Modeling, Measuring and Managing Risk by Georg Ch Pflug

By Georg Ch Pflug

This publication is the 1st out there to regard unmarried- and multi-period possibility measures (risk functionals) in an intensive, entire demeanour. It combines the remedy of homes of the danger measures with the comparable facets of selection making less than chance. The publication introduces the speculation of threat measures in a mathematically sound manner. It includes homes, characterizations and representations of danger functionals for single-period and multi-period actions, and in addition exhibits the embedding of such functionals in selection types and the houses of those types.

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Reverse supermartingale ordering, if E(Y (1) |Y (2) ) ≤ Y (2) implies that A(Y (1) ) ≤ A(Y (2) ). 14 (iii). Another group of functionals is formed by the translation-invariant ones. We call them deviation-type functionals. Within these, we identify a subgroup of functionals which measure the pure risk, which is the dimension orthogonal to the expectation and call them deviation risk functionals. A risk averse decision maker tries to minimize these functionals. 21 Definition: Deviation risk functionals.

19) G [−1] (q) = sup{uq − G(u) : u ∈ R}. 20) and We are now ready to state several alternative formulations of the order relations. 13 Theorem. (i) G1 ≺F SD G2 is equivalent to G1 (u) ≥ G2 (u) for all u ∈ R. −1 (ii) G1 ≺F SD G2 is also equivalent to G−1 1 (p) ≤ G2 (p) for all p ∈ (0, 1). (iii) G1 ≺CCD G2 is equivalent to G1 ≺SSD G2 and at the same time E{G1 } = E{G2 }. (iv) G1 ≺CCD G2 is also equivalent to the existence of a Markov kernel K with the properties G1 (u) = K(u|v) dG2 (v) and u dK(u|v) = v for all v.

The covariance and correlation are statistical parameters, which give some, but not a complete information about the dependence of two random variables with second moments. The covariance between the two random variables Y (1) and Y (2) is defined as Cov(Y (1) , Y (2) ) = E[(Y (1) − EY (1) )(Y (2) − EY (2) )], while their correlation is Corr(Y (1) , Y (2) ) = Cov(Y (1) , Y (2) ) . Var(Y (1) )Var(Y (2) ) MERI˙WS June 3, 2007 11:37 World Scientific Book - 9in x 6in Modeling uncertain outcomes MERI˙WS 25 For a random vector (Y (1) , .

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