权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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，我得到了许多与二次利率模型相关的数学结果。第一个结果是与Hara教授共同工作的结果，该结果已发表在《数学金融学》上。结果表明，无穷维二次结构在利率建模中起着核心作用。从这一观察出发，我与Y. Nitta和T Matsusita提出了一种新的仿射李代数的不可约表示方法，即所谓的反对称Malliavin演算，利用它在Wiener空间上构造了仿射李代数的不可约表示。本研究将阐明二次维纳泛函与KdV方程孤子解之间的神秘联系。受此研究的启发，我也一直在尝试建立随机微分方程的伽罗瓦规范理论，在这个方向上的第一个结果包括在与K.Yano和C.上西本文给出了群在所有解空间上的传递作用，并且群控制解的性质。此外，与J. Teichmann和T. Tsuchiya，我建立了一个新的建模方案，我们称之为“热核方法”。这在某种意义上是二次利率模型的推广，同时也是状态价格密度利率模型的一个子类。我们发现，一个因果结构，我们称之为“传播属性”起着核心作用。本征函数展开和θ函数也是我们方法中的两个关键角色。我还与H.青木湾，澳-地Nagata，Y. Morimura，Y. Kanishi和L.石井。通过对主成分分析的研究，我们得出了二次模型比线性模型更稳健的结论。