期刊名称：Sankhya. Series A, mathematical statistics and probability
印刷版ISSN：0976-836X
电子版ISSN：0976-8378
出版年度：2018
卷号：80
期号：2
页码：267-279
DOI：10.1007/s13171-018-0128-8
语种：English
出版社：Indian Statistical Institute
摘要：The stochastic Fourier transform, or SFT for short, is an application that transforms a square integrable random function f ( t , ω ) to a random function defined by the following series; \({\mathcal T}_{\epsilon , \varphi }f(t,\o ):= {\sum }_{n} \epsilon _{n} \hat {f}_{n}(\o )\varphi _{n}(t)\) where { 𝜖 n } is an ℓ 2-sequence such that 𝜖 n ≠ 0, ∀ n and \(\hat {f}_{n}\) is the SFC (short for “stochastic Fourier coefficient”) defined by \(\hat {f}_{n}(\o )={{\int }_{0}^{1}} f(t,\o )\overline {\varphi _{n}(t)}dW_{t}\) , a stochastic integral with respect to Brownian motion W t . We have been concerned with the question of invertibility of the SFT and shown affirmative answers with concrete schemes for the inversion. In the present note we aim to study the case of a special SFT called “natural SFT” and show some of its basic properties. This is a follow-up of the preceding article ( Ogawa,S.,“A direct inversion formula for SFT”, Sankhya-A 77-1 (2015) ).
关键词：Brownian motion ; Stochastic integrals ; Fourier series
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