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Discretizing Malliavin calculus

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  • Bender, Christian
  • Parczewski, Peter

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.

Suggested Citation

  • Bender, Christian & Parczewski, Peter, 2018. "Discretizing Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 128(8), pages 2489-2537.
  • Handle: RePEc:eee:spapps:v:128:y:2018:i:8:p:2489-2537
    DOI: 10.1016/j.spa.2017.09.014
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    Cited by:

    1. Privault, N. & Yam, S.C.P. & Zhang, Z., 2019. "Poisson discretizations of Wiener functionals and Malliavin operators with Wasserstein estimates," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3376-3405.

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