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

:\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 = E−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).,

Examples

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

References

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Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

actuarial-science transforms

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