Numerical analysis and density functions path-dependent/non-colliding stochastic differential equations with non-bounded coefficients

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K14552
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Okayama University
• Principal Investigator: Taguchi Dai 岡山大学, 異分野基礎科学研究所, 准教授 (70804657)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 確率微分方程式 / Euler-Maruyama近似 / Multilevel Monte Carlo法 / Avikainenの不等式 / 非衝突確率過程 / 後退確率Volterra積分方程式 / Polynomial diffusions / CIR過程

Outline of Final Research Achievements

In this research, the following seven results were obtained.
(1) Density estimates in the case of unbounded coefficients and its application to numerical analysis (2) Numerical analysis of irregular functionals of stochastic differential equations and its application to the Multilevel Monte Carlo method. (3) Discrete approximation for backward stochastic Volterra integral equations. (4) Discrete approximation for Polynomial diffusions. (5) Numerical analysis of stochastic differential equations with irregular diffusion coefficient.(6) Numerical analysis of stochastic differential equations driven by α-stable process with irregular diffusion coefficient. (7) Numerical analysis for Cox-Ingersoll-Ross (CIR) process. The results of (1)-(5) have already been published, and the result of (6)(7)(8) is in preparation for submission.

Free Research Field

確率数値解析

Academic Significance and Societal Importance of the Research Achievements

本研究の成果により、これまで数値解析が難しかった、もしくは精度が保証されていなかった確率過程に対して、精度保証付きの数値計算を行うことができるようになった。特に、多次元の確率過程に対する数値解析手法を導入し、強収束の誤差評価を精密に与えた。また、確率密度関数の解析を行い、Avikainenの不等式を多次元の確率過程の場合にまで拡張することによって、通常のモンテカルロ法よりも効率的に数値計算が可能となる　Multilevel Monte Carlo methodを適用できるようになり、計算量が大幅に改善できるようになった。