 Maximum aggregate loss in a classical risk model

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.