= Ruin integro-differential equation
{title2=$\frac c\lambda\psi'(u)=\psi(u)-(f_X*\psi)(u)-\overline F_X(u)$}
In the <classical risk model>, condition on the first arrival of the <Poisson process>. If available capital just before it is $v$, ruin has probability $\overline F_X(v)+\int_0^v f_X(x)\psi(v-x)\,dx$. Integrate this expression at $v=u+ct$ against $\lambda e^{-\lambda t}$ and change variables from $t$ to $v$. Differentiating the lower limit gives the displayed equation. The claim-density <convolution> is continuous for bounded $\psi$, so the first-arrival representation also supplies the differentiability required by the equation.
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