= Ultimate ruin with exponential claims
{title2=$\psi(u)=\frac{\lambda}{\alpha c}e^{-(\alpha-\lambda/c)u}$}
For rate-$\alpha$ <exponential distribution> claims and $c>\lambda/\alpha$, the <ruin integro-differential equation> reduces to $\psi''+(\alpha-\lambda/c)\psi'=0$. Differentiate the convolution and use its equation $A'=\alpha\psi-\alpha A$ to eliminate it. Positive <relative safety loading> implies $\psi(u)\to0$ as $u\to\infty$, and the <zero-capital survival probability> gives $\psi(0)=\lambda/(\alpha c)$. These two conditions determine the displayed solution. The <adjustment coefficient> is $\alpha-\lambda/c$.
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