Integro-PDEs: Numerical methods, Analysis, and Applications to Finance

This is an application for a Researcher project for Associate Professor E. R. Jakobsen (NTNU, Trondheim) and Professor K. H. Karlsen (University of Oslo). The 3 year Researcher project consists of one doctoral position (3 years, at NTNU), one post-doctora l position (2 years), and funding for equipment, travels, and guests. The goal is to study integro partial differential equations (integro-PDEs) appearing in optimal stochastic control theory. Today the most important application of optimal stochastic control theory is to finance. Mathematical finance is internationally one of the fastest growing areas of research and its practical applications to the problem of pricing derivatives and determining optimal portfolios are numerous. Since the discovery of the famous Black-Scholes equation in the 1970's there has been a rush in the development of sophisticated models for options and optimal portfolio selections. Due to their complexity these models cannot be solved analytically, and therefore we must resor t to numerical methods. Solving such integro-PDEs numerically is computationally demanding due to the fact that they can be non-local, non-linear, degenerate, and multi-dimensional at the same time. Numerical methods are the main focus of this project. We plan to develop, implement, and analyze efficient and robust general purpose numerical methods for such equations; we plan to develop a framework for error estimates for such methods; and we plan to solve numerically specific problems from finance. Nex t, we also want to pursue related theoretical studies into regularity of solutions and boundary value problems. We explain in the project description why these questions are related to, and important for, numerical analysis.