Lévy processes in actuarial ruin theory and exotic option pricing

最近由各国违约引发的金融市场动荡，使人们重新认识到风险衡量对于维持充足的资本要求和有效定价金融产品的重要性。大多数工业化国家已经制定了一个全面的框架，用于执行识别、衡量和分析风险的程序。精算破产理论旨在建立保险公司财富的模型，并衡量其风险。非常相似的是，信用风险的结构模型对公司和金融机构的偿付能力感兴趣。另一方面，从业人员和学者都认识到，简单的金融资产价格随机模型并不能解释市场上观察到的经验事实。与此同时，以这些资产为抵押的金融衍生品的复杂性也在不断增加。在所有情况下，都需要为风险测量、金融工具定价和对冲而设计的复杂金融模型和强大的数学技术。这个研究项目的重点是概率和随机过程之间的相互作用，精算科学和金融。精算科学和金融中出现的问题催生了概率论和随机过程的创新研究。另一方面，精算师和投资银行家感兴趣的模型分析需要对基础模型中使用的数学有很好的了解。预计将发展新的分析和概率数学技术来进行上述分析。

The recent turmoil in financial markets, triggered by defaults in various countries, has shed new lights on the importance of risk measurement for maintaining adequate capital requirements and efficiently pricing financial products. A comprehensive framework for implementing procedures to identify, measure, and analyze risk has been developed in most industrialized countries.Actuarial ruin theory aims at modelling, and measuring the risk of, the wealth of an insurance company. Very similarly, structural models in credit risk are interested in the solvency of firms and financial institutions. On the other hand, it has been acknowledged by practitioners and academics that simple stochastic models for the price of financial assets can not explain empirical facts observed on the markets. At the same time, the complexity of financial derivatives written on those assets is constantly increasing. In all cases, there is a need for sophisticated financial models and powerful mathematical techniques designed for risk measurement and for the pricing and hedging of financial instruments. This research program focuses on the interactions between probability and stochastic processes, with actuarial science and finance. Problems arising in actuarial science and finance generate innovative research in the theory of probability and stochastic processes. On the other hand, the analysis of models of interest for actuaries and investment bankers requires a good knowledge of the mathematics used in the underlying model. It is expected that new analytic and probabilistic mathematical techniques will be developed to perform the analyses alluded to above.