Title: Direct Inversion Formulas for the Natural SFT

Author(s): Shigeyoshi Ogawa
Issue: Volume 80 Series A Part 2 Year 2018
Pages: 267 -- 279
The stochastic Fourier transform, or SFT for short, is an application that transforms a square integrable random function $f(t, \omega)$ to a random function defined by the following series; ${\cal T}_{\epsilon, \phi} f(t, \omega) := \sum_n \epsilon_n f_n(\omega) \phi_n(t)$ where $\{\epsilon_n\}$ is an ${\cal l}^2$ -sequence such that $\epsilon_n \neq 0$, $\forall n$ and $\hat{f}_n$ is the SFC (short for “stochastic Fourier coefficient”) defined by $\hat{f}_n(\omega) = \int_0^1 f (t, \omega) \overline{\phi_n(t)} dW_t$, 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 $({\it Ogawa,S.,``\mbox{A direct inversion formula for SFT}”, Sankhya}$-${\it A 77}$-${\it 1 (2015)}).$
AMS (2000) subject classification. Primary: 60H05, 60H07; Secondary: 42A61.
Keywords and phrases: Brownian motion, Stochastic integrals, Fourier series.