Collaborative Research: High-Performance Computational Methods for Continuous-Time Markov Processes in Financial Engineering

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The project focuses on the development of high-performance computational tools for financial engineering. The goal is to develop computational methods to evaluate complex financial products used to manage foreign exchange, interest rate, equity, commodity and energy price risks and credit risk, and manage large portfolios of assets. The methodology is based on extensions to financial engineering of finite-element methods successfully used in diverse branches of engineering to solve numerically multi-dimensional partial differential and integral equations. Partial integro-differential equations arise in the study of Markov jump-diffusion processes and associated optimal stopping and stochastic control problems in financial engineering. The aim of the present proposal is to develop both the necessary mathematical theory to extend finite element methods to jump-diffusion processes and develop high-performance computational tools based on these methods that can be effectively implemented and used by industry practitioners in the financial services, as well as researchers in financial engineering, applied probability and branches of operations research that use continuous-time Markov processes. Specific challenges in financial engineering to be addressed in the project include high dimensionality and jumps.Methodologies developed in this project will help financial institutions, corporate treasuries and energy companies accurately value complex financial instruments, efficiently manage risk of financial transactions, and dynamically manage portfolios of assets. In addition to financial engineering, we anticipate that this project will have a broader impact on research and application areas that use continuous-time Markov processes as a modeling framework. Constructive approximations and computational algorithms for jump-diffusion processes developed in this project should prove useful for diverse areas of application that use jump-diffusion processes. This proposal will support the new Ph.D. major in financial engineering at Northwestern University. This new Ph.D. major will result in training of highly qualified researchers in financial engineering. This project is a part of the long-term development effort at Northwestern University in the area of financial engineering. This project will also help the Department of Mathematical Sciences at the University of Nevada Las Vegas establish a research program in financial mathematics.