In financial mathematics and economics, a distortion risk measure is a type of risk measure which is related to the cumulative distribution function of the return of a financial portfolio.
Mathematical definition.
The function formula_1 associated with the distortion function formula_2 is a "distortion risk measure" if for any random variable of gains formula_3 (where formula_4 is the Lp space) then
 formula_5
where formula_6 is the cumulative distribution function for formula_7 and formula_8 is the dual distortion function formula_9.
If formula_10 almost surely then formula_11 is given by the Choquet integral, i.e. formula_12 Equivalently, formula_13 such that formula_14 is the probability measure generated by formula_15, i.e. for any formula_16 the sigma-algebra then formula_17.
Properties.
In addition to the properties of general risk measures, distortion risk measures also have:
References.
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