Grant amount：\10010000 （ Direct Cost: \7700000 、 Indirect Cost：\2310000 ）

In the present project, we applied Malliavin Calculus and operator decomposition techniques to study various problems in applied mathematics. In particular, we provided new simulation techniques for the approximation of solutions of stochastic differential equations driven by jumps. We studied the mathematical properties of these methods and proved that they provide a more accurate and fast method in comparison with past methods of simulation. This type of equation is also used as a model in Mathematical Finance. In this area we provided various formulas for the so called Greeks which measure risk in financial products. Through the use of the infinite dimensional integration by parts formula, we provide various alternatives and we also provided simulations to show the applicability of the results. We also obtain a lower bound estimate for Asian type random variables. We are thinking of applying these results to various problems in filtering.

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Grant amount：\36010000 （ Direct Cost: \27700000 、 Indirect Cost：\8310000 ）

We have carried out theoretical research into mathematical technology of finance and financial engineering including three main topics ‘ optimization and operations research’,‘stochastic numerical analysis’, and ‘information network security’. We opened software of them to the internet. We also designed and developed a financial numerical calculation system. As a result, it makes possible to easily access our highly specialized mathematical technology.

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Grant amount：\3630000 （ Direct Cost: \3000000 、 Indirect Cost：\630000 ）

(1) We have succeeded in finding and constructing new higher-order numerical approximation algorithms for diffusion processes. The algorithms are based on the theory of Kusuoka approximation. They enjoy both the numerical robustness and the universality, that is, we can apply them for almost all diffusions. (2) We applied the algorithms to some finance problems and achieved remarkable improvements in : (a) very fast calculation (at least 100 times faster than the state of the art methods) (b) robust discretization (c) universality. (3) We have developed the computer program of the algorithms.

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Grant amount：\10320000 （ Direct Cost: \8700000 、 Indirect Cost：\1620000 ）

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Grant amount：\48490000 （ Direct Cost: \37300000 、 Indirect Cost：\11190000 ）

We investigated and developed new IT models for financial risk management in three directions of financial risk, optimization technology, and IT environment.
・Refinement of Financial risk measurement
We constructed a financial database of small-to-medium sized enterprises for quantitative analyses on credit risk at first.
We used the database to develop a new-type relevant loan pricing model. The main characteristics of our model is a hybrid of the methodology for pricing credit derivatives with stochastic calculus and the traditional statistical approach based on the financial database. We also studied an application of multi-objective genetic algorithms to tuning of a credit scoring model.
・Optimization technology
We constructed a model which minimizes a tracking error of a portfolio under uncertainty and developed an algorithm for computing the optimal solution of the model efficiently. We conducted basic researches into minimax classification problem for avoiding financial risk. We also succeeded in developing algorithms of Kusuoka-approximation which are capable of high order approximation and can be applied to stochastic differentiable equations without any limitation. It is demonstrated that the algorithm is very effective for mathematical financial problems.
・IT environment
We developed a method called structured aggregate signature with respect to digital signature, which is important in digital business transaction. We got a correction method for error of safety proof of Rabin signature. We also conducted researches into XBRL and a cyber financial system as a fundamental technology for sharing basic credit information.
As results of the research, we published 19 research papers and a book. We organized 6 symposiums or workshops for collecting the information of financial risk management and for announcing our results to inside and outside researchers.

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Grant amount：\14400000 （ Direct Cost: \14400000 ）

The principal aim of this project was to develop the integrated researches on the comteporain problems in stochastic numericas as follows (names in paratheses ( ) are main investigators occupying each session);
(1) Numerical integration by means of Monte Carlo Methods and random numbers (S.Ito, M.Mori, K.Fukuyama, H.Sugita, and S.Ogawa)
(2) Numerical solution of the SDEs (S.Ninomiya and S.Ogawa)
(3) Noncausal calculus based on the Ogawa integral and its applications (S.Ogawa)
(4) Probabilistic solution of functional equations (K.Nishioka, S.Ogawa)
(5) Development of stochastic methods and algorithms for problems in OR, Finance, Computer sciences and Mathematical economy. (S.Ninomiya, K.Nishioka and S.Ogawa)
Through the whole period of the research activity, from 2003 to 2006, each research group cited above has pursuived each research plan, in a pararel way or sometimes in collaboration with other groups and as an integrated research group got many fruitful results not only theoretical but also the results of pratical importance, like the establishment of some computational algorithms in mathematical finance.

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Grant amount：\3000000 （ Direct Cost: \3000000 ）

The object of this project is to establish the Kusuoka approximation. Kusuoka approximation is a new scheme that approximate E[f(X(T)] where X(t) denotes a diffusion process and f a function with some regularity. This problem is called weak approximation. By using the Kusuoka approximation, it is expected that we can reduce the number of the dimension of the domain of integration which arises in the last step of the approximation. This integral dimension is a very critical factor if we use quasi-Monte Carlo techniques. In this project, we have achieved the following successes :
1. The discovery of a versatile algorithm that enables us to apply the Kusuoka approximation easily to any diffusion processes described by SDEs.
2. The algorithm above also is compatible with quasi-Monte Carlo method.
3. We have applied the algorithm to financial derivative pricing problem and showed that our new algorithm makes at least 800 times faster calculation than existing methods.

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Grant amount：\3500000 （ Direct Cost: \3500000 ）

本年度は、拡散過程のシミュレーション技術の開発において成果を得た。
確率微分方程式で記述された拡散過程X(t, x)と関数fが与えられている時、その期待値E[f(X(t, x))]を求める問題は、確率数値解析の大きなテーマである。金融リスクの計算は、まさにこの問題に他ならない。この計算を行う方法として金融リスクの分野では次の方法がもっとも有用であり広く行われている。それは、拡散過程X(t, x)をEuler-丸山近似と呼ばれる方法で近似するという方法である。現在、拡散過程の近似は殆どこのEuler-丸山近似で行われており、また、この近似計算の研究もEuler-丸山近似を前提としたものが殆どである。研究代表者は、Euler-丸山近似とは別の近似方法である楠岡近似を用いることによって金融リスク計算の劇的な高速化を実現した。楠岡近似を用いると近似計算を数値積分に帰着させたとき、その積分空間の次元を非常に小さくすることが出来る。その結果として、数値積分の速度が非常に高速になる。本年度の結果によれば、ある種の金融派生商品の価格計算は楠岡近似を用いるとEuler-丸山近似を用いた場合に比べて約6000倍高速である。
この結果は、今後、金融リスク計算の分野においては、シミュレーションの技法がEuler-丸山近似から楠岡近似によるものにとって代られるであろうことを示しており、非常に重要な結果である。

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Grant amount：\13100000 （ Direct Cost: \13100000 ）

1. Failure Discriminant Analysis
2. Estimation of Failure Probability
3. Bounding Option Price
4. Maximal Predictability Portfolio
5. Pricing Derivatives by Simulation
6. Global Optimization Methods for Financial Optimization
7. Business Method Patent for Financial Engineering
8. Algorithm for Semi-Definite Programming Arising in Financial Optimization

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