published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

In the literature of ruin theory, there have been extensive studies trying to generalize the classical insurance risk model. In this thesis, we look into two particular risk processes considering multi-dimensional risk and dependent structures respectively. The first one is a bivariate risk process with a dividend barrier, which concerns a two-dimensional risk model under a barrier strategy. Copula is used to represent the dependence between two business lines when a common shock strikes. By defining the time of ruin to be the first time that either of the two lines has its surplus level below zero, we derive a discrete approximation procedure to calculate the expected discounted dividends until ruin under such a model. A thorough discussion of application in proportional reinsurance with numerical examples is provided as well as an examination of the joint optimal dividend barrier for the bivariate process. The second risk process is a semi-Markovian dual risk process. Assuming that the dependence among innovations and waiting times is driven by a Markov chain, we analyze a quantity resembling the Gerber-Shiu expected discounted penalty function that incorporates random variables defined before and after the time of ruin, such as the minimum surplus level before ruin and the time of the first gain after ruin. General properties of the function are studied, and some exact results are derived upon distributional assumptions on either the inter-arrival times or the gain amounts. Applications in a perpetual insurance and the last inter-arrival time before ruin are given along with some numerical examples. / published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

Traditionally, in the context of ruin theory, most judgements are made on an immediate sense. An example would be the determination of ruin, in which a business is declared broke right away when it attains a negative surplus. Another example would be the decision on dividend payment, in which a business pays dividends whenever the surplus level overshoots certain threshold. Such scheme of decision making is generally being criticized as unrealistic from a practical point of view. The Parisian concept is therefore invoked to handle this issue. This idea is deemed more realistic since it allows certain delay in the execution of decisions. In this thesis, such Parisian concept is utilized on two different aspects. The first one is to incorporate this concept on defining ruin, leading to the introduction of Parisian ruin time. Under such a setting, a business is considered ruined only when the surplus level stays negative continuously for a prescribed length of time. The case for a fixed delay is considered. Both the renewal risk model and the dual renewal risk model are studied. Under a mild distributional assumption that either the inter arrival time or the claim size is exponentially distributed (while keeping the other arbitrary), the Laplace transform to the Parisian ruin time is derived. Numerical example is performed to confirm the reasonableness of the results. The methodology in obtaining the Laplace transform to the Parisian ruin time is also demonstrated to be useful in deriving the joint distribution to the number of negative surplus causing or without causing Parisian ruin. The second contribution is to incorporate this concept on the decision for dividend payment. Specifically, a business only pays lump-sum dividends when the surplus level stays above certain threshold continuously for a prescribed length of time. The case for a fixed and an Erlang(n) delay are considered. The dual compound Poisson risk model is studied. Laplace transform to the ordinary ruin time is derived. Numerical examples are performed to illustrate the results. / published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

published_or_final_version / abstract / Statistics and Actuarial Science / Master / Master of Philosophy

published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

Cheung, Oi Lam Eunice. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 43-45). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Risk Theory --- p.1 / Chapter 1.2 --- Hybrid Numerical Scheme --- p.3 / Chapter 2 --- The Model --- p.5 / Chapter 2.1 --- Model --- p.5 / Chapter 2.2 --- Integro-Differential Equations --- p.8 / Chapter 2.3 --- Explicit Formulas and Asymptotic Properties --- p.13 / Chapter 3 --- Numerical Method --- p.16 / Chapter 3.1 --- From Integro-Differential Equations to Integral Equations --- p.17 / Chapter 3.2 --- Prom Integral Equations to Linear Equations --- p.19 / Chapter 3.3 --- Boundary Conditions --- p.20 / Chapter 3.4 --- Importance Sampling --- p.23 / Chapter 4 --- Numerical Study --- p.27 / Chapter 4.1 --- Exponential Claims with Equal Means --- p.28 / Chapter 4.1.1 --- Importance Sampling --- p.28 / Chapter 4.1.2 --- System of Linear Equations --- p.31 / Chapter 4.2 --- Exponential Claims with Unequal Means --- p.32 / Chapter 5 --- Conclusion --- p.40 / Bibliography --- p.43

Leung, Kit Hung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 47-48). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Classical Model --- p.1 / Chapter 1.2 --- Diffusion-perturbed model --- p.3 / Chapter 1.3 --- Hybrid computational scheme --- p.5 / Chapter 2 --- Integro-differential Equations --- p.7 / Chapter 2.1 --- Integro-differential equation of Chiu and Yin (2003) --- p.7 / Chapter 2.2 --- Integro-differential equations for ψs(u) and ψd(u) --- p.16 / Chapter 3 --- Numerical Method --- p.17 / Chapter 3.1 --- Trapezoidal approximation --- p.18 / Chapter 3.2 --- Boundary Conditions --- p.19 / Chapter 4 --- Importance Sampling --- p.22 / Chapter 4.1 --- Simulation Recipe --- p.25 / Chapter 4.2 --- Discussion --- p.26 / Chapter 5 --- Numerical Examples --- p.28 / Chapter 5.1 --- Probabilities of ruin: Oscillation and claim --- p.28 / Chapter 5.2 --- Comparison with the asymptotic results --- p.32 / Chapter 5.2.1 --- Ruin Probability --- p.38 / Chapter 5.2.2 --- Surplus before ruin --- p.40 / Chapter 5.2.3 --- Deficit after ruin --- p.42 / Chapter 6 --- Conclusion --- p.45 / References --- p.47

Graduation date: 2004

In this thesis, a systematic study is carried out for effectively approximating Gerber-Shiu functions under L´evy subordinator models. It is a hardly touched topic in the recent literature and our approach is via the popular Fourier-cosine method. In theory, classical Gerber-Shiu functions can be expressed in terms of an infinite sum of convolutions, but its inherent complexity makes efficient computation almost impossible. In contrast, Fourier transforms of convolutions could be evaluated in a far simpler manner. Therefore, an efficient numerical method based on Fourier transform is pursued in this thesis for evaluating Gerber-Shiu functions. Fourier-cosine method is a numerical method based on Fourier transform and has been very popular in option pricing since its introduction. It then evolves into a number of extensions, and we here adopt its spirit to insurance risk theory. In this thesis, the proposed approximant of Gerber-Shiu functions under an L´evy subordinator model has O(n) computational complexity in comparison with that of O(n log n) via the usual numerical Fourier inversion. Also, for Gerber-Shiu functions within the proposed refined Sobolev space, an explicit error bound is given and error bound of this type is seemingly absent in the literature. Furthermore, the error bound for our estimation can be further enhanced under extra assumptions, which are not immediate from Fang and Oosterlee’s works. We also suggest a robust method on the estimation of ruin probabilities (one special class of Gerber-Shiu functions) based on the moments of both claim size and claim arrival distributions. Rearrangement inequality will also be adopted to amplify the use of our Fourier-cosine method in ruin probability, resulting in an effective global estimation. Finally, the effectiveness of our result will be further illustrated in a number of numerical studies and our enhanced error bound is apparently optimal in our demonstration; more precisely, empirical evidence exhibiting the biggest possible error convergence rate agrees with our theoretical conclusion. / published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy