由随机傅里叶系数识别非因果有限变分过程
Identification of noncausal finite variation processes from the stochastic Fourier coefficients
摘要：
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的答案，给出了随机函数的几个具体的推导公式。本文通过引入元数学的构造性概念，重申了我们在“从随机傅里叶系数推导非因果有限变分过程的公式“（即将出版，J Ind Appl Math，2020）中给出的结果.
英文摘要：
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.
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