Novel development on stochastic differential equations and Malliavin calculus

2016 Fiscal Year Final Research Report

• Project/Area Number: 25800054
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Okayama University (2015-2016); Tohoku University (2013-2014)
• Principal Investigator: Kusuoka Seiichiro 岡山大学, 異分野基礎科学研究所, 准教授 (20646814)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: 確率微分方程式 / マリアヴァン解析 / 推移確率密度関数 / 放物型偏微分方程式 / 基本解 / ヘルダー連続性 / ランダム環境 / シュタインの手法

Outline of Final Research Achievements

I studied the continuity in the initial value and the parameter of the density component of the transition probability density function of the solutions to stochastic differential equations with non-regular coefficients, and obtained the Hoelder continuity in the initial value under a very weak assumption on the coefficients. Moreover, I obtained the continuity which is the almost same level as that of the equation without drift term in the parameter of the density component under a weak asummption on the coefficients. The continuity of the transition probability density function is associated with the continuity in the space parameters of the fundamental solutions to second-order parabolic partial differential equations. Besides, I also obtained some results on the recurrence of the Brownian motion in random environments, and on the convergence of the distributions of random variables by Malliavin calculus and Stein's method.

Free Research Field

確率解析