Published also as Thesis (Ph. D.)--Columbia University.

Thesis (Ph. D.)--University of Hong Kong, 2009. / Includes bibliographical references (leaves 121-138). Also available in print.

Published also as Thesis (Ph. D.)--Columbia University.

Thesis (M. Phil.)--University of Hong Kong, 2008. / Includes bibliographical references (leaf 80-86) Also available in print.

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

Thesis (M. Phil.)--University of Hong Kong, 2007. / Title proper from title frame. Also available in printed format.

Inaug.-diss.--Berlin. / Lebanslauf. "Sendersbdruck aus den 'Schriften des Mathematischen institute und des institute für angewandte mathematik der Universität B̈erlin'/band 4." Includes bibliographical references.

Regarding the relationships among different insurance claims, especially in non-life insurance, the dependence behaviour in various models has been studied extensively. In this thesis, some discrete-time risk models with dependence structures would be investigated. One traditional discrete-time risk model is the time series risk model, in which the dependence would be on two aspects: time correlated claims and dependent business classes. A general vector (multivariate) autoregressive moving average (VARMA) model would be adopted to analyze the ruin probability of a surplus process. An upper bound for the ruin probability is derived for the general order of multivariate time series models in claims. Simulation studies are carried out for model comparison for finite time ruin probabilities. Another class of risk model is the compound binomial risk model, where the dependence structure would be based on the existence of a so-called by-claim in the claim process. The by-claim could be incurred in the same period as the main insurance claim, or it would be incurred in the next period, depending on a certain probability. A randomized dividend payment scheme with some fixed threshold value in surplus level would also be considered in this thesis. A methodology is discovered to obtain the Gerber-Shiu expected penalty function for the extended model. The final model investigated in this thesis is the periodic time series risk model. The periodic structure of the model gives a practical interpretation of the business cycle, in which there are high season and low season for the business. Some lower order periodic time series models are considered for the claim structures. / 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