Designing tailor-made risk measures for insurance and financial applications

在过去的二十年里，风险衡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产生和应用的两个主要经济领域。尽管在这一领域进行了广泛的研究，但仍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带来了问题，需要量身定做更适合特定目标的风险措施。这不仅仅是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，在设计基于破产的风险措施以期适用于保险应用的问题中就发现了一种这样的情况。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，它关注于所谓的破产问题的研究。破产相关的量，如破产概率和破产时的赤字，现在被广泛的随机模型所理解。然而，目前还没有关于设计有意义的基于破产的风险度量的重要研究。这个项目的第一个目标是在有界相依过程空间上设计新的风险度量，它能够以类似于破产相关随机变量的方式捕捉与模型的路径属性相关的风险，但具有自下而上的公理构造，产生连贯的或凸的风险度量。这个项目的另一个目标恰恰是研究更适合于设计公理风险度量的保险或金融模型的新路径性质。一般而言，这些新风险措施的设计将分两个层面进行。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Levy模型提供的最大上下文中导出。第二步将是基于我们模型的路径依赖属性的风险度量的实际设计。有许多可以从主要目标中脱颖而出的附带项目，使其成为一个雄心勃勃但有价值的项目。

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.