Interior adjustment coefficient ruin prefactor ($C=(c-\lambda \mu )/(\lambda M^{\prime }(R)-c)$)

When the adjustment coefficient lies inside the finite domain of the claim moment-generating function, the tilted kernel has finite mean $(\lambda M^{\prime }(R)-c)/(cR)$. The forcing integral is $(c-\lambda \mu )/(cR)$. Their ratio is the constant $C$ in $\psi (u)\sim C{e}^{-Ru}$, by the key renewal theorem. An extra exponential moment controls the forcing tails and proves direct Riemann integrability.

For unit-rate Erlang distribution of shape two claims put $q=\lambda /c<1/2$. The adjustment coefficient satisfies $q(2-R)=(1-R{)}^2$, so $1-R=(q+\sqrt{q^2+4q})/2$. In the interior-adjustment prefactor $(1-2q)/(2q(1-R{)}^{-3}-1)$, substituting $q=(1-R{)}^2/(2-R)$ gives the displayed expression. It is positive for $0<R<1$ and tends to one as the loading tends to zero.