On risk theory and its applications

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The seminal work of Gerber and Shiu on the expected discounted penalty function in the classical risk model has caught the interest of the risk theory research community. The first natural reaction has been to extend the theory of these Gerber-Shiu (G-S) functions to more general classes of risk models. Recent works all represent valuable generalization efforts in different directions. These generalizations represent important advances that reduce some of the limitations of classical ruin theory and get closer to a realistic setting. Part of this project subscribes to a continuation of these efforts in extending the theory. At the same time, I plan also to investigate some applications of G-S functions. Criticisms are emerging to the effect that, like with ruin theory, G-S functions will not be useful in the study of practical actuarial solvency problems. Our first results seem to refute this idea. In any case, this question in certainly worth of scientific investigation. Classical ruin theory models claim arrivals by a homogeneous Poisson process. The exponential is thus the only possible distribution for the independent and identically distributed claim inter-arrival times. The theory studies the effect of the claim severity on ruin probabilities, as well as the effect of other model parameters (premiums, reinsurance, dividends, investment returns). The above extensions to more general claim arrival models, it is now also possible to investigate the effect of the claim frequency, not only on ruin probabilities, but also to the distribution of the other variables that the G-S function characterizes. For instance, we compare the ruin probabilities for Erlang(n) inter-arrival times, with constant mean but decreasing variances (in n). These are seen to reduce dramatically as n takes values 1,  2, and on. This reduction is significant in comparison to that obtained by varying the claim severity within a distributional class. Such studies can now be extended to wider frequency classes and for the distribution of the surplus before ruin, or the deficit at ruin. Perhaps this can help show if ruin theory with G-S functions is now closer to practical solvency studies, and can finally serve as an early warning system.