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(x; h) 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

$f(x;h)=\frac{e^{hx}f(x)}{\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{hx}f(x)dx}.$

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

$\frac{e^{hx}}{\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{hx}d\mu (x)}$

with respect to μ.

Basic properties

Combination

The Esscher transform of an Esscher transform is again an Esscher transform: E_h1 E_h2 = E_h1 + h2.

Inverse

The inverse of the Esscher transform is the Esscher transform with negative parameter: E⁻¹
_h = E_−h

Mean move

The effect of the Esscher transform on the normal distribution is moving the mean:

$E_h(𝒩(\mu ,{\sigma }^2))=𝒩(\mu +h{\sigma }^2,{\sigma }^2).$

Examples

Distribution	Esscher transform
Bernoulli Bernoulli(p)	$\frac{e^{hk}{p}^k(1-p{)}^{1-k}}{1-p+p{e}^h}$
Binomial B(n, p)	$\frac{(\genfrac{}{}{0ex}{}{n}{k})e^{hk}{p}^k(1-p{)}^{n-k}}{(1-p+p{e}^h{)}^n}$
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)	$\frac{e^{hk-\lambda e^h}{\lambda }^k}{k!}$

See also

• Esscher principle
• Exponential tilting