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

2022 Fiscal Year Final Research Report

• 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
• Keywords: SDE / mathematical finance / weak approximation / numerical method

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.

Free Research Field

確率論, 数理ファイナンス, 確率数値解析

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。