Ruin problems in non-standard risk models

The computation of ruin probabilities constitutes a central topic in risk theory. Even though the study of the ruin problem resulted in a huge amount of published articles (including in the closely related fields of queueing theory and dam and storage processes), some questions have never been studied in the literature, until recently. This thesis aims at contributing to the risk theory by looking into various problems in line with the real management of companies. This thesis is divided into two parts. In the first part, the purpose is to relax some classical assumptions to account for situations of practical relevance. The vast majority of existing results require independent increments for the stochastic process describing the aggregate claim amounts filed against the insurance company. There is a variety of situations where this independence assumption appears rather unrealistic. This is the case in presence of business cycles, delay in claim reporting or settlement, or as the result of the implementation of experience rating mechanisms by the insurance company. In such settings, the financial result of a given calendar year depends on the result of one or several preceding years. So, the first part of this thesis consists to deal with non-classical risk models allowing to reflect such basic insurance realities. In the second part, the aim is to study a risk measure derived from actuarial ruin theory. One historical purpose of studying ruin probabilities was to obtain the amount of initial capital needed to guarantee to the insurance company a given probability of solvency. The Value-at-Risk (VaR) and the expected shortfall are natural concepts in this context, and are widely used by the practitioners. However, these two notions are defined in terms of a given time horizon. Consequently, both the VaR and the expected shortfall do not reflect the possible adverse financial situations in between or beyond the specified time interval. The smallest capital allowing to ensure that the infinite-time ruin probability is less than some acceptance level appears as a good alternative to quantify the insured risk. It will suffice to cope with the risk at all times according to the given level. That is why the second part of this thesis studies the properties of this smallest capital.(ACTU 3) -- UCL, 201