On the design of customized risk measures in insurance, the problem of capital allocation and the theory of fluctuations for Lévy processes
The aim of this thesis is to study fundamental problems in financial and insurance mathematics particularly the problem of measuring risk and its application within financial and insurance frameworks. The main contributions of this thesis can be classified in three main axes: the theory of risk measures, the problem of capital allocation and the theory of fluctuation. In Chapter 2, we design new coherent risk measures and study the associated capital allocation in the context of collective risk theory. We introduce the family of Cumulative Entropic Risk Measures. In Chapter 3, we study the optimal portfolio problem for the Entropic Value at Risk coherent risk measure for particular return models which are based on relevant cases of Jump-Diffusion models. In Chapter 4, we extending the notion of natural risk statistics to the multivariate setting. This non-trivial extension will endow us with multivariate data-based risk measures that are bound to have applications in finance and insurance. In Chapter 5, we introduce the concepts of drawdown and speed of depletion to the ruin theory literature and study them for the class of spectrally negative Lévy risk processes.
Processus càdlàg Drawdown Coherent and convex risk measure Capital allocation Multivariate data-based risk measures Càdlàg Process Optimal portfolio problem Spectrally negative Lévy process Speed of depletion