AMC-SS: Problems in Mathematical Finance

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Quantifying, managing and pricing risk is not only important for the big players in the economy but has become increasingly important for every individual, who are trying to reduce the risk of outliving their wealth through their retirement funds, and the institutional investors representing them. The financial markets provide the necessary liquidity for the rest of the economy to function well, and not understanding the nature of the risk taken, the mispricing of the risk and an inadequate oversight, can have dramatic impacts on the welfare of our society, as the latest crisis demonstrated one more time. Consequently there has been an intense research effort, which is embodied in the field of financial mathematics, to solve the new mathematical challenges. This new field takes its roots in stochastic analysis and also frequently interacts with other fields of mathematics such as the theory of partial differential equations and functional analysis. This project resolves some fundamental questions concerning the important models of the field that remain unanswered. Additionally, new models which take into account the discrete, asynchronous and non-stationary nature of the observations will be developed and the mathematical and computational challenges for new problems of interest will be resolved. Optimization problems from a retiree's perspective (with an objective criterion) and an institutional investor perspective (with risk constraints imposed by regulators) are to be analyzed. Developing parsimonious models and analyzing the inverse problems for structured credit products, which were developed to provide insurance against (correlated) default of issuers, is another goal.More specifically, the first part of the proposed project will settle open problems in the theory of Optimal Stopping and related free boundary problems that arise in Financial Mathematics. The solutions in the second part have direct applications to many practical control problems since the nature of observations in many applications is discrete, asynchronous and non-stationary (i.e. the nature of the source of observation change with time). In the third part utility maximization problems with probability of lifetime ruin, occupation time, or other optimization and/or risk constraints will be considered. Optimal control/stopping problems with non-linear expectations will be discussed. These lead to new mathematical challenges which will lead to new methodological developments. The effects of uncertain investment environment and learning on the optimal investment strategies will also be investigated. The dramatic losses in the credit derivative markets last year shows that appropriate modeling and pricing of credit derivatives is a challenging open problem. In the fourth path, the investigator will develop effective models, solve the corresponding inverse problems and price over the counter options.