Gerber–Shiu Analysis of Multi-Layer Dividend Risk Models with Phase-Type Premiums and Claims: Theory, Special Cases, and Applications

We consider a compound-Poisson insurance reserve process with stochastic premium income, phase-type distributed claims and premiums and a segmented constant dividend outflow. We demonstrate that the non-local equations of ruin and dividend valuation admit a finite dimensional matrix representation which involves careful considerations of the boundary conditions. We find the free premium states and write the transversality equations. We extend the framework to accommodate discounted Gerber–Shiu penalties depending on both the reserve before ruin and the deficit. We explicitly keep the claim-penalty forcing term, including ruin as a side effect of continuous dividend payout. Instead of considering numerically observed spectral separation as a universal theorem we give a polynomial root-count analysis. The effects of fluctuations of premiums at constant mean, including Erlang-2 claims, are revealed by systematic comparisons and an autonomous numerical execution. We expand on the example of MTPL2: we see earned premiums, but not when the premium is paid. The MTPL2 earned-premium rate is distinguished from the income rate specified for a hypothetical 20% loading scenario. The empirical accounting figures and the stochastic cash-flow experiments are presented separately, since the dataset does not identify payment times or individual receipt sizes.