2007 Fiscal Year Final Research Report Summary
Mathematics of Quadratic Interest Rate Models
| Project/Area Number |
18540146
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Ritsumeikan University |
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Principal Investigator |
AKAHORI Jiro Ritsumeikan University, College of Science and Engineering, Professor (50309100)
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| Project Period (FY) |
2006 – 2007
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| Keywords | Mathematical Finance |
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| Research Abstract |
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.
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