We consider the identification problem of a noncausal Ito process from its stochastic Fourier coefficients with respect to the complete system of trigonometric functions. Here, a noncausal Ito process is the extension of Ito process whose diffusion and drift terms are assumed to be noncausal. If noncausal Ito processes are defined through Skorokhod integral, then authors obtained the affirmative answer in the previous article (Ogawa & Uemura,2014). In this note we generalize our result. We assume both diffusion and drift terms belong to the Watanabe space and identify the noncausal Ito process from its stochastic Fourier coefficients in the framework of Malliavin calculus.
Identification of noncausal finite variation processes from the stochastic Fourier coefficients
设$(B_t)_{t\in[0,\infty)}$是概率空间$(Ω,\mathcal{F},P)$上的布朗运动.我们关心的是，是否以及如何从它的随机傅里叶系数（简称SFC）$(e_n,dX):=\int_{0}^L\bar{e}_n(t)\,dX_t$确定一个非因果型随机微分$dX_t=a(t,ω)\,dB_t+b(t,ω)\,dt$相对于一个$L^2([0,L];\mathbb{C})$的CONS $(e_n)_{n\in\mathbb{N}}$。该问题由Ogawa提出（Stochastics（85）（2），286-294，2013），并已由Ogawa和Uemura研究（Ogawa in Ind J Stat 77-A（1）：30-45，2014，Ind J Stat 80-A：267-279，2018; Ogawa和Uemura in J Theor Probab 27：370-382，2014，Bull Sci Math 138：147-163，2014，RIMS Kôkyoungroku 1952：128-134，2015，J Ind Appl Math 35-1：373-390，2018）。本文给出了当$[0,L]$为有限或无限区间时Ogawa型和Skorokhod型随机微分问题的几个结果。具体地说，我们首先给出了随机函数从随机函数中鉴别出来的条件，并应用该条件得到了肯定…
Optimization of the Mean-Square Approximation Procedures for Iterated Ito Stochastic Integrals of Multiplicities 1 to 5 from the Unified Taylor-Ito Expansion Based on Multiple Fourier-Legendre Series
Exact Calculation of the Mean-Square Error in the Method of Expansion of Iterated Ito Stochastic integrals Based on Generalized Multiple Fourier Series
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
The Proof of Convergence with Probability 1 in the Method of Expansion of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
Implementation of Strong Numerical Methods of Orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito SDEs with Non-Commutative Noise Based on the Unified Taylor-Ito and Taylor-Stratonovich Expansions and Multiple Fourier-Legendre Series
Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals of Multiplicities 1 to 6 from the Taylor-Ito and Taylor-Stratonovich Expansions Using Legendre Polynomials
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
A new proof of the expansion of iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process based on generalized multiple Fourier series and Hermite polynomials
Strong Numerical Methods of Orders 2.0, 2.5, and 3.0 for Ito Stochastic Differential Equations Based on the Unified Stochastic Taylor Expansions and Multiple Fourier-Legendre Series
Explicit One-Step Strong Numerical Methods of Orders 2.0 and 2.5 for Ito Stochastic Differential Equations Based on the Unified Taylor-Ito and Taylor-Stratonovich Expansions
Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures and with Respect to Martingales Based on Generalized Multiple Fourier Series
Four New Forms of the Taylor-Ito and Taylor-Stratonovich Expansions and its Application to the High-Order Strong Numerical Methods for Ito Stochastic Differential Equations
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 Based on Generalized Multiple Fourier Series Converging in the Mean: General Case of Series Summation
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
Expansion of Iterated Stratonovich Stochastic Integrals of Fifth, Sixth, Seventh and Eighth Multiplicities Based on Generalized Multiple Fourier Series
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 4. Combained Approach Based on Generalized Multiple and Iterated Fourier series
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
New Simple Method of Expansion of Iterated Ito Stochastic integrals of Multiplicity 2 Based on Expansion of the Brownian Motion Using Legendre Polynomials and Trigonometric Functions
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
New representations of the Hu-Meyer formulas and series expansion of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
The University of Edinburgh（爱丁堡大学）；The Maxwell Institute for the Mathematical Sciences（麦克斯韦数学科学研究所）；CNRS, Laboratoire de mathématiques de Versailles, UVSQ, Université Paris-Saclay（法国国家科学研究中心，凡尔赛数学实验室，凡尔赛大学，巴黎萨克雷大学）；Monash University（莫纳什大学）；Beijing Institute of Technology（北京理工大学）
