Malliavin calculus in abstract Wiener space using infinitesimals
The author constructs a Malliavin calculus on \(C([0,1],{\mathbb B})\), where \({\mathbb B}\) is an abstract Wiener space with Cameron-Martin space \({\mathbb H}\), in a non-standard framework. Here, \([0,1]\) is replaced by \(T=\{1,\ldots ,H\}\) where \(H\) is a hyperfinite integer, and \({\mathbb H}\) is replaced with a hyperfinite dimensional space \({\mathbb F}\) with dimension \(\omega\). The Lebesgue measure on \([0,1]\) is replaced with the Loeb measure over the uniform probability on \(T\), and the basic probability space is now \({\mathbb F}^H\), endowed with the Loeb measure over the \(\omega \cdot H\)-dimensional standard Gaussian distribution of variance \(1/H\). In this framework a construction of Brownian motion in abstract Wiener spaces, iterated Itô integrals, chaos decompositions, Malliavin derivative, and Skorokhod integrals, is provided. The absolute continuity of certain anticipating shifts (called adequate shifts) of \({\mathbb B}\)-valued Brownian motion is studied, and as an application a Girsanov theorem for adapted shifts of Brownian motion by Bochner integrals is obtained.
scientific article; zbMATH DE number 1915629