#### Thesis Format

Integrated Article

#### Degree

Doctor of Philosophy

#### Program

Statistics and Actuarial Sciences

#### Supervisor

Sendova, Kristina P.

#### Abstract

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.

#### Summary for Lay Audience

In ruin theory, the uncertainty faced by an insurance company is often described by a collective risk model. Under the classical assumptions, there are two sources of uncertainty: The first is the irregularity of when a claim would occur and the second is the unpredictability of how much a payment would be. We are interested in a wide range of quantifiable risk measures—the likelihood of ruin, the time of ruin and the severity of ruin, etc.

Miscellaneous extensions to the classical model are desired to better depict real-world phenomena. For instance, one may consider a more general distribution for inter-claim times; a dependence structure between inter-claim times and claim sizes; or a perturbation in premium income. These extensions aim to improve alignment with observations and require dedicated tools to provide actuaries with explicit solutions. Each model studied in this thesis is based on one or more aforementioned extensions. In particular, we focus on the following three different aspects of such extensions.

In the first article (Chapter 2), we consider a strategy in which the company pays out dividend whenever its surplus attains a constant level. We explore conditions that lead to certain ruin, and those that render ruin impossible.

In the second article (Chapter 3), we look at the company’s maximum revenue and study the boundary behavior of related risk measures. The structure of rational distributions is revealed by utilizing integral transforms. An application to the expected total dividend is discussed as well.

In the third article (Chapter 4), we investigate a general dependence model under perturbation and develop a set of algebraic tools for analyses on complex dependence structure. This enables us to build a general type of equations, which can then be evaluated under various special cases.

These additional considerations introduce some common obstacles: The distributional assumptions are either too relaxed or too complicated to be handled by traditional methods. In response to these issues, we adopt different strategies and develop new techniques. Our goal is to generalize the well-known results to their fullest potential while making exciting new discoveries down the road.

#### Recommended Citation

Zhang, Ruixi, "On the Sparre-Andersen Risk Models" (2019). *Electronic Thesis and Dissertation Repository*. 6756.

https://ir.lib.uwo.ca/etd/6756

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

#### Included in

Other Applied Mathematics Commons, Other Statistics and Probability Commons, Probability Commons