Project/Area Number  10440030 
Research Category 
GrantinAid for Scientific Research (B).

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

Research Institution  OSAKA UNIVERSITY 
Principal Investigator 
NAGAI Hideo Graduate School of Engineering Science, OSAKA UNIVERSITY Professor, 基礎工学研究科, 教授 (70110848)

CoInvestigator(Kenkyūbuntansha) 
亀山 敦 大阪大学, 大学院・基礎工学研究科, 助手 (00243189)
中村 佳正 大阪大学, 大学院・基礎工学研究科, 教授 (50172458)
亀高 惟倫 大阪大学, 大学院・基礎工学研究科, 教授 (00047218)
白川 浩 東京工業大学, 理財工学研究センター, 助教授 (10216187)
TAKANOBU Satoshi Kanazawa University, Department of Mathematics, Associate Professor, 理学部, 助教授 (40197124)
SEKINE Jun Graduate School of Engineering Science, OSAKA UNIVERSITY Lecturer, 基礎工学研究科, 講師 (50314399)
AIDA Shigeki Graduate School of Engineering Science, OSAKA UNIVERSITY Associate Professor, 基礎工学研究科, 助教授 (90222455)
TAKEDA Masayoshi Tohoku Univesity, Graduate School of Science, Professor, 理学研究科, 教授 (30179650)
FUJIWARA Tsukasa Hyogo Education University, Associate Professor, 助教授 (30199385)

Project Fiscal Year 
1998 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥5,200,000 (Direct Cost : ¥5,200,000)
Fiscal Year 2000 : ¥1,700,000 (Direct Cost : ¥1,700,000)
Fiscal Year 1999 : ¥1,700,000 (Direct Cost : ¥1,700,000)
Fiscal Year 1998 : ¥1,800,000 (Direct Cost : ¥1,800,000)

Keywords  risksensitive stochastic control / ergodic Bellman equation / portfolio optimization / Schrodinger operator / large deviation principle / log Sobolev inequality / spectral gap / additive functional / リスク鋭感的確率制御 / エルゴード型ベルマン方程式 / ポートフォリオ最適化 / シュレーディンガー作用素 / 大偏差原理 / 対数ソボレフ不等式 / スペクトルギャップ / 加法的汎関数 / 対数ソポレフ不等式 / エルコンド型ベルマン方程式 / 粘性解 / 特異極限 / ベルマン方程式 / エルゴード型確率制御問題 
Research Abstract 
(a) It is important to know the conditions for no breakdown in risksensitive stochastic control problems since the value function has not always a finite value. In this research we have obtained the condition as that of the size of risksensitive parameter in the case of a finite time horizon and also shown the solvability of the corresponding Bellman equation under the condition. It is applicable for the case that the risksensitive parameter is large and has great meaning in application. In the case of infinite time horizon we have shown existence and uniqueness of the corresponding ergodic type Bellman equation under a similar condition by noticing the relationships between the problems and the eigenvalue problems for Schrodinger operator. Besides, we have derived a first order partial differential equation relating to game theoretical approach to nonlinear H_∞ control as its singular limit. (b) We have considered portfolio optimization problem for a factor model as application to m
… More
athematical finance of risksensitive stochastic control and got the results giving the explicit representation to optimal portfolio for the problem in the case of partial information. In the case of infinite time horizon we obtained the condition under which the solution of corresponding ergodic type Bellman equation defines the optimal portfolio and constructed it by the solution. We have also found that the solution does not always give the optimal portfolio without any condition. (c) We have shown existence of the spectral gap of Schrodinger opeartor by using log Sobolev inequality and gave the estimate. (d) We have shown that the large deviation principle holds for additive functional of Brownian motion corresponding to measures in Kato class. As its application we obtained necessary and sufficient condition under which additive functionals converges exponentially fast. (e) We have shown Trotter product formula with respect to L_p and trace norm and their error estimates for the Schrodinger operator with a potential bounded below. (f) We have shown that the sequence of symmetric statistics defined by Weyl transformation on infinite dimensional torus converges to the limit represented by multiple Wiener integral under the probability measures. Less
