This thesis develops several strategies for calculating ruin-related quantities for a variety of extended risk models. We focus on the Sparre-Andersen risk model, also known as the renewal risk model. The idea of arbitrary distribution for the waiting time between claim payments arose in the 1950’s from the collective risk theory, and received many extensions and modifications in recent years. Our goal is to tackle model assumptions that are either too relaxed for traditional methods to apply, or so complicated that elaborate algebraic tools are needed to obtain explicit solutions.
In Chapter 2, we consider a Lévy risk process and a Sparre-Andersen risk process with Parisian ruin in the presence of a constant dividend barrier. We demonstrate that with few exceptions, ruin occurs with certainty. Generalizations to certain dependent risk processes are discussed. We also provide a reinsurance contract in which the certainty of ruin can be avoided.
In Chapter 3, we investigate a class of Sparre-Andersen risk processes in which the inter-claim time is rational-distributed. A key property of the rational class is derived, which allows for direct derivation of an integro-differential equation satisfied by a probability concerning the maximum surplus. The solution is constructed using a set of linearly independent functions, one of which is obtained by a standard technique through a defective renewal equation while the rest are obtained via a homogeneous equation. The necessary boundary conditions are presented. We also provide examples involving rational claim sizes as well as an application to the total dividends paid under a threshold strategy.
In Chapter 4, we extend an exponential-combination dependence structure to an Erlang-combination for the Sparre-Andersen risk models in presence of diffusion. A set of tools are developed for establishing certain integro-differential equations in Gerber–Shiu analysis. This new technique lifts previous constraint on the multiplicities of parameters of the inter-claim times. We then illustrate applications of these equations under a variety of special dependence models. Results are compared with existing literature, including the diffusion-free cases.
Finally, in Chapter 5, we collect various results and provide conclusions. We also give an outline of potential future research
