Multidimensional ruin theory with an emphasis on continuous time

Specialeforsvar: Astrid Søby Ovesen (kl. 15:15) og Victoria Graverholt Søkilde (kl. 16:15)

Titel: Multidimensional ruin theory with an emphasis on continuous time

Abstract: In this thesis, we extend the theory of the classical ruin problem, the Cramér-Lundberg model, to higher dimensions. This makes sense in an insurance context where the different dimensions of the total capital process fStg correspond to different lines of business, i.e. several coverages. First, we motive the multidimensional result for the ruin probability by considering the total capital loss process as a discrete-time process and discuss what it means to have ruin in higher dimensions. We then proceed into the proof of this thesis’ main theorem, namely an asymptotic estimate for how the ruin probability of a non-life insurance company decays with the initial capital of the respective company. For this, among other things, we make significant use of the theory of large deviations and convex analysis. Later, we present two examples of the use of the asymptotic ruin estimate. These we present in two dimensions, and they are each given a real-life perspective in the form of a financial and
insurance-related problem, respectively. We complete the ’practical’ use of the ruin estimate as far as numerical and analytical approaches allow us to. This includes an application of importance sampling. We finish off by considering different types of time dependence between the lines of business, namely examples of the so-called Gärtner-Ellis sequences, for which there exists an asymptotic ruin probability estimate as well. We study the Markov-additive, AR(1), and the compound Hawkes processes. In the examination of the dependent processes we check that they are, in fact, Gärtner-Ellis sequences. This leads to a brief discussion of the use of the ruin estimate in general.

Vejleder: Jeffrey Collamore
Censor:    Christian Tarp, PwC Danmark