In actuarial science, the Esscher transform (Gerber & Shiu 1994) is a transform that takes a probability density f(x) and transforms it to a new probability density f(x; h) with a parameter h. It was introduced by F. Esscher in 1932 (Esscher 1932).
Let f(x) be a probability density.
Its Esscher transform is defined as More generally, if μ is a probability measure, the Esscher transform of μ is a new probability measure Eh(μ) which has density with respect to μ.