Distortion risk measure

In financial mathematics, 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. In financial mathematics, 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. The function ρ g : L p → R {displaystyle ho _{g}:L^{p} o mathbb {R} } associated with the distortion function g : [ 0 , 1 ] → [ 0 , 1 ] {displaystyle g: o } is a distortion risk measure if for any random variable of gains X ∈ L p {displaystyle Xin L^{p}} (where L p {displaystyle L^{p}} is the Lp space) then where F − X {displaystyle F_{-X}} is the cumulative distribution function for − X {displaystyle -X} and g ~ {displaystyle { ilde {g}}} is the dual distortion function g ~ ( u ) = 1 − g ( 1 − u ) {displaystyle { ilde {g}}(u)=1-g(1-u)} . If X ≤ 0 {displaystyle Xleq 0} almost surely then ρ g {displaystyle ho _{g}} is given by the Choquet integral, i.e. ρ g ( X ) = − ∫ 0 ∞ g ( 1 − F − X ( x ) ) d x . {displaystyle ho _{g}(X)=-int _{0}^{infty }g(1-F_{-X}(x))dx.} Equivalently, ρ g ( X ) = E Q [ − X ] {displaystyle ho _{g}(X)=mathbb {E} ^{mathbb {Q} }} such that Q {displaystyle mathbb {Q} } is the probability measure generated by g {displaystyle g} , i.e. for any A ∈ F {displaystyle Ain {mathcal {F}}} the sigma-algebra then Q ( A ) = g ( P ( A ) ) {displaystyle mathbb {Q} (A)=g(mathbb {P} (A))} .