PhD Thesis Proposal - Public Lecture - Ruixi Zhang

Title: A new structure in a Sparre-Andersen risk process with applications

Abstract:  The study of risk processes takes the center stage of ruin theory. Started by the work of Filip Lundberg in 1903 and Harald Cramér in the 1930s, the theoretical framework assumes independent exponential inter-claim times within the surplus evolution of an insurance company. E. Sparre Andersen proposed in 1957 a more general model, in which the inter-claim times follow arbitrary distribution. Explicit results are known
for Erlang(2), Erlang(n), generalized Erlang(n) and phase-type inter-claim times.

In the first project, we investigate a class of Sparre-Andersen risk processes in which the inter-claim times are rational-distributed. We explore a key property of the rational distribution class. This property allows us to propose a new form of integro-differential equation. This newly discovered structure will enable us to study the maximum surplus and related topics such as dividend payments under a constant barrier strategy.

In the second project, we introduce a new source of randomness to the surplus process, a Wiener process modeling the uncertainty of the premium income. We will use the valuable insight provided by the first project to develop a new tool to study the perturbed risk process, in which the strong Markov property no longer holds.

We hope we will resolve the difficulties with the absence of Markov property, and we will find useful applications for these models.

Supervisor: Kristina Sendova