Mathematics of Quadratic Interest Rate Models

二次利率模型的数学

基本信息

批准号：
18540146
负责人：
AKAHORI Jiro
金额：
$ 2.62万
依托单位：
Ritsumeikan University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2006
资助国家：
日本
起止时间：
2006 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540146/
关键词：
Mathematical Finance

项目摘要

In this research project, I have obtained many mathematical results related to quadratic interest rate models. First result is the one in the joint work with Prof Hara, which has published in Mathematical Finance. The result shows that infinite dimensional quadratic structure plays a central role in interest rate modeling. Starting from this observation, I have established, together with Y. Nitta and T Matsusita, so-called anti-symmetric Malliavin calculus, by which an irreducible representation of Affine Lie algebra is constructed on Wiener space. This study will clarify the mysterious connection between quadratic Wiener functionals and soliton solution of KdV equation. Motivated by the study, I have also been trying to establish a Galois-Gauge theory of stochastic differential equations, and in this direction the first results are included in the joint work with K.Yano and C. Uenishi. In the paper we have hind a transitive action of a group on the space of all solutions and the group controls the property of solutions. Further, jointly working with J. Teichmann and T. Tsuchiya, I have established a new modeling scheme which we call "heat kernel approach". This may be a generalization of quadratic interest rate models in a sense, and at the same time it is a subclass of state price density interest rate models. We have found that a causal structure which we call "propagation property" plays a central role. The eigenfunction expansion and theta functions are also two of key player in our approach. I have also done a more practical oriented study on interest rates, together with H. Aoki, Y. Nagata, Y. Morimura, Y. Kanishi, and L. Ishii. Starting from the careful study of principal component analysis, we have concluded that quadratic models are more robust than linear models.

在这个研究项目中，我获得了许多与二次利率模型相关的数学成果。第一个成果是与 Hara 教授合作的成果，已发表在《数学金融》杂志上。结果表明，无限维二次结构在利率建模中发挥着核心作用。从这一观察出发，我与 Y. Nitta 和 T Matsusita 一起建立了所谓的反对称 Malliavin 演算，通过它在维纳空间上构造了仿射李代数的不可约表示。这项研究将阐明二次维纳泛函与 KdV 方程的孤子解之间的神秘联系。在这项研究的推动下，我也一直在尝试建立随机微分方程的 Galois-Gauge 理论，并且在这个方向上的第一个结果包含在与 K.Yano 和 C. Uenishi 的联合工作中。在本文中，我们给出了群在所有解的空间上的传递作用，并且群控制了解的性质。此外，我与 J. Teichmann 和 T. Tsuchiya 合作，建立了一种新的建模方案，我们称之为“热核方法”。这在某种意义上可能是二次利率模型的推广，同时它也是状态价格密度利率模型的一个子类。我们发现，我们称之为“传播属性”的因果结构起着核心作用。本征函数展开和 theta 函数也是我们方法中的两个关键因素。我还与 H. Aoki、Y. Nagata、Y. Morimura、Y. Kanishi 和 L. Ishii 一起对利率进行了更实用的研究。从对主成分分析的仔细研究开始，我们得出结论，二次模型比线性模型更稳健。