Distortion function

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

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

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