Discretizing Malliavin calculus

Bender, Christian ; Parczewski, Peter

DOI: https://doi.org/10.1016/j.spa.2017.09.014
URL: https://www.sciencedirect.com/science/article/abs/...
Additional URL: https://arxiv.org/abs/1602.08858
Document Type: Article
Year of publication: 2018
The title of a journal, publication series: Stochastic Processes and Their Applications
Volume: 128
Issue number: 8
Page range: 2489 - 2537
Place of publication: Amsterdam [u.a.]
Publishing house: Elsevier
ISSN: 0304-4149
Publication language: German
Institution: School of Business Informatics and Mathematics > Wirtschaftsmathematik II (Neuenkirch 2013-)
Subject: 510 Mathematics
Keywords (English): Malliavin calculus , Strong approximation , Stochastic integrals , S-transform , Chaos decomposition , Invariance principle
Abstract: Suppose B is a Brownian motion and Bn is an approximating sequence of rescaled random walks on the same probability space converging to B pointwise in probability. We provide necessary and sufficient conditions for weak and strong L2-convergence of a discretized Malliavin derivative, a discrete Skorokhod integral, and discrete analogues of the Clark–Ocone derivative to their continuous counterparts. Moreover, given a sequence (Xn) of random variables which admit a chaos decomposition in terms of discrete multiple Wiener integrals with respect to Bn, we derive necessary and sufficient conditions for strong L2-convergence to a σ(B)-measurable random variable X via convergence of the discrete chaos coefficients of Xn to the continuous chaos coefficients.

Dieser Eintrag ist Teil der Universitätsbibliographie.

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