## MEMBERS

### Program Members

There are many examples where mathematics and industry amalgamate, which is what the Math-for-Industry (hereafter referred to as MI) strives to achieve. One may recall wavelet analysis, cryptography, and CT scanning. Another celebrated example is the encounter of economics and probability theory in mathematical finance. The origin of Probability theory goes back to the letters on gambling exchanged between Pascal and Fermat, and hence the amalgamation of economics and probability theory may be promised from the very beginning. Even though, the close relation between them, which we see today, is a bonne fortune.

The importance of probability theory in mathematical finance is based on stochastic analysis created by Kiyosi I＾to in 1942, which is a calculus corresponding to the Brownian motion. In 1827 Robert Brown, a Scottish botanist, discovered an irregular motion, which was named after him and is now known as the Brownian motion. The Brownian motion was formulated with mathematical vigor in the 1920s. Einstein’s celebrated paper on Brownian motion was published in 1905. Five years before Einstein’s paper, a stock price model using the Brownian motion was proposed by a French mathematician Bachelier, and the current amalgamation between mathematical finance and probability theory started with his paper. The Black-Scholes model, which is recently often cited even in newspapers, is a theoretical model using the Brownian motion to provide a pricing formula of derivatives. The model was established by Black, Sholes, Merton, and others, who, in 1960s and 1970s, were working around Samuelson, the Novel laureate in economy. The rapid development since 1940s of stochastic analysis plays an important role in building and developing the models. In generalizing these models, stochastic analyses based on stochastic processes other than Brownian motion are essential, and there have been active mutual stimulations between theory and practice.

The recent research in stochastic analysis has been progressing hand in hand between so-called the Itô calculus, which is the calculus related to stochastic integrals and stochastic differential equations, and the Malliavin calculus, which was formulated in the latter half of the 1970s. I have been working mainly on stochastic differential equations and the Malliavin calculus. My research achievements are on probability density functions of random variables obtained from solutions to stochastic differential equations, stochastic flows determined by stochastic differential equations, a complex change of variables on path spaces using the Malliavin calculus, stochastic oscillatory intetgrals (Fourier-Laplace type transformations on infinite-dimensional spaces), applications of stochastic analysis to KdV equations, and so on. A stochastic oscillatory integral is a probabilistic counterpart to the Feynman path integral, and the effect of the study of stationary phase method for stochastic oscillatory integrals goes beyond the probability theory. The probabilistic research on KdV equations, which is an approach via stochastic analysis to non-linear equations, will open a door to a new interdisciplinary research area between theory of partial differential equations and probability theory.

While all of my research projects concern pure mathematics, by utilizing my knowledge of stochastic analysis, I have been working hard at training MI researchers who possess advanced knowledge, which is closely tied to the needs in the industry and the government. For example, in lecturing and supervising graduate and undergraduate students, I have been teaching topics in stochastic analysis, which relates to mathematical finance. Moreover, I have written books on mathematical finance. Many of my former students, undergraduate and graduate alike, have gained mathematical knowledge in financial engineering, and have found employment in financial corporations and the government.

I am responsible for the joint research program between the Nisshin Fire & Marine Insurance Co., Ltd. and the Faculty of Mathematics, Kyushu University. This program is called the Nisshin Fire Program, and since April 2008 I have been conducting our part of the joint research project by employing two postdoctoral fellows. In the insurance industry, new regulations on internationalization (International Accounting Standards) and financial competence (solvency margin) are to be introduced in 2011. Objectives of the joint research project are to educate people with advanced mathematical knowledge to tackle new insurance business, and to develop mathematical models for insurance. This project is a part of collaborative research projects between industrial technology and mathematics, which the Faculty of Mathematics is conducting, and, in addition to research, the objectives are to educate and train young research personnel at both institutes. With this project as a launching pad, we hope to expand opportunities to conduct directly collaborative researches by research organizations with other financial institutes and non-manufacturing businesses.