On some topics in Levy insurance risk models
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Risk management has long been the central focus within actuarial science. There are various risks a typical actuarial company would look into, solvency risk being one of them. This falls under the scope of surplus analysis. Studying of an insurer's ability to maintain an adequate surplus level in order to fulfill its future obligation would be the subject matter, which requires modeling of the underlying surplus process together with de fining appropriate matrices to quantity the risk. Ultimately, it aims at accurately reflecting the solvency status to a line of business, which requires developing realistic models to predict the evolution of the underlying surplus and constructing various ruin quantities depending on the regulations or the risk appetite set internally by the company. While there have been a vast amount of literature devoted to answering these questions in the past decades, a considerable amount of effort is devoted by different scholars in recent years to construct more accurate models to work with, and to develop a spectrum of risk quantities to serve different purposes. In the meantime, more advanced tools are also developed to assist with the analysis involved. With the same spirit, this thesis aims at making contributions in these areas. In Chapter 3, a Parisian ruin time is analyzed under a spectrally negative L evy model. A hybrid observation scheme is investigated, which allows a more frequent monitoring when the solvency status to a business is observed to be critical. From a practical perspective, such observation scheme provides an extra degree of realism. From a theoretical perspective, it uni es analysis to paths having either bounded or unbounded variations, a core obstacle for analysis under the context of spectrally negative L evy model. Laplace transform to the concerned ruin time is obtained. Existing results in the literature are also retrieved to demonstrate consistency by taking appropriate limits. In Chapter 4, the toolbox of discrete Poissonian observation is further complemented under a spectrally negative L evy context. By extending the classical definition of potential measures, which summarizes the law of ruin time and de cit at ruin under continuous observation, to its discrete counterpart, expressions to the Poissonian potential measures are derived. An interesting dual relation is also discovered between a Poissonian potential measure and the corresponding exit measure. This further strengthens the motivation for studying the Poissonian potential measures. To further demonstrate its usefulness, several problems are formulated and analyzed at the end of this chapter. In Chapter 5, motivated from regulatory practices, a more conservative risk matrix is constructed by altering the traditional definition to a Parisian ruin time. As a starting point, analysis is performed using a Cram er-Lundberg model, a special case of spectrally negative L evy model. The law of ruin time and its de cit at ruin is obtained. An interesting ordering property is also argued to justify why it is a more conservative risk measure to work with. To ensure that the thesis flows smoothly, Chapter 1 and 2 are devoted to the background reading. Literature reviews and existing tools necessary for subsequent derivations are provided at the beginning of each chapters to ensure self-containment. A summary and concluding remarks can be found in Chapter 6.
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Jeff Wong (2019). On some topics in Levy insurance risk models. UWSpace. http://hdl.handle.net/10012/14805