Time Changes of Markov Processes: Applications in Financial Mathematics

该项目关注的是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，以及金融衍生工具的定价，作为减轻金融市场和信用风险的工具的金融工具。我们的目标是开发经验现实和分析听话的模型描述金融变量的随机动态和分析和计算方法来实现这些模型。该项目侧重于以下领域。(i)开发一个包含状态依赖跳跃、随机波动和违约的金融模型的丰富工具箱。这将是基于马尔可夫过程的时间变化的方法。建议的模型架构是配对马尔可夫过程与解析易处理的半群与时间变化的解析易处理的拉普拉斯变换设计解析易处理的过程所需的属性。(ii)在这些模型中，基于拉普拉斯变换反演和对称马尔可夫过程的谱扩展方法，开发衍生证券的有效定价工具。(iii)开发下一代统一的信贷资产模型，包括跳跃，随机波动和违约，为与给定公司相关的所有金融义务提供统一的处理方法，包括股票，债务，股票期权和信贷衍生品，在马尔可夫过程的时间变化框架内。在这个统一的框架下研究期权定价和公司债务估值。(iv)通过使均值回复扩散服从跳跃的时间变化，为利率、商品和能源市场建立一类均值回复跳跃的新模型。该项目的目的是为金融实践开发复杂的数学工具。这项研究预计将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。主要细分市场包括利率、货币、股票、商品、能源和用于管理金融风险的信用衍生品。预计拟议的研究将在所有这些金融市场部门中找到应用。在这个项目中开发的经验上现实的，同时，分析和计算上易于处理的模型将促进用于管理市场和信用风险的金融工具的一致，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。该项目还预计将有一个更广泛的随机分析和应用概率的数学影响。在这个项目中开发的马尔可夫过程的分析易处理的时间变化将提供一个有用的实验室，在随机分析的理论发展，以及在各种领域，采用马尔可夫过程的物理，生物，工程和经济现象的建模中找到应用。该项目将对教育和人力资源开发产生影响。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。浓度.它将为学术界和工业界培养高素质的研究人员。

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.