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統計数学セミナー
過去の記録 ～04/19｜次回の予定｜今後の予定 04/20～
| 担当者 | 吉田朋広、荻原哲平、小池祐太 |
|---|---|
| セミナーURL | http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/ |
| 目的 | 確率統計学およびその関連領域に関する研究発表, 研究紹介を行う. |
2016年08月09日(火)
13:00-16:30 数理科学研究科棟(駒場) 117号室
本ワークショップ・レクチャーは統計数学セミナー共催であり，JST CRESTによってサポートされています.
David Nualart 氏 (Kansas University)
Malliavin calculus and normal approximations
http://www2.ms.u-tokyo.ac.jp/probstat/?page_id=180
本ワークショップ・レクチャーは統計数学セミナー共催であり，JST CRESTによってサポートされています.
David Nualart 氏 (Kansas University)
Malliavin calculus and normal approximations
[ 講演概要 ]
The purpose of these lectures is to introduce some recent results on the application of Malliavin calculus combined with Stein's method to normal approximation. The Malliavin calculus is a differential calculus on the Wiener space. First, we will present some elements of Malliavin calculus, defining the basic differential operators: the derivative, its adjoint called the divergence operator and the generator of the Ornstein-Uhlenbeck semigroup. The behavior of these operators on the Wiener chaos expansion will be discussed. Then, we will introduce the Stein's method for normal approximation, which leads to general bounds for the Kolmogorov and total variation distances between the law of a Brownian functional and the standard normal distribution. In this context, the integration by parts formula of Malliavin calculus will allow us to express these bounds in terms of the Malliavin operators. We will present the application of this methodology to derive the Fourth Moment Theorem for a sequence of multiple stochastic integrals, and we will discuss some results on the uniform convergence of densities obtained using Malliavin calculus techniques. Finally, examples of functionals of Gaussian processes, such as the fractional Brownian motion, will be discussed.
[ 参考URL ]The purpose of these lectures is to introduce some recent results on the application of Malliavin calculus combined with Stein's method to normal approximation. The Malliavin calculus is a differential calculus on the Wiener space. First, we will present some elements of Malliavin calculus, defining the basic differential operators: the derivative, its adjoint called the divergence operator and the generator of the Ornstein-Uhlenbeck semigroup. The behavior of these operators on the Wiener chaos expansion will be discussed. Then, we will introduce the Stein's method for normal approximation, which leads to general bounds for the Kolmogorov and total variation distances between the law of a Brownian functional and the standard normal distribution. In this context, the integration by parts formula of Malliavin calculus will allow us to express these bounds in terms of the Malliavin operators. We will present the application of this methodology to derive the Fourth Moment Theorem for a sequence of multiple stochastic integrals, and we will discuss some results on the uniform convergence of densities obtained using Malliavin calculus techniques. Finally, examples of functionals of Gaussian processes, such as the fractional Brownian motion, will be discussed.
http://www2.ms.u-tokyo.ac.jp/probstat/?page_id=180
