| The calculation of the exact "closed form" solution for the infinite time ruin probability under different risk models is a hard task. Finding exact closed form solutions for finite time ruin probabilities is an even harder problem, sometimes impossible to solve. Therefore, finding an approximation is the alternative way to look for ruin probabilities.; The calculation of the finite and infinite time ruin probabilities is the main problem addressed by the thesis. Some of the chapters present new exact closed form solutions for finite time ruin probabilities. When this is not possible, new approximations for the ruin probabilities are presented.; One of the most popular approximations for the ultimate probability of ruin in the classical risk model is the De Vylder approximation. The present work generalizes De Vylder's approach, to achieve even greater accuracy, by replacing the exponential claim size assumption with a Coxian distribution of order two. The obtained approximation is always at least as good as De Vylder's original approximation, which is itself always one solution to the described methodology.; Three main approaches to calculate finite time ruin probabilities are discussed in the thesis. The first one is based on the Laplace transform of the time until ruin for a fairly general risk model that allows for correlated arrival processes, and even claim sizes that can depend upon environmental factors such as periods of contagion. A Gaver-Stehfest numerical inversion technique is applied to determine the finite time ruin probability.; The second approach calculates the ruin probability before a phase-type distributed horizon in the renewal model. Using Erlang horizons, approximations for finite time ruin probabilities are then obtained.; The last approach is based on a saddlepoint approximation, a relatively new technique in the actuarial field. A modified saddlepoint approximation for the finite time ruin probabilities, based on an Inverse Gaussian distribution is obtained in the classical risk model with claims phase-type distributed. |
