) over Y (2) , while the decision makers respecting the probability functional R would prefer Y (2) over Y (1) .

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