By Georg Ch Pflug

This ebook is the 1st out there to regard unmarried- and multi-period danger measures (risk functionals) in a radical, entire demeanour. It combines the remedy of houses of the danger measures with the similar facets of selection making below danger. The publication introduces the speculation of possibility measures in a mathematically sound method. It includes houses, characterizations and representations of danger functionals for single-period and multi-period actions, and in addition indicates the embedding of such functionals in determination types and the houses of those types.

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**Example text**

R3) Positive homogeneity. ρ(λY ) = λρ(Y ), for λ ≥ 0. (R4) Pointwise antimonotonicity. Y (1) ≤ Y (2) implies that ρ(Y (1) ) ≥ ρ(Y (2) ). ρ is a coherent risk capital functional, if and only if A(Y ) = −ρ(Y ) is a positively homogeneous acceptability functional. The notion of acceptability functionals is just the mirror image of the concept of risk (capital) functionals. A coherent risk (capital) functional may be interpreted as the amount of risk capital needed to make a profit distribution Y acceptable.

A version-independent functional R is called recursive, if the following relationship holds for compound distributions K ◦ G R{K ◦ G} = R{H}, where H is the distribution of R{K(·|X)}, for an X, which has distribution G. 16 Example. Here is a general method for constructing recursive functionals: Let U be a strictly monotonic utility function and let U −1 its inverse. The functional R{G} = U −1 U (v) dG(v) is recursive. Notice first that R{K ◦ G} = U −1 U (w) dK(w|v) dG(v) . 2 U (w) dK(w|v) dG(v) = R{K ◦ G}.

We denote by dom R the domain of R, that is the set, where R is finite. Examples. Examples of probability functionals are the expectation E(Y ), the median Med(Y ) or the functional Y → E(Y 2 Z) for some fixed Z. The natural domain for the expectation is L1 and for the median is the whole Y. There is an important distinction to make: Some functionals are determined by the distribution function G of Y , G(u) = P{Y ≤ u}, some are 27 MERI˙WS June 3, 2007 11:37 World Scientific Book - 9in x 6in 28 Modeling, Measuring and Managing Risk not.

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