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

摘要

Let be a real Brownian motion on a probability space Our concern is whether and how a noncausal type stochastic differential determined from its stochastic Fourier coefficients (SFCs for short) with respect to a CONS of This problem was proposed by Ogawa (Stochastics and has been studied by Ogawa and Uemura (Ogawa in Ind J Stat 77-A(1):3045, 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 Kokyuroku 1952:128134, 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.