Distortion function

A distortion function g : [ 0 , 1 ] → [ 0 , 1 ] is a non-decreasing function such that g ( 0 ) = 0 and g ( 1 ) = 1 . The dual distortion function is g ~ ( x ) = 1 − g ( 1 − x ) . Distortion functions are used to define distortion risk measures.

Given a probability space ( Ω , F , P ) , then for any random variable X and any distortion function g we can define a new probability measure Q such that for any A ∈ F it follows that