Designing tailor-made risk measures for insurance and financial applications

在过去的二十年里，风险度量的综合理论得到了发展。金融和保险市场是现代风险度量理论的主要动因和应用领域。尽管在该领域的广泛研究，仍然有许多有趣的方向有待探索。事实上，保险和金融领域出现的特殊情况会带来一些问题，需要采取更适合特定目标的量身定制的风险措施。这超出了现有方法和工具的简单应用，而是需要开发新的数学结构。例如，在设计以破产为基础的风险计量办法以便应用于保险的问题上就存在这种情况。在过去的二十年里，破产理论已经发展成为一个大型的数学框架，它关注于研究所谓的破产问题。破产相关的量，如破产概率和破产赤字，现在很好地理解了广泛的随机模型。然而，目前还没有重大的研究设计有意义的破产为基础的风险措施。这个项目的第一个目标是在有界càdlàg过程空间上设计新的风险度量，它可以像与破产相关的随机变量那样以类似的方式捕获与模型的路径属性相关的风险，但是具有自底向上的公理化构造，从而产生一致或凸的风险度量。这个项目的另一个目标正是研究更适合于公理化风险度量设计的保险或金融模型的新路径属性。一般而言，这些新风险措施的设计将在两个层面上进行。首先，有必要定义有意义的路径依赖量，这些量的表达式可以在利维模型提供的最大范围内推导出来。第二步将是基于我们模型的路径依赖特性的风险度量的实际设计。有许多次要项目可以从主要目标中跳出来，使其成为一个雄心勃勃但值得的项目。

The last two decades have seen the development of a comprehensive theory of risk measures. The financial and insurance markets are the two main economic sectors where the modern theory of risk measures mainly finds its motivation and application. Despite the extensive research in the field, there are still many interesting directions to be explored. In fact, particular situations arising in insurance and finance pose problems that call for tailor-made risk measures that are better suited for a given particular goal. This goes beyond a simple application of the existing methods and tools and it requires instead the development of new mathematical constructs. For instance, one such situation is found in the problem of designing ruin-based risk measures with a view towards insurance applications. In the last twenty years, ruin theory has grown into a large mathematical framework that concerns itself with the study of the so-called ruin problem. Ruin related quantities, such as the probability of ruin and the deficit at ruin, are now well understood for a wide range of stochastic models. Yet there exist no significant research on designing meaningful ruin-based risk measures. The first goal of this project is then to design new risk measures on the space of bounded càdlàg processes that can capture the risk associated with the path-properties of the model in a similar fashion as the ruin-related random variables do, but with a bottom-up axiomatic construction yielding coherent or convex risk measures. Another objective of this project is precisely the study of new path properties of an insurance or financial model that are more suitable for the design of axiomatic risk measures. In general, the designing of these new risk measures will be carried out in two levels. First, it will be necessary to define meaningful path-dependent quantities for which expressions can be derived in the largest context provided by Levy models. The second step will be the actual design of risk measures based on path-dependent properties of our models. There are numerous side projects that can spring out of the main objectives, making it an ambitious yet worthwhile project.