Ruin integro-differential equation ($\frac{c}{\lambda }{\psi }^{\prime }(u)=\psi (u)-(f_X\ast \psi )(u)-{\overline{F}}_X(u)$)

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.

In the classical risk model with density $3e^{-4x}+e^{-2x}/2$ and relative safety loading $3/5$, the claim mean is $5/16$ and $\lambda /c=2$. The convolution terms $A_a(u)={\int }_0^u\varphi (u-x)e^{-ax}dx$ satisfy $A_a^{\prime }=\varphi -a{A}_a$. Applying $(D+4)(D+2)$ to ${\varphi }^{\prime }=2\varphi -6A_4-A_2$ yields ${\varphi }^{(3)}+4{\varphi }^{\prime \prime }+3{\varphi }^{\prime }=0$. Its solution is fixed by $\varphi (\infty )=1$, $\varphi (0)=3/8$ and ${\varphi }^{\prime }(0)=3/4$, giving the displayed probability. The derivative condition comes from the original integral equation and excludes spurious solutions of the higher-order equation.