By Georg Ch Pflug

This ebook is the 1st available in the market to regard unmarried- and multi-period possibility measures (risk functionals) in an intensive, accomplished demeanour. It combines the therapy of houses of the chance measures with the comparable points of selection making below chance. The e-book introduces the speculation of danger measures in a mathematically sound method. It includes houses, characterizations and representations of probability functionals for single-period and multi-period actions, and likewise exhibits the embedding of such functionals in choice versions and the homes of those versions.

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**Sample text**

Vi) and (vii) are obvious. (viii) is analogous to (iii). (ix) The functional E(Y )−D(Y ) has the dual representation inf{E(Y Z)+D(1−Z) : Z ∈ ZD }. If Z ≤ 1, then 1 − Z ≥ 0, and the monotonicity follows from (iv). (x) Suppose that D(Y ) is version-independent. 14) that Y (1) = E(Y (2) |Y (1) ). Since E[Y (1) Z] = E[E(Y (2) |Y (1) ) Z] = E[Y (2) E(Z|Y (1) )] and since by assumption D(E(Z|Y (1) )) ≤ D(Z) we have D(Y (1) ) = sup{E[Y (1) Z] − D(Z) : Z ∈ ZD } = sup{E[Y (2) E(Z|Y (1) )] − D(Z) : Z ∈ ZD } ≤ sup{E[Y (2) E(Z|Y (1) )] − D(E(Z|Y (1) )) : Z ∈ ZD } ≤ sup{E[Y (2) Z] − D(Z) : Z ∈ ZD } = D(Y (2) ).

Another name for ZA resp. ZD is risk envelope. The following theorem demonstrates how most properties of functionals can be derived from their dual representation. We formulate it both for acceptability functionals and for deviation risk functionals. 30 Theorem. (A) Suppose that A has the representation A(Y ) = inf{E(Y Z) − A(Z) : Z ∈ ZA ⊆ Z}, where A(Z) > −∞ for Z ∈ ZA . Suppose further that the probability space is non-atomic. Then (i) A is translation-equivariant (A1), if for all Z ∈ ZA , E(Z) = 1.

Compound linearity follows from the following equations R{G} = U (v) dK ◦ G(v) = U (v) dK(v|w) dG(w). The same relation, expressed in terms of random variables reads R(Y ) = E[U (Y )] = E[E(U (Y )|F1 )] = E[R(Y |F1 )]. 14 Example. 3) where H is some class of functions and A is some functional, is compound convex. This follows from R{K ◦ G} = sup{ = sup{ h(u) dK(u|v) dG(v) − A(h) : h ∈ H} h(u) dK(u|v) − A(h) dG(v) : h ∈ H} ≤ sup{h(u) dK(u|v) − A(h) : h ∈ H} dG(v) = R{K(·|v)} dG(v). On the other hand, a functional of the form inf{ h(v) dG(v) − A(h) : h ∈ H} is compound concave.