Designing tailor-made risk measures for insurance and financial applications

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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.