= Finite-claim ruin probability
{title2=$\psi_n(u)=\mathbb P(\min_{1\le j\le n}U_{T_j}<0)$}
In a <classical risk model>, $T_j$ is the time of the $j$th claim. The displayed probability restricts ruin to the first $n$ claims and increases to the <ultimate ruin probability>. Conditioning on the first arrival and severity gives $\psi_{n+1}(u)=\mathbb E[\overline F_X(u+cT_1)+\int_0^{u+cT_1}\psi_n(u+cT_1-x)f_X(x)\,dx]$, with $\psi_0=0$. For <exponential distribution> claims of mean $\mu$ and <relative safety loading> $\rho$, put $d=1+\rho$ and $h=d+1$. Then $\psi_1(u)=e^{-u/\mu}/h$ and $\psi_2(u)=e^{-u/\mu}(h^{-1}+(u/\mu)h^{-2}+dh^{-3})$. The recursion follows because a surviving first claim leaves fresh independent future arrivals and claims. The formulas follow by integrating the rate-$\lambda$ first-arrival density; the claim-arrival rate cancels with $c=d\lambda\mu$.
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