# 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

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

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

${\displaystyle {\frac {e^{hx}}{\int _{-\infty }^{\infty }e^{hx}d\mu (x)}}}$

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:
${\displaystyle E_{h}({\mathcal {N}}(\mu ,\,\sigma ^{2}))={\mathcal {N}}(\mu +h\sigma ^{2},\,\sigma ^{2}).\,}$

## Examples

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