In actuarial science, the Esscher transform 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 .
Definition
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.
Basic properties
Combination
The Esscher transform of an Esscher transform is again an Esscher transform: Eh1 Eh2 = Eh1 + h2.
Inverse
The inverse of the Esscher transform is the Esscher transform with negative parameter: E = E−h
Mean move
The effect of the Esscher transform on the normal distribution is moving the mean: