Browsing by Subject "Ladder Height Distribution"

In the field of insurance mathematics, it is critical to control the solvency of an insurance company. In particular, calculating the probability of ruin, which is the probability that the company’s surplus falls below zero. In this thesis a review of the fundamentals of ruin theory, the modelling process and some results and methods for the estimation of ruin probabilities is made. Most of the theorems are taken from different bibliographical sources, but a good amount of the proofs presented are original, in order to provide a more rigorous and detailed explanation. A central focus of this thesis is the Pollaczek-Khinchine formula. This formula provides a solution for the probability distribution of the maximum potential loss of an insurance company in terms of convolutions of a particular function related to the claim sizes. Apart from the theoretical results that may be derived from it and its elegance, its usefulness lies in the ideas underlying it. Specially, the idea to understand the maximum potential loss of the company as the biggest of the historical records in the loss process. Using these ideas, a recursive approach to estimating ruin probabilities is ex- plained. This approach results in an easy to program and efficient bounds method which allows for any type of claim sizes (that is, the random variables that model how big are the claims of the insureds). The only restrictions imposed come from the fact that this discussion takes place within the Poisson model. This framework allows for various claim size distributions and models the number of claims as a Poisson process. Finally, two examples of light and heavy-tailed claim size distributions are simu- lated using this recursive approach. This shows the applicability of the method and the differences between light and heavy-tailed distributions with regards to the ruin probabilities that emerge from them.