# Esscher transform

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(xh) with a parameter h. It was introduced by F. Esscher in 1932 (Esscher 1932).

## Definition

Let f(x) be a probability density. Its Esscher transform is defined as

$f(x;h)=\frac{e^{hx}f(x)}{\int_{-\infty}^\infty e^{hx} f(x) dx}.\,$

More generally, if μ is a probability measure, the Esscher transform of μ is a new probability measure Eh(μ) which has density

$\frac{e^{hx}}{\int_{-\infty}^\infty e^{hx} d\mu(x)}$

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−1
h
= Eh
Mean move
The effect of the Esscher transform on the normal distribution is moving the mean:
$E_h(\mathcal{N}(\mu,\,\sigma^2)) =\mathcal{N}(\mu + h\sigma^2,\,\sigma^2).\,$

## Examples

Distribution Esscher transform
Bernoulli Bernoulli(p)  $\,\frac{e^{hk}p^k(1-p)^{n-k}}{1-p+pe^h}$
Binomial B(np)  $\,\frac{{n\choose k}e^{hk}p^k(1-p)^{n-k}}{(1-p+pe^h)^n}$
Normal N(μ, σ2)   $\,\frac{1}{\sqrt{2 \pi \sigma^2}}e^{-\frac{(x-\mu-\sigma^2 h)^2}{2\sigma ^2}}$
Poisson Pois(λ)   $\,\frac{e^{h-\lambda e^h}\lambda^k}{k!}$