Mathematical applications in insurance
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