Stein’s Method, Malliavin Calculus,Relations and Applications

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Mirzaei, Keivan | 2021

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 54276 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Zohuri Zangeneh, Bijan; Tahmasebi, Mahdieh
Abstract:
In this thesis, after introducing some preliminary concepts, Stein’s method and Malliavin calculus is discussed. Our approach for introducing Malliavin calculus is based on isonormal Gaussian processes, which is more general and natural than Gaussian noises. After dealing with isonormal Gaussian processes, Wiener chaos and important operators of Malliavin calculus, namely differential, divergence and Ornstein-Uhlenbeck operators are discussed and some relation between them is studied. At last, some connections between Stein’s method and Malliavin calculus is developed. As a result, some exact asymptotics for central limit theorems on Gaussian functionals are obtained. These results are used, for instance, for Toeplitz quadratic functionals, Brownian sheet and fractional Brownian motion
Keywords:
Differential Operators ; Divergence Operator ; Central Limit Theorem ; Stochastic Integral ; Fractional Brownian Motion ; Gaussian process ; Wiener-Ito Choas Expansion ; Hermite Polynomials ; Hilbert Spaces Tensor Product ; Brownian Sheet ; Stein’s Method ; Ornstein-Uhlenbeck Operator