Английская Википедия:Esscher transform

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In actuarial science, the Esscher transform Шаблон:Harv 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 Шаблон:Harv.

Definition

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

<math>f(x;h)=\frac{e^{hx}f(x)}{\int_{-\infty}^\infty e^{hx} f(x) dx}.\,</math>

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

<math>\frac{e^{hx}}{\int_{-\infty}^\infty e^{hx} d\mu(x)} </math>

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: EШаблон:Su = E_−h

Mean move

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

<math>E_h(\mathcal{N}(\mu,\,\sigma^2)) =\mathcal{N}(\mu + h\sigma^2,\,\sigma^2).\,</math>

Examples

Distribution	Esscher transform
Bernoulli Bernoulli(p)	<math>\,\frac{e^{hk}p^k(1-p)^{1-k}}{1-p+pe^h}</math>
Binomial B(n, p)	<math>\,\frac{{n\choose k}e^{hk}p^k(1-p)^{n-k}}{(1-p+pe^h)^n}</math>
Normal N(μ, σ2)	<math>\,\frac{1}{\sqrt{2 \pi \sigma^2}}e^{-\frac{(x-\mu-\sigma^2 h)^2}{2\sigma ^2}}</math>
Poisson Pois(λ)	<math>\,\frac{e^{hk-\lambda e^h}\lambda^k}{k!}</math>

See also

• Esscher principle
• Exponential tilting

References

• Шаблон:Cite journal

• Шаблон:Cite journal