Identification of a noncausal Itô process from the stochastic Fourier coefficients

Let Xt be a noncausal Itô process of Skorokhod type driven by the Brownian motion W., that is, a stochastic process of the form dXt=b(t,ω)dt+a(t,ω)dWt where the term a(⋅)dWt is understood as Skorokhod integral. For such an Itô process Xt we consider the Fourier coefficient Fn(dX) of the differential dXt by Fn(dX)=∫01en(t)¯dXt, en(t)=exp(2π−1nt) (n∈Z) and we are concerned with the elementary question: whether we can identify the two parameters a(⋅,ω), b(⋅,ω) from the complete set of the stochastic Fourier coefficients {Fn(dX),n∈Z}. In this note we study this problem in a framework of noncausal calculus, as we did in the previous articles (Ogawa, 2013; Ogawa and Uemura, in press), and we give an affirmative answer with a concrete scheme for the reconstruction of the parameters a(⋅,ω), b(t,ω). Our result will give another light to the theoretical background of the method of Fourier series for the volatility estimation proposed by P. Malliavin et al. (Malliavin and Mancino, 2002; Malliavin and Thalmaier, 2009).