First-claim decomposition for survival probability
= First-claim decomposition for survival probability
In a <classical risk model>, before the first claim at time $t$, available capital is $u+ct$. A claim of size $x\le u+ct$ leaves future survival probability $\varphi(u+ct-x)$ by the <Markov property>. Integrating over the independent first-arrival <exponential distribution> and claim density gives $\varphi(u)=\int_0^\infty\lambda e^{-\lambda t}\int_0^{u+ct}\varphi(u+ct-x)f(x)\,dx\,dt$.