Structure-preserving methods for stochastic differential equations

2020 Fiscal Year Final Research Report

• Project/Area Number: 17K18736
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Research Field: Analysis, Applied mathematics, and related fields
• Research Institution: Osaka University
• Principal Investigator: Furihata Daisuke 大阪大学, サイバーメディアセンター, 教授 (80242014)
• Project Period (FY): 2017-06-30 – 2021-03-31
• Keywords: 確率微分方程式 / 構造保存数値解法 / 離散微分幾何 / 非線形差分

Outline of Final Research Achievements

Focusing on the particular structure of the linear stochastic differential equation, which appears when based on the Ito integral, not the Stratonovich integral, we have developed a new method to conserve the quadratic structure by introducing the discrete square operator corresponding to the development operator of the stochastic differential equation. Via this method, we can construct new structure-preserving numerical methods.
Furthermore, by proposing a discretization of differential geometry appropriate for the structure-preserving methods, it is possible to make the target region space multidimensional. We also proposed new nonlinear difference operators as the discretization of differential operators, and we show that we can obtain numerically stable solutions without the calculation of the travelling-wave direction for wave-type equations.

Free Research Field

数値解析学

Academic Significance and Societal Importance of the Research Achievements

確率微分方程式は数学理論上の問題のみならず，気象分野，金融問題や伝染病感染予測等で現れる社会的にも重要な数学的ツールである．その基本たる伊藤積分と時間対称型のStratonovich積分との間に本質的な違いは無いものと認識されているが伊藤積分にのみ表出する数学構造がある．この事実を利用し，発展作用素に対応する離散的な平方作用素を導入して二次形式構造を保存する方法論を考案，構造保存数値解法を構成することで確率微分方程式の近似数値解計算の向上に本質的に寄与する成果を得た．