Esscher transform

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


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

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


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

with respect to &mu;.

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$&minus;1 h$ = E&minus;h


 * Mean move


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


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