Lundberg approximations for compound distributions with insurance applications
Lundberg approximations for compound distributions with insurance applications
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DOI：
10.1007/978-1-4613-0111-0
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
G. Willmot;X. Lin
中科院分区：
文献类型：
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作者：
G. Willmot;X. Lin
这本专著讨论了复合分布的Lundberg近似，特别强调了在保险风险建模中的应用。从分析的角度来看，这些分布有些笨拙，但在保险和其他应用概率建模领域(如排队论)中发挥着核心作用。因此，对这些领域感兴趣的研究人员和研究生对这些材料很感兴趣。材料是自成体系的，但保险风险理论的入门课程对潜在读者是有益的。在排队论中，关于破产概率和等待时间分布的Lundberg渐近性和界已经有很长的历史了，最近又被推广到复合分布。这种联系源于破产概率和等待时间分布的复合几何表示。在很大程度上借鉴了可靠性理论中的单调性思想，提供了对这些近似的系统处理。然后将所得结果应用于缺陷更新方程的求解、保险破产时间和严重程度的分析以及更新风险模型，这些模型也可以用G/G/1排队的均衡等待时间分布来看待。许多已知的结果都是经过推导和推广的，因此许多材料在文献中没有出现在其他地方。一个独特的特点涉及到基本分析技术的使用，这只需要本科数学作为先决条件。给出了许多结果的新证明，并提供了广泛的参考文献。戈登·威尔莫特是滑铁卢大学的统计学和精算学教授。他的研究兴趣是保险风险和排队论。他是《北美精算杂志》的副主编。
This monograph discusses Lundberg approximations for compound distributions with special emphasis on applications in insurance risk modeling. These distributions are somewhat awkward from an analytic standpoint, but play a central role in insurance and other areas of applied probability modeling such as queueing theory. Consequently, the material is of interest to researchers and graduate students interested in these areas. The material is self-contained, but an introductory course in insurance risk theory is beneficial to prospective readers. Lundberg asymptotics and bounds have a long history in connection with ruin probabilities and waiting time distributions in queueing theory, and have more recently been extended to compound distributions. This connection has its roots in the compound geometric representation of the ruin probabilities and waiting time distributions. A systematic treatment of these approximations is provided, drawing heavily on monotonicity ideas from reliability theory. The results are then applied to the solution of defective renewal equations, analysis of the time and severity of insurance ruin, and renewal risk models, which may also be viewed in terms of the equilibrium waiting time distribution in the G/G/1 queue. Many known results are derived and extended so that much of the material has not appeared elsewhere in the literature. A unique feature involves the use of elementary analytic techniques which require only undergraduate mathematics as a prerequisite. New proofs of many results are given, and an extensive bibliography is provided. Gordon Willmot is Professor of Statistics and Actuarial Science at the University of Waterloo. His research interests are in insurance risk and queueing theory. He is an associate editor of the North American Actuarial Journal.