Time Changes of Markov Processes: Applications in Financial Mathematics

马尔可夫过程的时间变化：在金融数学中的应用

• 批准号: 0802720
• 负责人: Vadim Linetsky
• 金额: $ 21万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0802720&HistoricalAwards=false
• 关键词: Time; Changes; Markov; Processes; Applications

This project is concerned with stochastic modeling of financial variables, such as equity prices, foreign exchange rates, interest rates, and commodity and energy prices, and the pricing of financial derivatives, financial instruments that serve as tools to mitigate financial market and credit risk. The goal is to develop empirically realistic and analytically tractable models describing stochastic dynamics of financial variables and analytical and computational methods to implement these models. The project focuses in the following areas. (i) Develop a rich toolbox of financial models with state-dependent jumps, stochastic volatility, and default. This will be based on the methodology of time changes of Markov processes. The proposed model architecture is to pair Markov processes with analytically tractable semigroups with time changes with analytically tractable Laplace transforms to design analytically tractable processes with desired properties. (ii) Develop efficient pricing tools for derivative securities in these models based on Laplace transform inversion and, for symmetric Markov processes, on the spectral expansion method. (iii) Develop the next generation of unified credit-equity models with jumps, stochastic volatility, and default that provide a unified treatment for all financial obligations related to a given firm, including stock, debt, stock options, and credit derivatives in the framework of time changes of Markov processes with killing. Study options pricing and corporate debt valuation in this unified framework. (iv) Develop a novel class of models with mean-reverting jumps for interest rates, commodities, and energy markets by subjecting mean-reverting diffusions to time changes with jumps. The aim of this project is to develop sophisticated mathematical tools for financial practice. This research is expected to have significant practical impact. According to the Bank for International Settlements (2006), the size of the global derivatives markets is $343 trillion in notional amounts. Major market segments include interest rate, currency, equity, commodity, energy, and credit derivatives used to manage financial risks. The proposed research is expected to find applications in all of these financial market sectors. Empirically realistic and, at the same time, analytically and computationally tractable models developed in this project will facilitate consistent, fast, and accurate modeling and pricing of financial instruments used to manage market and credit risk. This will help improve efficiency and stability of financial markets. The project is also expected to have a broader mathematical impact on stochastic analysis and applied probability. Analytically tractable time changes of Markov processes developed in this project will provide a useful laboratory for theoretical developments in stochastic analysis, as well as find applications in a variety of areas that employ Markov processes for the modeling of physical, biological, engineering, and economic phenomena. The project will have an impact on education and human resources development. It is part of the long-term development effort at Northwestern University in financial engineering, including the recently established Ph.D. concentration. It will train highly qualified researchers for academia and industry.

该项目关注金融变量的随机建模，如股票价格、外汇汇率、利率、商品和能源价格，以及金融衍生品的定价，金融衍生品是作为减轻金融市场和信贷风险工具的金融工具。目标是开发经验上现实的和分析上易于处理的模型，描述金融变量的随机动态以及实现这些模型的分析和计算方法。该项目侧重于以下领域。(i)开发具有状态依赖跳跃、随机波动和违约的丰富金融模型工具箱。这将基于马尔可夫过程的时间变化方法。提出的模型架构是将马尔可夫过程与具有时间变化的可解析处理半群与可解析处理拉普拉斯变换配对，以设计具有期望性质的可解析处理过程。（ii）为这些模型中的衍生证券开发基于拉普拉斯变换反演的有效定价工具，并为对称马尔可夫过程开发基于谱展开方法的有效定价工具。（iii）建立具有跳跃、随机波动和违约的新一代统一信用-权益模型，在马尔可夫过程的时间变化框架下，对与给定企业相关的所有金融义务，包括股票、债务、股票期权和信用衍生品，提供统一的处理。在这个统一的框架下研究期权定价和公司债务估值。（iv）通过使均值回归扩散随时间变化而变化，为利率、商品和能源市场开发一类具有均值回归跳跃的新模型。这个项目的目的是为金融实践开发复杂的数学工具。本研究可望产生重大的实际影响。根据国际清算银行（2006年）的数据，全球衍生品市场的名义规模为343万亿美元。主要细分市场包括用于管理金融风险的利率、货币、股票、商品、能源和信用衍生品。拟议的研究预计将在所有这些金融市场部门找到应用。本项目开发的具有经验现实性，同时具有分析和计算可操作性的模型，将有助于对用于管理市场和信用风险的金融工具进行一致、快速和准确的建模和定价。这将有助于提高金融市场的效率和稳定性。该项目还有望对随机分析和应用概率产生更广泛的数学影响。在这个项目中开发的分析可处理的马尔可夫过程的时间变化将为随机分析的理论发展提供一个有用的实验室，并在使用马尔可夫过程进行物理，生物，工程和经济现象建模的各种领域中找到应用。该项目将对教育和人力资源发展产生影响。这是西北大学金融工程长期发展努力的一部分，包括最近设立的博士学位。它将为学术界和工业界培养高素质的研究人员。