论文标题
从随机傅立叶系数中识别非因果有限变化过程
Identification of noncausal finite variation processes from the stochastic Fourier coefficients
论文作者
论文摘要
令$（b_t）_ {t \ in [0，\ infty）} $为概率空间上的布朗运动$（ω，\ mathcal {f}，p）$。我们担心的是，非因果类型的随机差异$ dx_t = a（t，ω）\，db_t+b（t，t，ω）\，从其随机傅立叶系数（sfcs for Short）$（简短）$（e_n，dx）$（e_n，dx）尊重$ l^2（[[0，l]; \ Mathbb {c}）$的cons $（e_n）_ {n \ in \ mathbb {n}} $。大瓦（Ogawa）提出了这个问题（随机（85）（2），286-294，2013），已由Ogawa和Uemura（Ind J Stat J Stat J Stat 77-A（1）：30-45，20alt in Ind J Stat in Ind J Station 77-A（1）：Ind J Stat 80-a：267-279，2018; Ogawa; Ogawa and j toir; Ogawa和Uemura in Ind Jat Stat 80-a（Ind J Stat J Station 77-a（1）：30-45，2014年3月27日SCI数学138：147-163，2014，RimsKôky-Roku 1952：128-134，2015，J Ind Appl Math 35-1：373-390，2018）。在本文中，当$ [0，l] $是有限的或无限的间隔时，我们就每种随机差异类型的随机差异和Skorokhod类型给出了几个结果。具体而言，我们首先给出了从SFC中识别随机函数的条件，并将其应用以获取问题的肯定答案，并使用随机函数的几个具体派生公式。本文重述了我们的文章“从随机傅立叶系数的非因果有限变化过程的推导公式”（出现，j ind appl Math，2020年），我们在此处引入的结构性概念。
Let $(B_t)_{t\in[0,\infty)}$ be a Brownian motion on a probability space $(Ω,\mathcal{F},P)$. Our concern is whether and how a noncausal type stochastic differential $dX_t=a(t,ω)\,dB_t+b(t,ω)\,dt$ is identified from its stochastic Fourier coefficients (SFCs for short) $(e_n,dX):=\int_{0}^L\bar{e}_n(t)\,dX_t$ with respect to a CONS $(e_n)_{n\in\mathbb{N}}$ of $L^2([0,L];\mathbb{C})$. This problem was proposed by Ogawa (Stochastics (85)(2), 286-294, 2013) and has been studied by Ogawa and Uemura (Ogawa in Ind J Stat 77-A(1):30-45, 2014, Ind J Stat 80-A:267-279, 2018; Ogawa and Uemura in J Theor Probab 27:370-382, 2014, Bull Sci Math 138:147-163, 2014, RIMS Kôkyûroku 1952:128-134, 2015, J Ind Appl Math 35-1:373-390, 2018). In this paper we give several results on the problem for each of stochastic differentials of Ogawa type and Skorokhod type when $[0,L]$ is an finite or infinite interval. Specifically, we first give a condition for a random function to be identified from the SFCs and apply it to obtain affirmative answers to the question with several concrete derivation formulas of the random functions. This paper restates the result given in our article "Derivation formulas of noncausal finite variation processes from the stochastic Fourier coefficients"(to appear, J Ind Appl Math, 2020) by a metamathematical notion of constructiveness we introduce here.
