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

本计画发展一类新的分析与计算方法，以谱分析为基础，解决一类马尔可夫过程的首达与最佳停止问题。最优停止问题的目标是在随机过程建模的不确定性面前确定最优决策时机，以最大化奖励。第一次通过问题的目的是确定随机过程第一次通过边界的概率分布。这些数学问题在金融数学中有着广泛的应用，包括信用风险建模（借款人拖欠债务的风险，如公司债券），评估具有提前行使权的金融合约，如美式期权、可赎回债券和可转换债券，以及真实的期权。所研究的一类马尔可夫过程是跳扩散过程，可以由一维扩散的随机时间变化构造。在这个项目中开发的方法是基于代表条件期望算子与马尔可夫过程的特征函数展开。有效的计算算法将发展的马尔可夫过程的特征函数表示的正交多项式。 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和常数方差弹性（CEV）扩散，以及纯粹的跳跃和跳跃-扩散过程所产生的时间改变这些扩散。新的分析方法和计算算法在这个项目中开发的将适用于一系列问题的金融数学领域在包括债券市场、股票市场、商品和能源市场在内的各种市场中，以及在真实的期权和不可逆转的投资决策中，建立信用风险模型和评估具有提前行使权的金融合同。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。它们还将通过促进真实的期权分析的应用，帮助非金融企业做出更好的管理决策。预计该项目将对最优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。这是西北大学金融数学和工程长期努力的一部分，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。

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.