A class of bivariate Erlang distributions and ruin probabilities in multivariate risk models

This dissertation is devoted to modeling dependence with potential applications in actuarial science and is divided in two parts: the first part considers dependence in the context of bivariate survival data analysis and the second, related to risk theory, deals with dependence between classes of an insurance business.;The first part is presented in the form of a research paper in Chapter 3. In this contribution, we introduce a new class of bivariate distributions of Marshall-Olkin type, called bivariate Erlang distributions. It is shown that the bivariate Erlang distribution has both an absolutely continuous and a singular part. The Laplace transform, product moments and conditional densities are derived and also, the finite mixture of the bivariate Erlang distributions is described. Potential applications of bivariate Erlang distributions in life insurance and finance are considered. The maximum likelihood estimators of the parameters are computed via an Expectation-Maximization algorithm. Simulations are carried out to measure the performance of the estimator.;The second part related to risk theory is presented in Chapters 4 and 5 of this thesis and is devoted to the study of multivariate risk processes, which may be useful in analyzing ruin problems for insurance companies with a portfolio of dependent classes of business. We apply results from the theory of piecewise deterministic Markov processes in order to derive exponential martingales needed to establish computable upper bounds for the ruin probabilities, as their exact expressions are intractable.;As an extension of the multivariate risk model proposed by Asmussen and Albrecher (2010), we first consider an m-dimensional risk process obtained by modeling the dependence through the number of claims using the Poisson model with common shocks. We assume that in addition to the individual shocks, a common shock affects all classes of business and that another common shock has an impact on each pair of classes. Also, dependence between claims sizes across classes is allowed. The asymptotic behavior of the the probability that ruin occurs in all classes simultaneously before a fixed time t, in both cases of dependent heavy-tailed claims and independent heavy-tailed claims, is investigated.;Inspired by the work of Dufresne and Gerber (1991) and of Li, Liu and Tang (2007), we embrace the idea of adding a diffusion process characterized by an m-dimensional correlated Brownian motion.;For each of these two multivariate models an expression for the probability that ruin occurs in at least one class of business and an upper bound for the probability that ruin occurs in all classes simultaneously are derived. Numerical results regarding the upper bounds are reported for these models assuming three classes of insurance business, where the dependence between claims sizes is modeled using the notion of copula. It is established that adding a diffusion process leads to increasing these upper bounds.;Further, in a more realistic setting, our research project is outlined by investigating ruin probabilities associated to an m-dimensional risk process which assumes that in addition to the individual claim arrivals for each class of business, governed by Poisson processes, there are aggregate claims produced by a common renewal counting process that affects all classes of business.;In this multivariate context, the surplus vector process is Markovianized by introducing a supplementary process, and tools from the theory of piecewise deterministic Markov processes are applied in order to obtain exponential martingales. Based on these martingales, we derive an upper bound for the probability that ruin occurs in all classes simultaneously. Also, an upper bound for this type of ruin probability is derived in a special case where the individual shocks are absent and the claims across classes are produced only by the renewal process. The latter upper bound is illustrated by numerical results, where a bivariate version is considered and the dependence in claim sizes is captured using copula techniques.;Keywords: Erlang distribution, Expectation-Maximization algorithm, Piecewise deterministic Markov processes, Multivariate risk model, Ruin probability, Poisson model with common shocks, Renewal processes, Copulas.