On risk theory and its applications

Gerber和Shiu对经典风险模型中期望折现惩罚函数的开创性研究引起了风险理论研究界的兴趣。第一个自然反应是将这些Gerber-Shiu （G-S）函数的理论扩展到更一般的风险模型类别。最近的作品都代表了不同方向的有价值的概括努力。这些概括代表了重要的进步，减少了经典破产理论的一些局限性，并更接近于现实环境。这个项目的一部分是为了继续这些努力来扩展这个理论。同时，我还打算研究G-S函数的一些应用。批评正在出现，其效果是，与破产理论一样，G-S函数在实际精算偿付能力问题的研究中不会有用。我们的初步结果似乎反驳了这一观点。无论如何，这个问题肯定值得科学研究。经典破产理论模型声称到达是一个齐次泊松过程。因此，指数是独立且相同分布的索赔间隔到达时间的唯一可能分布。该理论研究了索赔严重程度对破产概率的影响，以及其他模型参数（保费、再保险、股息、投资回报）的影响。上述扩展到更一般的索赔到达模型，现在也有可能调查索赔频率的影响，不仅对破产概率，而且对G-S函数表征的其他变量的分布。例如，我们比较了Erlang(n)到达间隔时间的破产概率，平均值恒定，但方差（在n中）减小。当n取1、2等值时，这些值会急剧减少。与通过改变分布类内的索赔严重性所获得的减少相比，这种减少是显著的。这种研究现在可以扩展到更广泛的频率类别，以及破产前盈余或破产时赤字的分布。也许这有助于表明，具有G-S函数的破产理论现在是否更接近实际的偿付能力研究，并最终可以作为一个早期预警系统。

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.