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

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

with respect to μ.

Basic properties

Combination

The Esscher transform of an Esscher transform is again an Esscher transform: E_h_₁ E_h_₂ = E_h_{₁ + 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

See also

• Esscher principle
• Exponential tilting

References