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.
ρ
associated with the distortion function
is a distortion risk measure if for any random variable of gains
is the Lp space) then where
is the cumulative distribution function for
is the dual distortion function
almost surely then
ρ
is given by the Choquet integral, i.e.
ρ
[1][2] Equivalently,
ρ
is the probability measure generated by
[3] In addition to the properties of general risk measures, distortion risk measures also have: