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:
Examples Distribution Esscher transform Bernoulli Bernoulli(p) Binomial B(n, p) Normal N(μ, σ2) Poisson Pois(λ)
Esscher principle
The Esscher principle is an insurance premium principle used in actuarial sciences that derives from the Esscher transform. It is given by , where is a strictly positive parameter. This premium is the net premium for a risk , where denotes the moment generating function. This risk measure does not respect the positive homogeneity property of coherent risk measure for .
See also Exponential tilting
References