Risk models with dependence: construction, properties, and risk measurement

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It is crucial for insurers and financial institutions to develop efficient risk management tools to have an appropriate assessment of the global risk associated to a portfolio of risks. In actuarial science, effective risk management relies on appropriate risk models, adequate risk measures that summarize the global risk of the portfolio through the distribution of the aggregate claim amount, and efficient methods to compute such a distribution. Contributions of the components of the portfolio to the risk measure according to a specific allocation rule is another key element. Besides modeling individual risks, the problem of finding the appropriate dependence structure describing their interrelations is challenging. My research program over the next few years is to propose and study risk models with dependence and, within them, to define appropriate risk measures, study their properties and propose methods to compute them. These innovations will be better suited for modern tasks of actuaries. In short-term, I will propose single-period risk models with dependence, and examine risk aggregation and risk allocation in those models. To this end, for portfolios with a large number of risks, I will propose new approaches to build hierarchical copulas, and investigate their properties and their estimation. I will also introduce multivariate counting distributions, based on the Poisson distribution and its extensions. Then, I will define and investigate collective risk models with different dependence structures that link the number of claims to each claim amount and each claim amount to the others. I will use hierarchical copulas to define such structures. Afterwards, I will examine desirable properties for ruin-based risk measures such as homogeneity, subadditivity, convexity, and consistency regarding different stochastics orders for stochastic processes. Notably, I will investigate the properties of an index of riskiness defined as the multiplicative inverse of the Lundberg coefficient. I will introduce discrete-time risk models, allowing both temporal dependence and dependence between lines of business. I will examine the impact of these two forms of dependence on ruin-based risk measures. I will also investigate risk allocation based on Euler's principle. My research program promises to have a significant impact on the construction and investigation of dependent risk models in actuarial science, leading to contributions in top journals in actuarial science and related fields. In the long term, I anticipate that the accomplishment of the objectives of my research program will significantly contribute to important advances and innovations in actuarial science.  My research program will also offer excellent training opportunities for HQP.