Browsing by Subject "Ruin Probabilities"
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(2022)Large deviations theory is a branch of probability theory which studies the exponential decay of probabilities for extremely rare events in the context of sequences of probability distributions. The theory originates from actuaries studying risk and insurance from a mathematical perspective, but today it has become its own field of study, and is no longer as tightly linked to insurance mathematics. Large deviations theory is nowadays frequently applied in various fields, such as information theory, queuing theory, statistical mechanics and finance. The connection to insurance mathematics has not grown obsolete, however, and these new results can also be applied to develop new results in the context of insurance. This paper is split into two main sections. The first presents some basic concepts from large deviations theory as well as the Gärtner-Ellis theorem, the first main topic of this thesis, and then provides a fairly detailed proof of this theorem. The Gärtner-Ellis theorem is an important result in large deviations theory, as it gives upper and lower bounds relating to asymptotic probabilities, while allowing for some dependence structure in the sequence of random variables. The second main topic of this thesis is the presentation of two large deviations results developed by H. Nyrhinen, concerning the random time of ruin as a function of the given starting capital. This section begins with introducing the specifics of this insurance setting of Nyrhinen’s work as well as the ruin problem, a central topic of risk theory. Following this are the main results, and the corresponding proofs, which rely to some part on convex analysis, and also on a continuous version of the Gärtner-Ellis theorem. Recommended preliminary knowledge: Probability Theory, Risk Theory.
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(2024)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.
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