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

该项目专注于金融工程高性能计算工具的开发。其目标是开发计算方法，以评估用于管理外汇、利率、股票、大宗商品和能源价格风险以及信用风险的复杂金融产品，并管理大型资产组合。这种方法是基于金融工程的扩展，有限元方法在工程的不同分支中成功地用于数值求解多维偏微分方程组和积分方程式。偏积分-微分方程组是研究金融工程中的马尔可夫跳跃扩散过程及其相关的最优停止和随机控制问题时产生的。本提案的目的是发展必要的数学理论，将有限元方法扩展到跳跃扩散过程，并开发基于这些方法的高性能计算工具，这些方法可以有效地实施并由金融服务行业从业人员以及金融工程、应用概率和使用连续时间马尔可夫过程的运筹学分支的研究人员使用。金融工程中需要解决的具体挑战包括高维和跳跃。本项目开发的方法将帮助金融机构、公司金库和能源公司准确评估复杂的金融工具，有效管理金融交易风险，并动态管理资产组合。除了金融工程，我们预计该项目将对使用连续时间马尔可夫过程作为建模框架的研究和应用领域产生更广泛的影响。本项目开发的跳跃扩散过程的构造性近似和计算算法应被证明对使用跳跃扩散过程的不同应用领域有用。这项提议将支持西北大学金融工程专业的新博士学位。这一新的博士专业将培养出高素质的金融工程研究人员。该项目是西北大学在金融工程领域的长期发展努力的一部分。该项目还将帮助内华达大学拉斯维加斯分校数学科学系建立一个金融数学研究项目。

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.