On the stochastic differentiability of noncausal processes with respect to the process with quadratic variation

Let be a stochastic process with quadratic variation on a probability space and a dense subset of , where is regarded as the infinite interval when . First, we introduce the -module of V-differentiable noncausal processes on Q and V-derivative operator defined on , which enjoys the modularity: for any and . Second, we show that the class forms an -module, where stands for the quadratic variation on Q. As a result, we have the isometry: for any , where stands for the quadratic covariation on Q. Finally, we present universal properties and examples of the stochastic integral I with . This result is essentially used for solving the identification problem from the stochastic Fourier coefficients.