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

Definition

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

f ( x ; h ) = e h x f ( x ) ∫ − ∞ ∞ e h x f ( x ) d x . {\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 Eh(μ) which has density

e h x ∫ − ∞ ∞ e h x d μ ( x ) {\displaystyle {\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
h = E−h
Mean move
The effect of the Esscher transform on the normal distribution is moving the mean:
E h ( N ( μ , σ 2 ) ) = N ( μ + h σ 2 , σ 2 ) . {\displaystyle E_{h}({\mathcal {N}}(\mu ,\,\sigma ^{2}))={\mathcal {N}}(\mu +h\sigma ^{2},\,\sigma ^{2}).\,}

Examples

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

See also

References

  • 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.
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