Discretizing Malliavin calculus

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:128

Article

Bender, Christian¹  Parczewski, Peter² 

[1] Saarland Univ, Dept Math, POB 151150, D-66041 Saarbrucken, Germany

[2] Univ Mannheim, Inst Math, A5,6, D-68131 Mannheim, Germany

关键词: Malliavin calculus;    Strong approximation;    Stochastic integrals;    S-transform;    Chaos decomposition;    Invariance principle;   

DOI  :  10.1016/j.spa.2017.09.014

来源: Elsevier

【 摘 要 】

Suppose B is a Brownian motion and B-n 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 L-2-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 (X-n) of random variables which admit a chaos decomposition in terms of discrete multiple Wiener integrals with respect to B-n, we derive necessary and sufficient conditions for strong L-2-convergence to a sigma (B)-measurable random variable X via convergence of the discrete chaos coefficients of X-n to the continuous chaos coefficients. (C) 2017 Elsevier B.V. All rights reserved.

【 授权许可】

Free