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My specialty is probability theory and mathematical finance. Mathematical finance is a field that seeks to solve financial problems using mathematical methods, and I mainly analyze financial markets using stochastic differential equations.

As a premise for the discussion of what a stochastic differential equation is, first of all, there is something called a differential equation. Natural and social phenomena behave with some regularity, so by finding that regularity and modeling it, we can predict the future. The way to express such a model is a differential equation. However, predictions are not perfect, and there is a certain degree of "fluctuation" (uncertainty) in actual phenomena. Stochastic differential equations incorporate this "fluctuation" that is added probabilistically into differential equations for calculations, and in the world of mathematical finance, stochastic differential equations are used to build mathematical formulas and models that predict future stock prices, including the fluctuation. To explain using familiar temperature, for example, let's say you predict that the average temperature in August in Tokyo 10 years from now will be 30°C. However, since it will not be exactly as predicted, the "fluctuation" in the prediction becomes important. If the fluctuation is ±2°C, the average temperature will be 30°C ±2°C, which means 28°C to 32°C, and people who need to prepare for a heatwave should prepare for 32°C. Just knowing the steady 30 degrees Celsius doesn't tell us whether we should prepare for temperatures that are even hotter than that, nor does it tell us how much of a temperature change we should be prepared for.

Mathematical finance is a field that contributes to people's asset formation by analyzing financial trends using stochastic differential equations, and research has been progressing for about 50 years. When buying and selling stocks, even if there is a tendency for the stock price to move positively in the long term, various uncertain factors are added in the short term, and there are times when you gain and times when you lose. Therefore, if you can predict that the stock price will be 40,000 yen in 10 years, and even if there is a fluctuation, it will be plus or minus 10,000 yen, investors can prepare for risks. The desire to increase assets that underpins buying and selling behavior is common to many people, and since most investors act rationally, it is actually easier to model than natural phenomena, and it can be said that this field has a high level of accuracy in analysis using stochastic differential equations.

In stochastic differential equations, trends are analyzed from two perspectives: "trends" and "uncertainties," which can be found from patterns in the trends of things. "Trends" are small in the short term but progress steadily in the long term, while "uncertainties" are large in the short term but average zero and cancel each other out in the long term, making the impact small. This two-pronged approach can be applied to a variety of phenomena other than analyzing financial trends. Take climate change as an example: global warming is progressing steadily, but when we look at the weather each year, there are hot years and cold years. The long-term trend is that global warming is progressing steadily, and short-term weather fluctuations are an additional element of uncertainty (= fluctuations).

Stochastic differential equations can be expressed through numerical simulations. When the progress of global warming is predicted and graphed, it can be seen to steadily increase to the right, showing a rough predicted value. When uncertainties are added to this, a certain range is created above and below the center line. By using this range, it becomes possible to envision the best and worst case scenarios for the future, making it easier to come up with concrete measures.

Simulation example diagram

When considering stock price trends in mathematical finance, it is relatively easy to calculate long-term, steady trends, but calculating short-term uncertainty is a difficult problem. The parameter that is the basis of uncertainty is called volatility, and volatility increases when people who buy and sell stocks feel anxious about the future. I speculate that automated stock trading by AI, which has become widespread in recent years, is also having a significant impact on the increase in volatility. This is because AI does not hesitate to buy and sell based on trading data, etc., so it is thought that there will be an increase in follow-up behavior, such as "I will buy because someone else bought (I will sell because someone else sold)." If this kind of behavior, which can be called information cascade ^*, spreads, it could cause a sudden increase in volatility (a major fluctuation in stock prices), which could lead to a bubble economy or a financial crisis. In addition, it is possible that AI (artificial intelligence) will take similar buying and selling actions using similar algorithms, which will result in behavior that is close to following.

In my research, I would like to analyze people's psychology and behavioral mechanisms to improve the accuracy of volatility predictions and present indicators that can suppress overreactions in the stock market. Specifically, I aim to suppress uncertainty within an appropriate range by lecturing people who actually trade stocks on the points and points to be careful of when trading, and by appealing to financial authorities to publish specific data as a basis for determining volatility. If more data is published by financial authorities, the mechanism of volatility will be elucidated and the appropriate fluctuation range will become clearer. This will prevent a situation in which someone starts selling stocks for some reason, causing an information cascade, and AI-based trading will further spur a stock price crash and lead to corporate bankruptcy. It is very dangerous if the stock market becomes dominated by non-subjective trading such as information cascades. It is necessary to encourage diverse trading based on individual judgment and keep stock price fluctuations within an appropriate range close to the center line that shows a long-term and steady trend, in other words, to correct it to the way it should be, where everyone makes calm judgments based on accurate information and buys and sells.

Specifically, we are investigating the relationship between volatility and follow-up behavior from ultra-short-term data on large amounts of stock trading, and creating a procedure for adjusting published volatility to an appropriate level. We hope to continue our analysis and obtain results that will lead to the stabilization of financial markets in the future.

* Information cascade: A phenomenon in which behavior that imitates others without being based on sufficient information is repeated and spreads.

At Aoyama Gakuin University, there is the Aoyama Finance Research Group, where professors specializing in finance work together across departmental boundaries. They regularly hold research presentations and exchange opinions, and it is not uncommon for them to apply their findings to their own research, and it is not uncommon for them to develop into joint research involving professors from other universities. The direction of observing the aforementioned follow-up behavior from a short-term perspective was decided upon with confidence from discussions with the professors in the research group, and the research has progressed at an accelerated pace since then. In addition, I feel that the visceral sense of those working at the forefront of the financial industry is important in the research process, and I am also very excited to be able to hear stories from the traders at securities companies who participate in the study group. Theory is not omnipotent. It is a valuable opportunity to learn factors that cannot be captured by theory alone, and the stories of those in the field are a treasure trove of information. I am very grateful for the unexpected hints I sometimes receive.

I am a mathematician, but I would say that I am currently more of an economist. Mathematical finance is a field that lies between mathematics and economics, but in recent years I have been working in a field that is closer to economics. While mathematics is a world of strict theory, the real social phenomena that economics deals with do not often proceed according to theory. This is what makes it interesting, and I feel great joy when I discover a theory that leads to an understanding of real social phenomena. I may be a rare type of mathematician. My long-held desire to "do research that is useful to society" is what gives me motivation today.

As a researcher, I mainly focus on finance, but stochastic differential equations can be applied to simulating all kinds of social phenomena. In addition to simulating climate change, as mentioned above, they can also be used to predict trends in the number of infectious disease patients, maximum precipitation, and flood damage, and can even be used as a basis for policy decision-making.

In my seminar, students not only study financial products and asset management methods, but also make future predictions about social phenomena that interest them after studying finance. The topics are very diverse, including the spread of My Number cards, the penetration of cashless payments, the proportion of eco-cars and the effects of policies, and more. Using stochastic differential equations, students identify trends by separating uncertain factors from those that steadily increase over time, and then complete their reports. In addition, we also place importance on discussing social issues and learning about society as a whole. Through discussions, each student deepens their thoughts about what social phenomenon they should choose as their theme. I hope that students will use the analytical methods for financial trends using stochastic differential equations in a wide range of fields in the future. I expect them to be active in many fields, including finance, consulting, product development, and crisis management.

School of Social Informatics is a faculty that combines the humanities and sciences, and since there are many students from the humanities, it is not rare for them to struggle with mathematics at first. However, students who enroll with an interest in the theme of understanding social phenomena gradually become aware of the fascination of unraveling various phenomena with mathematics. We hope that all new students who have overcome the rigorous entrance exam preparation will also discover the charm and depth of mathematics.