Hans Bühlmann Mathematical Methods in Risk Theory

$S$: loss amount $g(S)$: retention (loss held by the insurer)

Proportional reinsurance with retained share of $a$ $g(S)=a S$.
Excess of loss: $g(S)= S $ for $S\le M$, $g(S)= M$ for $S >M$

Basic mathematical model is a compund Poisson process:

$Z_t = \sum_{i \in A_t}^N X_i$.

$Z_t$ total claim amount
$A_t$ jump time, claim occurence time, Poisson process with intensity $\lambda$
$X_i$ claim amount, iid distributed

Tools: Moment generating functions, cumulant generating function

From total claim amount: calculate premium, retention and reserve. F Fr