Malliavin calculus in abstract Wiener space using infinitesimals |
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Authors: | Horst Osswald |
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Affiliation: | Mathematisches Institut der Universität München, Theresienstr. 39, München D-80333, Germany |
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Abstract: | Using infinitesimals, we develop Malliavin calculus on spaces which result from the classical Wiener space by replacing with any abstract Wiener space .We start from a Brownian motion b on a Loeb probability space Ω with values in the Banach space 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 , where is a ∗finite-dimensional linear space over between the Hilbert space and its ∗-extension .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 of -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). |
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Keywords: | Primary 60H07 Secondary 03H05 |
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