Financial Mathematics: Nonlinear Problems and Systemic Risk

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In the first project, it is proposed to establish 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 faced by practitioners in the financial industry. These problems include risk management under uncertain volatility, behavior of implied volatilities at short maturities, and portfolio optimization under stochastic volatility. In the second project, it is proposed to develop new models based on systems of interacting diffusions where the coupling is modeled in the drifts through lending preferences. Combined with sophisticated Monte Carlo methods, including interacting particle system methods, these models will allow to study the stability of the system and the various statistical quantities relevant to systemic risk of a network.Challenging nonlinear problems arising naturally in the context of risk management under uncertain or randomly fluctuating volatility are addressed in the first project. This research has direct applications to practical problems faced by practitioners in the financial industry. The second project is a new direction of research on 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. It is proposed to develop new models which will allow to study the stability of the system and the various statistical quantities relevant to systemic risk of a network. This research, including its training component, is expected to contribute to the effort started by the regulators in the recent creation of the Office of Financial Research.