##### Abstract

In the literature of risk theory, two risk models have emerged to have been studied extensively. These are the Classical Compound Poisson risk model and the Continuous Time Renewal risk model also referred to as Sparre-Andersen model. The focus of this thesis is the latter and in the Sparre-Andersen risk model, the distribution of claim inter-arrival times (T) and the claim sizes (amounts) (X) are assumed to be independent. This assumption is not only too restrictive but also inappropriate for some real world situations like collision coverages under automobile insurance in cities with hash weather conditions. In this thesis, the assumption of independence between T and X for different classes of bivariate distributions is relaxed prior to computing the ultimate ruin probability. Also, the effect of correlation on ruin probability is investigated. Correlation is introduced through the use of the Moran and Downton's bivariate Exponential distribution. The underlying method in the entire modeling process is the Wiener-Hopf factorization technique, details of which are covered in chapter 2. The main results are covered in chapters 3, 4, 5, 6 and 7. In all of these chapters, we assume T to follow an Exponential distribution while X is Gamma distributed with shape parameter 2 in chapter 3. In chapter 4, X follows Gamma distribution with shape parameter (m) where m is greater than 2. In chapter 5, a mixture of Exponential (Hyper Exponential) distributions is used to model X. The Pareto distribution is used to model X in chapter 6, in the modeling process however, the cummulative distribution function (CDF) of W=X-cT where (X,T) is Exponential-Pareto distribution is approximated by the CDF of Y=X-cT where (X,T) is Exponential-Hyper Exponential distribution and Exponential- Gamma (m) distribution with the main estimation approach being method of moments estimation. Chapter 7 follows somewhat a different approach by employing asymmetric finite mixture of Double Exponential distributions and the EM algorithm prior to computing the ruin probability. In some of these situations, an explicit expression for the ultimate ruin probability is obtained and in chapters 3 and 5, we show that increasing correlation between T and X diminishes the impact of ruin probability. Also, in situations where an explicit expression for the ultimate ruin probability is not possible, excellent approximation is obtained. Chapter 8 summarizes the entire thesis, provides concluding remarks and future research.