Spectral Methods for Optimal Stopping and First Passage Problems with Applications in Financial Mathematics

最优停止和首次通过问题的谱方法及其在金融数学中的应用

• 批准号: 1109506
• 负责人: Vadim Linetsky
• 金额: $ 20万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1109506&HistoricalAwards=false
• 关键词: Spectral; Methods; Optimal; Stopping; First

This project develops a novel class of analytical and computational methods based on spectral analysis to solve first passage and optimal stopping problems for a class of Markov processes. The objective of an optimal stopping problem is to determine optimal decision timing to maximize reward in the face of uncertainty modeled by a stochastic process. The objective of a first passage problem is to determine the probability distribution of the first time a stochastic process passes through a boundary. These mathematical problems arise in a wide variety of applications in financial mathematics, including modeling credit risk (the risk of default of a borrower on its debt, such as a corporate bond), evaluating financial contracts with early exercise rights, such as American-style options and callable and convertible bonds, and real options. The class of Markov processes under study are jump-diffusion processes that can be constructed by stochastic time changes of one-dimensional diffusions. The methods developed in this project are based on representing conditional expectation operators associated with Markov processes by eigenfunction expansions. Efficient computational algorithms will be developed for Markov processes whose eigenfunctions are expressed in terms of orthogonal polynomials. Many of the most important stochastic processes in financial mathematics belong to this category, including the Ornstein-Uhlenbeck, Cox-Ingersoll-Ross, and constant elasticity of variance (CEV) diffusions, as well as pure jump and jump-diffusion processes arising from time changing these diffusions.The novel analytical methods and computational algorithms developed in this project will be applied to a range of problems in financial mathematics in the areas of modeling credit risk and evaluating financial contracts with early exercise rights in a variety of markets, including bond markets, equity markets, commodities and energy markets, and to real options and irreversible investment decisions. The mathematical methods developed in this project will help financial institutions to accurately evaluate and manage the risk of a variety of financial transactions. They will also help non-financial firms make better managerial decisions by facilitating applications of real options analysis. The project is expected to have a broader mathematical impact on research on optimal stopping and first passage problems. The project will also have an impact on education and human resources development. It is part of the long-term effort at Northwestern in financial mathematics and engineering, including the Ph.D. concentration in this area. It will train highly qualified researchers for academia and industry.

该项目开发了一类基于谱分析的新型分析和计算方法，以解决一类马尔可夫过程的首次通过和最优停止问题。最佳停止问题的目标是确定最佳决策时机，以在面对随机过程建模的不确定性时最大化奖励。首次通过问题的目标是确定随机过程第一次通过边界的概率分布。这些数学问题出现在金融数学的各种应用中，包括信用风险建模（借款人债务违约的风险，例如公司债券），评估具有早期行使权的金融合同，例如美式期权、可赎回和可转换债券以及实物期权。正在研究的一类马尔可夫过程是跳跃扩散过程，可以通过一维扩散的随机时间变化来构造。该项目中开发的方法基于通过特征函数展开来表示与马尔可夫过程相关的条件期望算子。将为马尔可夫过程开发有效的计算算法，其特征函数用正交多项式表示。 金融数学中许多最重要的随机过程都属于这一类，包括 Ornstein-Uhlenbeck、Cox-Ingersoll-Ross 和恒定方差弹性 (CEV) 扩散，以及因时间改变这些扩散而产生的纯跳跃和跳跃扩散过程。该项目中开发的新颖分析方法和计算算法将应用于金融数学中的一系列问题 在信用风险建模和评估各种市场（包括债券市场、股票市场、大宗商品和能源市场）以及实物期权和不可逆转投资决策中具有早期行使权的金融合同领域。该项目开发的数学方法将帮助金融机构准确评估和管理各种金融交易的风险。它们还将通过促进实物期权分析的应用来帮助非金融公司做出更好的管理决策。该项目预计将对最佳停止和首次通过问题的研究产生更广泛的数学影响。该项目还将对教育和人力资源开发产生影响。这是西北大学在金融数学和工程领域长期努力的一部分，包括博士学位。集中在这个领域。 它将为学术界和工业界培养高素质的研究人员。