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

该项目的重点是为金融工程开发高性能计算工具。目标是开发计算方法，以评估用于管理外汇，利率，股票，商品和能源价格风险和信贷风险的复杂金融产品，并管理大型资产组合。该方法是基于扩展到金融工程的有限元方法成功地用于不同的工程分支，以解决多维偏微分和积分方程的数值。偏积分微分方程出现在研究马尔可夫跳扩散过程和相关的最优停止和随机控制问题的金融工程。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，并根据这些方法开发高性能的计算工具，这些方法可以有效地实施和金融服务行业从业人员以及金融工程研究人员，应用概率和使用连续时间马尔可夫过程的运筹学分支。该项目将解决金融工程中的具体挑战，包括高维性和跳跃性。该项目开发的方法将帮助金融机构，企业财务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，这个项目将有一个更广泛的研究和应用领域，使用连续时间马尔可夫过程作为建模框架的影响。在这个项目中开发的跳跃扩散过程的建设性近似和计算算法应该证明有用的不同领域的应用，使用跳跃扩散过程。该提案将支持新的博士学位。在西北大学主修金融工程。这个新博士主要将导致高素质的金融工程研究人员的培训。该项目是西北大学金融工程领域长期发展努力的一部分。该项目还将帮助内华达州拉斯维加斯大学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.