This proposal is on nonlinear analysis with an emphasis on applied mathematics problems in material science and financial mathematics. Particular applications in material science include modeling grain boundaries, super cooled solidification, and superconductivity. In financial mathematics, applications include pricing contingent claims in incomplete markets and consumption-investment problems. As an analytical tool, the PI proposes to develop a theory of bounded variation to suitable for vector-valued functions. This theory of functions of type BnV may have a wide impact in real analysis. In financial mathematics, the PI will look at problems with portfolio constraints, transaction costs, and models with stocastic volatility.
The sophisticated mathematical models arising in the study of new materials and in financial transactions require new methods of mathematics to show that the models are consistent with reality and give reasonable results. The PI will be investigating new methods of analysis to apply to such problems as superconducting materials and financial models of derivative transactions
