Project/Area Number  09440074 
Research Category 
GrantinAid for Scientific Research (B).

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  ATHE UNIVERSITY OF TOKYO 
Principal Investigator 
KUSUOKA Shigeo GARADUATE SCHOOL OF MATHEMATICAL SCIENCE, THE UNIVERSITY OF TOKYO, PROFESSOR, 大学院・数理科学研究科, 教授 (00114463)

CoInvestigator(Kenkyūbuntansha) 
上木 直昌 姫路工業大学, 理学部, 助教授 (80211069)
YOSHIDA Nakahiro GARADUATE SCHOOL OF MATHEMATICAL SCIENCE, THE UNIVERSITY OF TOKYO, ASSOCIATE PROFESSOR, 大学院・数理科学研究科, 助教授 (90210707)
長田 博文 東京大学, 大学院・数理科学研究科, 助教授 (20177207)
TAKAHASHI Akihiko GARADUATE SCHOOL OF MATHEMATICAL SCIENCE, THE UNIVERSITY OF TOKYO, ASSOCIATE PROFESSOR, 大学院・数理科学研究科, 助教授 (50313226)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥12,400,000 (Direct Cost : ¥12,400,000)
Fiscal Year 1999 : ¥3,200,000 (Direct Cost : ¥3,200,000)
Fiscal Year 1998 : ¥4,700,000 (Direct Cost : ¥4,700,000)
Fiscal Year 1997 : ¥4,500,000 (Direct Cost : ¥4,500,000)

Keywords  floating interest rate / interest rate model / default risk / mathematical finance / diffusion model / 変動金利 / 金利モデル / デフォルト リスク / 数理ファイナンス / 拡散モデル / 確率偏微分方程式 / ハザード率 / 確率解析 / 数値計算 / マリアバン・カリキュラス / リー環 / 確率テイラー展開 / デリバディブ / 変動金利モデル / クレジットデリバティブ / ハザードレート / 確率変動 / 確率偏微分方程 
Research Abstract 
The purpose of this research was to do theoretical and positive research on stochastic partial differential equation models begun recently. We were planning to think of only government bond or LIBOR which are supposed to be riskless. However, quite recently people got interested in socalled credit derivatives concerning defaultable bonds. So we slightly changed our plan and thought of model containing default probability. First we showed the existence and uniqueness of solution to stochastic partial differential equations related to interest rate under certain conditions for boundary conditions and coefficients. Also, we gave a condition so that the solution remains in positive range. Next we did theoretical research on prices of defaultable bonds. Here we gave a counterexample for a formula which people widely believed on the relationship between hazard rate processes and conditional default probabilities. Also, we gave a formula on hazard rate process in continuoustime filtering models. In mathematical finance, practically important formula for prices are given in terms of expectations. In estimates of statistical parameters, it is important to compute such expectations precisely and rapidly. We introduced a new numerical computation method in diffusion models, which are rather restrictive but widely used, and we showed that it is quite effective theoretically. In computing hedging strategies, we need more complicated expectations, but the research of them are postponed to the future. The statistical consideration for stochastic process models will be getting important more and more. Our plan contained the construction of a theory for it. In this respect we only got an asymptotic expansion formula related to convergence of probability law of additive functionals for hypoelliptic diffusion processes.
