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.
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机译：在不同的风险模型下，计算无限时间毁灭概率的精确“封闭形式”解是一项艰巨的任务。为有限的时间毁坏概率找到精确的闭合形式的解是一个甚至更困难的问题，有时是无法解决的。因此，找到近似值是寻找破产概率的另一种方法。有限和无限时间毁灭概率的计算是本文解决的主要问题。有些章节介绍了针对有限时间毁坏概率的新的精确封闭形式解决方案。如果不可能的话，将给出毁灭概率的新近似值。 De Vylder逼近是经典风险模型中最常见的毁灭最终概率近似值之一。本工作概括了De Vylder的方法，通过用二阶的Coxian分布替换指数索赔额假设来获得更高的准确性。所获得的近似值始终至少与De Vylder的原始近似值一样好，它本身始终是所描述方法的一种解决方案。本文讨论了计算有限时间毁灭概率的三种主要方法。第一个是基于一个相当通用的风险模型直至破产的时间的拉普拉斯变换，该模型允许相关的到达过程，甚至索赔大小可能取决于环境因素，例如传染期。应用Gaver-Stehfest数值反演技术确定有限时间破坏概率。第二种方法是在更新模型中计算阶段型分布层位之前的毁灭概率。然后，使用Erlang视界获得有限时间毁灭概率的近似值。最后一种方法基于鞍点近似，这是精算领域中的一种相对较新的技术。在索赔相位类型为分布的经典风险模型中，获得了基于逆高斯分布的有限时间破产概率的修正鞍点近似。
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