= Erlang claim-size differential equation for survival probability
{c}
{title2=$\phi'''+(2\beta-r)\phi''+(\beta^2-2r\beta)\phi'=0$}
For an <Erlang distribution> claim density $f(x)=\beta^2xe^{-\beta x}$ and $r=\lambda/c$, write $H=\phi*f$. The <survival integro-differential equation> gives $\phi'=r(\phi-H)$. The convolution satisfies $(D+\beta)^2H=\beta^2\phi$, because $H=\beta^2e^{-\beta u}\int_0^u(u-t)\phi(t)e^{\beta t}dt$. Applying $(D+\beta)^2$ to the first equation eliminates $H$ and gives the displayed third-order equation. The original integral equation also supplies $\phi'(0)=r\phi(0)$ and $\phi''(0)=r^2\phi(0)$; these conditions must not be lost when solving the higher-order equation.
Back to article page
