Since the pioneering works of Black & Scholes, Merton and Markowitch, sophisticated quantitative methods are being used to introduce more complex financial products each year. However, this exciting increase in the complexity forces the industry to engage in proper risk management practices. The recent financial crisis emanating from risky loan practices is a prime example of this acute need. This proposal focuses exactly on this general problem. We will develop mathematical techniques to measure and assess the financial risk of new instruments. In the theoretical direction, we will expand the scope of recent studies on risk measures of Artzner et-al., and the stochastic representation formulae proved by the principal investigator and his collaborators. The core research team consists of mathematicians and the finance faculty. The newly created state-of-the-art finance laboratory at the host institution will have direct access to financial data. Moreover, executive education that is performed in this unit enables the research group to have close contacts with high level executives of the financial industry. The theoretical side of the project focuses on nonlinear partial differential equations (PDE), backward stochastic differential equations (BSDE) and dynamic risk measures. Already a deep connection between BSDEs and dynamic risk measures is developed by Peng, Delbaen and collaborators. Also, the principal investigator and his collaborators developed connections to PDEs. In this project, we further investigate these connections. Chief goals of this project are theoretical results and computational techniques in the general areas of BSDEs, fully nonlinear PDEs, and the development of risk management practices that are acceptable by the industry. The composition of the research team and our expertise in quantitative methods, well position us to effectively formulate and study theoretical problems with financial impact.