= Survival renewal equation for a classical risk model
{title2=$\varphi=1-\lambda\mu/c+(\lambda/c)\varphi*(1-F)$}
The ultimate <survival probability> satisfies $\varphi(u)=\varphi(0)+(\lambda/c)\int_0^u\varphi(u-x)(1-F(x))\,dx$. Condition on the first arrival of the <Poisson process>, differentiate the resulting exponentially weighted integral, and integrate the convolution derivative equation from zero. <Tonelli theorem> converts the claim-density convolution to the tail convolution. The kernel mass is $\lambda\mu/c<1$, so this is a <defective renewal equation>.
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