Embedding the Abstract Wiener Space in a Probability Space

AbstractIn the first part, of this paper it is pointed out that for certain applications of the stochastic calculus of variations it is useful to replace the classical domain of definition—the Wiener space—with a general probability space in which the Wiener space is embedded. This yields a certain “conditional Malliavin calculus” and is applicable to “signal” and “noise” problems. In a somewhat analogous way, it is pointed out in the second part of the paper that formulating the Itô calculus in a setup of an abstract Wiener space embedded in a general probability space endowed with a filtration has certain useful applications. In particular it enables the formulation and derivation of a dimension-free form of the Girsanov theorem as well as a dimension free form of the representation of Lp-Wiener functionals as Itô integrals