Distortion risk measure

Mathematical definition

The function ${\displaystyle {\rho }_g:L^p\to \mathbb{R}}$ associated with the distortion function ${\displaystyle g:[0,1]\to [0,1]}$ is a distortion risk measure if for any random variable of gains ${\displaystyle X\in L^p}$ (where ${\displaystyle L^p}$ is the L^p space) then

${\displaystyle {\rho }_g(X)=-{\int }_0^1{F}_{-X}^{-1}(p)d\tilde{g}(p)={\int }_{-\mathrm{\infty }}^0\tilde{g}(F_{-X}(x))dx-{\int }_0^{\mathrm{\infty }}g(1-F_{-X}(x))dx}$

where ${\displaystyle F_{-X}}$ is the cumulative distribution function for ${\displaystyle -X}$ and ${\displaystyle \tilde{g}}$ is the dual distortion function ${\displaystyle \tilde{g}(u)=1-g(1-u)}$.

If ${\displaystyle X\le 0}$ almost surely then ${\displaystyle {\rho }_g}$ is given by the Choquet integral, i.e. ${\displaystyle {\rho }_g(X)=-{\int }_0^{\mathrm{\infty }}g(1-F_{-X}(x))dx.}$ Equivalently, ${\displaystyle {\rho }_g(X)={\mathbb{E}}^{\mathbb{Q}}[-X]}$ such that ${\displaystyle \mathbb{Q}}$ is the monotone and normalized set function generated by ${\displaystyle g}$, i.e. for any ${\displaystyle A\in \mathcal{F}}$ the sigma-algebra then ${\displaystyle \mathbb{Q}(A)=g(\mathbb{P}(A))}$.

Properties

In addition to the properties of general risk measures, distortion risk measures also have:

1. Law invariant: If the distribution of ${\displaystyle X}$ and ${\displaystyle Y}$ are the same then ${\displaystyle {\rho }_g(X)={\rho }_g(Y)}$.
2. Monotone with respect to first order stochastic dominance.
   1. If ${\displaystyle g}$ is a concave distortion function, then ${\displaystyle {\rho }_g}$ is monotone with respect to second order stochastic dominance.
3. ${\displaystyle g}$ is a concave distortion function if and only if ${\displaystyle {\rho }_g}$ is a coherent risk measure.