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(xh) with a parameter h. It was introduced by F. Esscher in 1932 (Esscher 1932).


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 Eh(μ) which has density

with respect to μ.

Basic properties[edit]

The Esscher transform of an Esscher transform is again an Esscher transform: Eh1 Eh2 = Eh1 + h2.
The inverse of the Esscher transform is the Esscher transform with negative parameter: E−1
 = Eh
Mean move
The effect of the Esscher transform on the normal distribution is moving the mean:


Distribution Esscher transform
Bernoulli Bernoulli(p)  
Binomial B(np)  
Normal N(μ, σ2)  
Poisson Pois(λ)  

See also[edit]


  • Esscher, F. (1932). "On the Probability Function in the Collective Theory of Risk". Skandinavisk Aktuarietidskrift 15 (3): 175–195. doi:10.1080/03461238.1932.10405883.