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In [[actuarial science]], the '''Esscher transform''' {{harv|Gerber|Shiu|1994}} is a transform that takes a [[probability density function|probability density]] ''f''(''x'') and transforms it to a new probability density ''f''(''x'';&nbsp;''h'') with a parameter ''h''. It was introduced by F. Esscher in 1932 {{harv|Esscher|1932}}.

==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 ''&mu;'' is a [[probability measure]], the Esscher transform of ''&mu;'' is a new probability measure ''E<sub>h</sub>''(''&mu;'') which has [[Radon–Nikodym derivative|density]]

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

with respect to ''&mu;''.

== Basic properties ==

; Combination

: The Esscher transform of an Esscher transform is again an Esscher transform: ''E<sub>h</sub>''<sub><sub>1</sub></sub>&nbsp;''E<sub>h</sub>''<sub><sub>2</sub></sub>&nbsp;=&nbsp;''E<sub>h</sub>''<sub><sub>1</sub>&nbsp;+&nbsp;''h''<sub>2</sub></sub>.

; Inverse

: The inverse of the Esscher transform is the Esscher transform with negative parameter: ''E''{{su|b=''h''|p=&minus;1}}&nbsp;=&nbsp;''E''<sub>&minus;''h''</sub>

; 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 ==
{|class="wikitable"
|-
! Distribution
! Esscher transform 
|-
| [[Bernoulli distribution|Bernoulli]] Bernoulli(''p'')
| &nbsp;<math>\,\frac{e^{hk}p^k(1-p)^{n-k}}{1-p+pe^h}</math>
|-
| [[Binomial distribution|Binomial]] B(''n'',&nbsp;''p'')
| &nbsp;<math>\,\frac{{n\choose k}e^{hk}p^k(1-p)^{n-k}}{(1-p+pe^h)^n}</math>
|-
| [[Normal distribution|Normal]] ''N''(''μ'', ''σ''<sup>2</sup>'')
| &nbsp; <math>\,\frac{1}{\sqrt{2 \pi \sigma^2}}e^{-\frac{(x-\mu-\sigma^2 h)^2}{2\sigma ^2}}</math>
|-
| [[Poisson distribution|Poisson]] Pois(''λ'')
| &nbsp; <math>\,\frac{e^{h-\lambda e^h}\lambda^k}{k!}</math>
|- 
|}

==See also==
* [[Esscher principle]]

== References ==
* {{cite journal
 |title=Option Pricing by Esscher Transforms
 |last=Gerber
 |first=Hans U.
 |first2=Elias S. W.
 |last2=Shiu
 |journal=Transactions of the Society of Actuaries
 |volume=46
 |year= 1994
 |pages=99–191
 | url = http://pages.stern.nyu.edu/~dbackus/Disasters/Gerber_Shiu_94.pdf
 |ref=harv}}

* {{cite journal
 |title=On the Probability Function in the Collective Theory of Risk
 |journal=Skandinavisk Aktuarietidskrift
 |volume=15
 |issue = 3
 |year=1932
 |pages=175–195
 |last=Esscher
 |first=F.
 |doi = 10.1080/03461238.1932.10405883
 |ref=harv}}

[[Category:Actuarial science]]
[[Category:Transforms]]