Distortion function

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

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

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