Numerical solution of differential equations with applications to computational finance and numerical weather forecasting

微分方程的数值解及其在计算金融和数值天气预报中的应用

• 批准号: 8073-2006
• 负责人: Jackson, Kenneth
• 金额: $ 3.64万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2006
• 资助国家: 加拿大
• 起止时间: 2006-01-01 至 2007-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=338355
• 关键词: Numerical; solution; differential; equations; applications

I am engaged in several research projects to develop, analyze, test and evaluate numerical methods and to construct robust mathematical software.  I hope these projects will ultimately lead to improvements in mathematical software, enabling scientists, engineers and financial modelers either to solve previously intractable problems or to solve problems more efficiently and/or reliably than is currently possible. Previously, my research focused primarily on initial-value problems (IVPs) for ordinary differential equations (ODEs), but it has since broadened to include boundary value problems (BVPs) for ODEs as well as the application of ODE techniques to partial differential equations (PDEs) and to the solution of practical problems in science, engineering and finance.  I have a secondary interest in numerical linear algebra, particularly problems arising from differential equations.  Over the next few years, I will concentrate on the following two major projects: (1) the study of guaranteed error bounds for the numerical solution of IVPs for ODEs, with the long term goal of developing a robust, reliable, easy-to-use package incorporating these schemes; (2) the study of numerical methods for finance, with particular emphasis on effective computational techniques for computing the price of convertible bonds and the price of credit derivatives.

我参与了几个研究项目，以开发、分析、测试和评估数值方法，并构建强大的数学软件。我希望这些项目最终将导致数学软件的改进，使科学家、工程师和金融建模人员能够解决以前难以解决的问题，或者比目前可能的情况更有效和/或更可靠地解决问题。以前，我的研究主要集中在常微分方程组的初值问题，但后来扩大到包括常微分方程组的边值问题，以及将常微分方程组的边值问题应用于偏微分方程组和解决科学、工程和金融中的实际问题。我对数值线性代数，特别是来自微分方程的问题有次要的兴趣。在接下来的几年里，我将专注于以下两个主要项目：(1)常微分方程组数值解的保证误差界的研究，长期目标是发展一个健壮的，包含这些计划的可靠、易于使用的一揽子计划；(2)金融数值方法的研究，特别强调计算可转换债券价格和信用衍生品价格的有效计算技术。