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 the actuarial literature, dependence structures in risk models have been extensively studied. The main theme of this thesis is to investigate some discrete-time risk models with claim numbers modeled by integer-valued time series processes. The first model is a common shock risk model with temporal dependence between the claim numbers in each individual class of business. Specifically the Poisson MA(1) process and Poisson AR(1) process are considered for the temporal dependence. To study the ruin probability, the equations associated with the adjustment coefficients are derived. Comparisons are also made to assess the impact of the dependence structures on the ruin probability. Another model involving both the correlated classes of business and the time series approach is then studied. Thinning dependence structure is adopted to model the dependence among classes of business. The Poisson MA(1) and Poisson AR(1) processes are used to describe the claim-number processes. Adjustment coefficients and ruin probabilities are examined. Finally a discrete-time risk model with the claim number following a Poisson ARCH process is proposed. In this model, the mean of the current claim number depends on the previous observations. Within this framework, the equation for finding the adjustment coefficient is derived. Numerical studies are also carried out to examine the effect of the Poisson ARCH dependence structure on several risk measures including ruin probability, Value at Risk, and conditional tail expectation. / published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

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

Thesis (doctoral)--Universität Göttingen, 1968. / Includes bibliographical references (viii-xxvi).

Thesis (doctoral)--Göteborgs universitet, 2002. / Extra t.p. with thesis statement inserted. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin, 1966. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Bibliography: leaves 246-251.

Thesis (Ph. D.)--University of California, San Diego, 2002. / Vita. Includes bibliographical references (leaves 268-289).

Thesis (M. Phil.)--University of Hong Kong, 2005. / Title proper from title frame. Also available in printed format.

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