| In the actuarial literature on ruin theory, two general classes of risk models are proposed to describe the behavior of the surplus of an insurance company. These two classes are the discrete-time and the continuous-time risk models. Historically, most results have been derived in the continuous-time risk models class, partly due to the simplicity of various calculations under the classical compound Poisson risk model in ruin theory. In this thesis, our attention is shifted to the class of discrete-time risk models for which continuous-time risk models can be viewed as limiting cases. In this sense, results in the discrete-time risk models class are stronger than their counterparts in the continuous-time risk models class.; Among the discrete-time risk models, there is the well-known compound binomial model in which the number of insurance claims is governed by a binomial process. In the compound binomial model, it is assumed that the increments of the aggregate claim amount process are independent. However, in many real life situations, this independence assumption seems no longer reasonable and leads to an underestimation of ruin probability. Therefore, we present in this monograph three extensions of the compound binomial model, all three introducing time dependency in the increments of the aggregate claim amount process. These three extensions are the compound Markov binomial model, the compound binomial model defined in a Markovian environment and the discrete-time renewal risk model. For these three risk models, we study in detail some properties of the surplus process in order to help characterize its behavior and improve our knowledge of the riskiness for an insurance company. As a key indicator of the riskiness of a surplus process, we rely mostly on ruin probability which corresponds to the probability that the surplus process falls at least once below the level 0. In the compound Markov binomial model, a recursive algorithm is proposed to compute ruin probabilities whereas in the two other extensions, we rely on numerical algorithms for their computations. Finally, two of these "extended" discrete-time risk models will also be used as approximations to extensions of Lundberg's compound Poisson risk model. |
