On a Stochastic Fourier Coefficient: Case of Noncausal Functions

Abstract

Given a random function \(f(t,\omega )\) and an orthonormal basis \(\{\varphi _n \}\) in \(L^2(0,1),\) we are concerned with the basic question whether the function can be reconstructed from the complete set of its stochastic Fourier coefficients \(\{{\hat{f}}_n(\omega )\}\) which are defined by the following stochastic integral with respect to the Brownian motion \(W.\): \({\hat{f}}_n(\omega ):=\int _0^1 f(t,\omega ) \overline{\varphi _n(t)}{\text{ d}}_*W_t\), where the symbol \(\int {\text{ d}}_*W_t\) stands for the stochastic integral of noncausal type. In an earlier article (Stochastics, doi: 10.1080/17442508.2011.651621, 2012), Ogawa studied the question in the limited framework of homogeneous chaos and gave some affirmative answers when the random functions are causal and square integrable Wiener functionals for which the Ito integral is used for the definition of the stochastic Fourier coefficient. In this note, we aim to extend those results to the more general case where the functions are free from the causality restriction and the Skorokhod integral is employed instead of the Ito integral.