Application of the theory of SDE to real-time analysis of high-frequency data

Research Project

Project/Area Number	18K18718
Research Category	Grant-in-Aid for Challenging Research (Exploratory)
Allocation Type	Multi-year Fund
Review Section	Medium-sized Section 12:Analysis, applied mathematics, and related fields
Research Institution	Tokyo Institute of Technology
Principal Investigator	ninomiya syoiti 東京工業大学, 理学院, 教授 (70313377)
Project Period (FY)	2018-06-29 – 2023-03-31
Project Status	Completed (Fiscal Year 2022)
Budget Amount *help	¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000) Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000) Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000) Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords	SDE / mathematical finance / weak approximation / numerical method / 確率論 / 数理ファイナンス / 確率微分方程式 / 確率数値解析 / 高頻度取引データ / 前方積分 / 高頻度データ / 金融取引 / 確率微分方程式の数値解法 / 非因果的確率積分 / forward integral / 高頻度取引
Outline of Final Research Achievements	Of the objectives of this study (1-4), results were obtained for (1), (3), and (4). Although (2) could not be fully verified due to two reasons: the problem of access to sensitive data of financial institutions in the Corona Disaster and changes in the market environment, we were able to confirm the theoretical results by substituting numerical simulations for a part of them. The following theoretical results were obtained for objectives (1), (2), and (3). [1]It is possible to treat theoretically the case in which the expectations of the higher order basis of the free Lie algebra generated by the iterated integrals do not disappear by using the theory of forward integral. [2]Calculations for the second-order case above and their interpretation in the market. The interpretation above is actually confirmed by numerical simulations.
Academic Significance and Societal Importance of the Research Achievements	高頻度取引市場の時系列データを(拡張された)確率微分方程式(以下SDE)で記述されているものであると見做してその高次反復積分の和への展開の係数として現われる確率変数で市場を調べるものであると一般化できる。この一般化は人工知能の中の所謂深層学習と整合性が高い。ファイナンス理論は市場データからヘッジ戦略を記述するSDEを発見するものと見做すことができるが, 深層学習の深層に相当する部分はこのSDEを記述するベクトル場を時間方向に並べることに相当するからである。この知見により、今後のファイナンス理論の研究に深層学習の理論を取り込む手段の有力な候補が発見された。