= Maximum aggregate loss in a classical risk model
{title2=$L=\sup_{t\geq0}(S(t)-ct),\quad\mathbb P(L>u)=\psi(u)$}
Starting from capital $u$, ruin occurs exactly when the aggregate loss exceeds $u$. Positive <relative safety loading> and the <strong law of large numbers> make $S(t)-ct$ tend to minus infinity, so its supremum is finite almost surely. For <exponential distribution> claims of rate $\alpha$, $L$ has an atom $1-\lambda/(\alpha c)$ at zero, and, conditional on $L>0$, has <exponential distribution> of rate $\alpha-\lambda/c$. The <ultimate ruin probability> determines this mixture directly.
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