Discretizing Malliavin calculus

Christian Bender and Peter Parczewski

Stochastic Processes and their Applications, 2018, vol. 128, issue 8, 2489-2537

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.

Keywords: Malliavin calculus; Strong approximation; Stochastic integrals; S-transform; Chaos decomposition; Invariance principle (search for similar items in EconPapers)
Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:128:y:2018:i:8:p:2489-2537

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DOI: 10.1016/j.spa.2017.09.014

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