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

这笔赠款为金融工程高性能计算工具的开发提供资金。目标是开发计算方法来评估用于管理外汇、利率、股权、大宗商品、能源和信贷风险的复杂金融产品，管理资产组合，以及评估制造业的金融合同和设备租赁。该方法基于有限元方法在金融工程中的扩展，用于数值求解偏微分方程组。偏积分-微分方程解存在于马尔可夫跳跃扩散模型以及金融工程中的最优停止和随机控制问题中。有限元方法将应用于跳跃扩散过程，以开发计算工具，供金融服务业从业者以及金融工程、应用概率和运筹学研究人员使用。应用包括各种金融合约(股权和外汇期权、利率衍生品、商品合约和制造业设备租赁)的定价算法，以及考虑交易成本和交易限制的投资组合优化。如果成功，该项目开发的方法将帮助金融机构、制造和服务公司的公司金库以及能源公司准确评估复杂的金融工具，并有效地管理金融风险。该项目还将对使用连续时间马尔可夫过程的研究和应用领域产生更广泛的影响，如排队理论、库存控制、调度和制造中的重交通限制。关于估值、最优停止和随机控制问题的构造性逼近的结果也将有助于仿真和随机优化研究，为仿真和随机优化提供可靠的基准。这笔助学金支持西北大学金融工程博士学位的攻读。这将为学术界和工业界培养高素质的金融工程研究人员。这笔拨款还将帮助内华达大学数学科学系建立一个金融数学研究项目。

This grant provides funding for 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, energy, and credit risks, manage portfolios of assets, and evaluate financial contracts and equipment leases in manufacturing industries. The methodology is based on the extension to financial engineering of finite element methods used to solve numerically partial differential equations. Partial integro-differential equations arise in Markov jump-diffusion models and associated optimal stopping and stochastic control problems in financial engineering. Finite element methods will be applied to jump-diffusion processes to develop computational tools to be used by practitioners in the financial services industry, as well as researchers in financial engineering, applied probability and operations research. Applications include pricing algorithms for a wide range of financial contracts (equity and foreign exchange options, interest rate derivatives, commodity contracts, and equipment leases in manufacturing), as well as portfolio optimization with transaction costs and trading restrictions. If successful, methodologies developed in this project will help financial institutions, corporate treasuries of manufacturing and service firms, and energy companies accurately value complex financial instruments and efficiently manage financial risks. This project will also have a broader impact on research and application areas that use continuous-time Markov processes, such as heavy traffic limits in queueing theory, inventory control, scheduling, and manufacturing. The results on constructive approximations to the valuation, optimal stopping and stochastic control problems will also help simulation and stochastic optimization research by providing reliable benchmarks for simulation and stochastic optimization. This grant supports the Ph.D. concentration in financial engineering at Northwestern. This will result in training highly qualified researchers in financial engineering for academia and industry. The grant will also help the Department of Mathematical Sciences at the University of Nevada establish a research program in financial mathematics.