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For the Classical Risk model, the survival probability, $\phi(u)$, satisfies the integro-differential equation: $$\phi(u)=1-\psi(u)=\int\limits_o^\infty \int\limits_o^{u+ct}\lambda e^{-\lambda t}\phi(u+ct-x)f_X(x)dxdt.$$ Where $\psi(u)$ is the ruin probability.

Using change of limits show that, for $u\geq0$, $$\int\limits_o^u \int\limits_o^x\phi(s)e^{s/2}dsdx=\int\limits_o^u\phi(x)(u-x)e^{x/2}dx.$$

I have managed to derive the integro-differential equation that the survival probability, $\phi(u)$, satisfies (first equation), but I am really struggling to derive the second. Any help would be appreciated. For more background information, please see the attached image.enter image description here

Amber
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1 Answers1

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I was looking for a related topic (integro-diff equations) and I've found your (already dated) question. Anyway, it's just Fubini:

\begin{align*} \int_{0}^{u}\int_{0}^{x}\phi (s)e^{s/2}ds dx & = \int_{0}^{u}\int_{0}^{u}\textbf{1}_{s \le x}\phi (s)e^{s/2}ds dx \\ & = \int_{0}^{u}\left ( \int_{0}^{u}\textbf{1}_{s \le x}dx \right ) \phi (s)e^{s/2}ds\\ &= \int_{0}^u (u-s) \phi (s)e^{s/2}ds \end{align*}

Just change $s$ by $x$.

Leonardo Videla
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