Distortion risk measure
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 ρ g : L p → R {\displaystyle \rho _{g}:L^{p}\to \mathbb {R} } associated with the distortion function g : [ 0 , 1 ] → [ 0 , 1 ] {\displaystyle g:[0,1]\to [0,1]} is a distortion risk measure if for any random variable of gains X ∈ L p {\displaystyle X\in L^{p}} (where L p {\displaystyle L^{p}} is the Lp space) then ρ g ( X ) = − ∫ 0 1 F − X − 1 ( p ) d g ~ ( p ) = ∫ − ∞ 0 g ~ ( F − X ( x ) ) d x − ∫ 0 ∞ g ( 1 − F − X ( x ) ) d x {\displaystyle \rho _{g}(X)=-\int _{0}^{1}F_{-X}^{-1}(p)d{\tilde {g}}(p)=\int _{-\infty }^{0}{\tilde {g}}(F_{-X}(x))dx-\int _{0}^{\infty }g(1-F_{-X}(x))dx} where F − X {\displaystyle F_{-X}} is the cumulative distribution function for − X {\displaystyle -X} and g ~ {\displaystyle {\tilde {g}}} is the dual distortion function g ~ ( u ) = 1 − g ( 1 − u ) {\displaystyle {\tilde {g}}(u)=1-g(1-u)} .