Identification of random functions from the SFCs defined by the Ogawa integral regarding regular CONSs (Probability Symposium)

Abstract

Let (B_{t})_{tin[0, 1]} be a Brownian motion on a probability space (Omega, mathcal{F}, P). Our concern is whether and how a noncausal type stochastic differential dX_{t}=a(t, omega)dB_{t}+b(t, omega)dt is identified from its stochastic Fourier coefficients (SFCs for short) (e_{n}, dX) := int_{0}^{1}overline{e_{n}(t)}dX_{t} with respect to a CONS (e_{n})_{nin mathbb{N}} of L^{2}([0, 1];mathbb{C}). This problem has been studied by S. Ogawa and H.Uemura (Ogawa (2013)[9], (2014)[10]; Ogawa, Uemura (2014)[12], [13], (2015)[14]). In this note we explain the result we obtained on the problem for the stochastic differentials by the Ogawa integral for regular CONSs. This note is an announcement of the author's full paper on this result.