Abstract
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.
Acknowledgments
The author would like to express his grate gratitude to Professor Tetsuya Kazumi for supervising to draw this paper and to Professor Shigeyoshi Ogawa and Professor Hideaki Uemura for paying kind attention and giving me some advice and comments to this study on several occasions and to Professor Masanori Hino for giving some comments and advice at an opportunity to present this study.
Disclosure statement
No potential conflict of interest was reported by the author(s).