= Survival integro-differential equation for a classical risk model
{title2=$\varphi'(u)=(\lambda/c)(\varphi(u)-(\varphi*f)(u))$}
= Survival integro-differential equation
{synonym}
In the <classical risk model> with premium rate $c>0$ and claim <Poisson process> rate $\lambda$, the <first-claim decomposition for survival probability> implies the displayed equation for ultimate <survival probability>. Change the first-claim integral to an integral over available capital and differentiate its lower limit. The bounded <survival probability> function convolved with the integrable claim <probability density function> is continuous, so this also establishes the needed differentiability. In particular $\varphi'(0)=(\lambda/c)\varphi(0)$.
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