Distortion function

A distortion function ${\displaystyle g:[0,1]\to [0,1]}$ is a non-decreasing function such that ${\displaystyle g(0)=0}$ and ${\displaystyle g(1)=1}$. The dual distortion function is ${\displaystyle {\tilde {g}}(x)=1-g(1-x)}$.^[1][2] Distortion functions are used to define distortion risk measures.^[2]

Given a probability space ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}$, then for any random variable ${\displaystyle X}$ and any distortion function ${\displaystyle g}$ we can define a new probability measure ${\displaystyle \mathbb {Q} }$ such that for any ${\displaystyle A\in {\mathcal {F}}}$ it follows that

${\displaystyle \mathbb {Q} (A)=g(\mathbb {P} (X\in A)).}$ ^[1]

References

1. Balbás, A.; Garrido, J.; Mayoral, S. (2008). "Properties of Distortion Risk Measures". Methodology and Computing in Applied Probability. 11 (3): 385. doi:10.1007/s11009-008-9089-z.
2. Julia L. Wirch; Mary R. Hardy. "Distortion Risk Measures: Coherence and Stochastic Dominance" (pdf). Retrieved March 10, 2012.