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.