Esscher transform

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).

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

${\displaystyle f(x;h)={\frac {e^{hx}f(x)}{\int _{-\infty }^{\infty }e^{hx}f(x)\,dx}}.\,}$

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

${\displaystyle {\frac {e^{hx}}{\int _{-\infty }^{\infty }e^{hx}\,d\mu (x)}}}$

with respect to μ.

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: E⁻¹
_h = E_−h

Mean move

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

${\displaystyle E_{h}({\mathcal {N}}(\mu ,\,\sigma ^{2}))={\mathcal {N}}(\mu +h\sigma ^{2},\,\sigma ^{2}).\,}$

Distribution	Esscher transform
Bernoulli Bernoulli(p)	${\displaystyle k\mapsto {\frac {e^{hk}p^{k}(1-p)^{1-k}}{1-p+pe^{h}}}{\text{ for }}k=0,1.}$
Binomial B(n, p)	${\displaystyle k\mapsto {\frac {{n \choose k}e^{hk}p^{k}(1-p)^{n-k}}{(1-p+pe^{h})^{n}}}{\text{ for }}k=0,1,\ldots ,n.}$
Normal N(μ, σ2)	${\displaystyle x\mapsto {\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu -\sigma ^{2}h)^{2}}{2\sigma ^{2}}}}{\text{ for }}x{\text{ real.}}}$
Poisson Pois(λ)	${\displaystyle k\mapsto {\frac {e^{hk-\lambda e^{h}}\lambda ^{k}}{k!}}{\text{ for }}k=0,1,2,\ldots \,.}$

The Esscher principle is an insurance premium principle used in actuarial sciences that derives from the Esscher transform. It is given by ${\displaystyle \pi [X,h]=E[Xe^{hX}]/E[e^{hX}]}$, where ${\displaystyle h}$ is a strictly positive parameter. This premium is the net premium for a risk ${\displaystyle Y=Xe^{hX}/m_{X}(h)}$, where ${\displaystyle m_{X}(h)}$ denotes the moment generating function. This risk measure does not respect the positive homogeneity property of coherent risk measure for ${\displaystyle h>0}$.