Harman Patil (Editor)

Esscher transform

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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).

Contents

Definition

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

f ( x ; h ) = e h x f ( x ) ∫ − ∞ ∞ e h x f ( x ) d x .

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

e h x ∫ − ∞ ∞ e h x d μ ( 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 = E−h
Mean move
The effect of the Esscher transform on the normal distribution is moving the mean:

References

Esscher transform Wikipedia
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