Dependence in risk theory

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An insurance company submits itself to an important risk by subscribing a high number of insurance policies. To avoid facing financial difficulties in the future, insurance companies must be able to have an adequate appraisal of the financial health of their institution. For that purpose, risk theory can be used to measure the global risk of a portfolio of insurance contracts. This evaluation can be made with short term models over a fixed period of time and with long term models. With long term models, the global risk can be assessed by examining the evolution of the surplus process over several periods, either on a discrete or continuous time basis. In such a case, ruin theory is most oftenly used to accomplish this task. The description of the characteristics of the surplus process has relied traditionally on an assumption of independence between the amount of claims and the interarrival times. In many applications, this assumption is inadequate and generalizations with a dependence structure is needed to avoid a misevaluation of the global risk faced by an insurance company. In my research program, I will propose and include different dependence constructions which can be observed in practice in risk models. Both discrete and continuous time risk models will be considered. Different quantities such as the ruin probability, the surplus immediately prior to ruin and the deficit at ruin are of interest in the assessment of the risk of an insurance portfolio. The unified approach based on the Gerber-Shiu penalty function will be used to study these ruin related quantities in the extensions proposed. The possibility of dividend payments to shareholders and investment income will also be studied within the suggested risk models with dependence.