Challenging nonlinear problems arising naturally in the context of risk management under uncertain or randomly fluctuating volatility and under liquidity constraints are addressed in the first part of the project. This work has direct applications to problems faced by practitioners in the financial industry, including pension funds managers. The second part of the project is a study of systemic risk and mathematical analysis of the stability (or instability) of our banking system. The recent financial crisis has revealed a lack of understanding of the risk of cascade of defaults in banking networks. To help assess measures of such systemic risks, the investigator develops and analyzes new stochastic games models of borrowing and lending among banks. A graduate student and a postdoctoral researcher are engaged in the project each year. This project, including its training component, contributes to the effort started by the regulators in the recent creation of the Office of Financial Research.
In the first part of the project, the investigator establishes new results in the theory of asymptotic analysis and homogenization of nonlinear partial partial differential equations, with direct applications to several important and practical problems in financial mathematics. These problems include risk management under uncertain volatility, portfolio optimization under stochastic volatility, and investment with co-integrated assets. In the second part of the project, he studies stochastic games models of inter-bank borrowing and lending with applications to measures of systemic risk. New mathematical problems arise naturally; they include backward-forward stochastic partial differential equations and stochastic games with delay. A graduate student and a postdoctoral researcher are engaged in the project each year.
