FRG: Collaborative Research on Mathematical Methods for Defaultable Instruments

奖项摘要dms-0456195/dms-0455982/dms-0456118普林斯顿大学Rene A.Carmona和K.Ronnie Sircar北卡罗来纳州立大学Jean-Pierre Fuque Thaleia Zariphopoulou，德克萨斯大学奥斯汀FRG：关于可违约工具的数学方法的合作研究这个项目调查由信贷市场推动的金融数学问题，其中一个主要风险来源是债务人潜在的偿付义务违约。具体地说，正在考虑的问题是：i)效用-违约风险的无差别估值；ii)设计工具以最佳地提高信用价值；iii)随机强度模型的渐近分析，以研究企业收益率利差的时间尺度内容；iv)与分析跨公司违约之间的相关性相关的计算问题，被建模为交互中的大系统。第一部分涉及与随机强度模型和无限维利率模型相关的随机控制问题。第二个也是重叠的，涉及对部分观测系统的过滤。第三种方法使用奇异和正则摄动技术处理在此背景下产生的一类相互作用势偏微分方程，第四种方法使用相互作用的粒子系统来计算对缺省相关性敏感的概率，以及为分析罕见事件而设计的蒙特卡罗计算。该项目的学术价值在于开发适用的科学工具，以解决对管理违约风险至关重要的特定类别的优化、设计、校准和计算问题。可违约工具或信用相关衍生品是指根据违约事件(如公司(或国家或市政当局)破产、无法偿还贷款或拖欠抵押贷款)向持有者支付金额的金融证券。近年来，与信用挂钩的衍生品市场增长了七倍多，从1997年的1700亿美元增长到2001年的近14000亿美元。这些工具在建模、分析、计算和估计方面提出了新的挑战，我们建议在这里通过召集一个具有应用数学、随机过程和计算统计专业知识的重点研究小组来研究其中一些挑战。该项目的更广泛影响是加深了对信贷风险的理解，这些风险影响着从大型商业机构到拥有养老基金和抵押贷款的个人，以及设计和正确评估控制信贷风险的工具。该项目还着重于培训五名研究生和一名博士后助理，他们将通过多次会议，特别是在三年结束时举行一次关于该研究领域的大型国际会议，与基础广泛的小组的所有成员进行互动，从中受益匪浅。他们所获得的经验将反映在他们的专业研究生和本科课程的教学中，并为这一领域的高级论文项目提供建议。除闭幕会议外，还将通过学术和行业会议、课程和为同行评议期刊撰写的文章来传播工作成果。

Award Abstract DMS-0456195 / DMS-0455982 / DMS-0456118Rene A. Carmona and K. Ronnie Sircar, Princeton UniversityJean-Pierre Fouque, North Carolina State UniversityThaleia Zariphopoulou, University of Texas at AustinFRG: Collaborative Research on Mathematical Methods for Defaultable Instruments This project investigates problems in financial mathematics motivated by credit markets in which a major source of risk is the potential default of debtors on their payment obligations. Specifically, the problems under consideration are i) utility-indifference valuation of default risk; ii) design of instruments to optimally enhance credit worthiness; iii) asymptotic analysis of stochastic intensity models to study the time-scale content of corporate yield spreads; iv) computational issues related to the analysis of correlation between defaults across firms, modeled as large systems in interaction. The first part involves stochastic control problems related to random intensity models and infinite dimensional interest rate models. The second also overlaps and involves filtering of partially observed systems. The third uses singular and regular perturbation techniques for the class of interacting potential partial differential equations arising in this context, and the fourth uses interacting particle systems to compute probabilities which are sensitive to correlation of defaults, as well as Monte Carlo computations designed for the analysis of rare events. The intellectual merit of this project is in developing applicable scientific tools to address the particular class of optimization, design, calibration and computation issues which are essential for managing default risk.Defaultable instruments, or credit-linked derivatives, are financial securities that pay their holders amounts that are contingent on the occurrence (or not) of a default event such as the bankruptcy of a firm (or a country or municipality), non-repayment of a loan or missing a mortgage payment. The market in credit-linked derivative products has grown more than seven-fold in recent years, from $170 billion outstanding notional in 1997, to almost $1400 billion through 2001. These instruments raise new challenges in modeling, analysis, computation and estimation, some of which we propose to study here by bringing together a Focused Research Group with expertise in applied mathematics, stochastic processes and computational statistics. The broader impact of the project is in deeper understanding of credit risks, which affect people from large commercial institutions to individuals with pension funds and mortgages, and designing and correctly valuing instruments to control for it. The project is also strongly geared towards training of five graduate students and one postdoctoral associate, who will benefit enormously from interaction with all parts of the broad-based group through many meetings, and in particular a large international conference on the research area at the end of the three years. The experience gained by the PI's will be reflected in their teaching of specialist graduate and undergraduate classes, and advising Senior Thesis projects in this field. As well as the closing conference, the results of the work will be disseminated through academic and industry meetings, classes and articles written for peer-reviewed journals.