Malliavin calculus in abstract Wiener space using infinitesimals

Abstract

Using infinitesimals, we develop Malliavin calculus on spaces which result from the classical Wiener space C([0, 1], ℝ) by replacing ℝ with any abstract Wiener space (ℍ, B). We start from a Brownian motion b on a Loeb probability space Ω with values in the Banach space B; b is the standard part of a * finite-dimensional Brownian motion B. Then we define iterated Itô integrals as standard parts of internal iterated Itô integrals. The integrator of the internal integrals is B and the values of the integrands are multilinear forms on Fn, where F is a * finite-dimensional linear space over * ℝ between the Hilbert space ℍ and its *-extension * ℍ. In the first part we prove a chaos decomposition theorem for L2-functionals on Ω that are measurable with respect to the σ-algebra generated by b. This result yields a chaos decomposition of L2-functionals with respect to the Wiener measure on the standard space CB of B-valued continuous functions on [0, 1]. In the second part we define the Malliavin derivative and the Skorohod integral as standard parts of internal operators defined on * finite-dimensional spaces. In an application we use the transformation rule for finite-dimensional Euclidean spaces to study time anticipating and non-anticipating shifts of Brownian motion by Bochner integrals (Girsanov transformations). Published by Elsevier Science (USA).