Mathematics of Quadratic Interest Rate 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结果。第一个成果是与原原诚司合作的成果，该成果已发表在《数学金融》杂志上。结果表明，无限维二次型结构在利率模型中起着核心作用。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，利用它在Wiener空间上构造了仿射李代数的一个不可约表示。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。在这项研究的激励下，我也一直试图建立随机微分方程的伽罗瓦规范理论，在这个方向上，第一个结果包括在与K.Yano和C.Uenishi的联合工作中。本文证明了群在所有解的空间上的传递作用，并且群控制解的性质。此外，我与J.Teichmann和T.Tuchiya共同建立了一个新的模型方案，我们称之为“热核法”。这在某种意义上可能是二次利率模型的推广，同时也是国家价格密度利率模型的一个子类。我们发现，一种我们称之为“传播性质”的因果结构起着核心作用。特征函数展开和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一起做了一项更实际的利率研究。通过对主成分分析的深入研究，我们得出结论：二次模型比线性模型更稳健。

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.