Derivation formulas of noncausal finite variation processes from the stochastic Fourier coefficients

Abstract

Let $$(B_t)_{t\in [0,\infty )}$$ be a real Brownian motion on a probability space $$(\varOmega ,{\mathcal {F}},P)$$. Our concern is whether and how a noncausal type stochastic differential $$dX_t=a(t,\omega )\,dB_t+b(t,\omega )\,dt$$ is determined from its stochastic Fourier coefficients (SFCs for short) $$(e_n,dX):$$$$=\int _{0}^L\overline{e_n(t)}\,dX_t$$ with respect to a CONS $$(e_n)_{n\in {\mathbb {N}}}$$ of $$L^2([0,L];{\mathbb {C}})$$. This problem was proposed by Ogawa (Stochastics (85)(2), 286–294, 2013) and has been studied by Ogawa and Uemura (Ogawa in Ind J Stat 77-A(1):30–45, 2014, Ind J Stat 80-A:267–279, 2018; Ogawa and Uemura in J Theor Probab 27:370–382, 2014, Bull Sci Math 138:147–163, 2014, RIMS K$${\hat{\mathrm{o}}}$$ky$${\hat{\mathrm{u}}}$$roku 1952:128–134, 2015, J Ind Appl Math 35-1:373–390, 2018). In this paper we give several results on the problem for each of stochastic differentials of Ogawa type and Skorokhod type when [0, L] is a finite or an infinite interval. Specifically, we first give a condition for a random function to be determined from the SFCs and apply it to obtain affirmative answers to the question with several concrete derivation formulas of the random functions.