Discretizing Malliavin calculus

作者：Bender Christian^*; Parczewski Peter

来源：Stochastic Processes and Their Applications, 2018, 128(8): 2489-2537.

DOI：10.1016/j.spa.2017.09.014

• Malliavin calculus
• Strong approximation
• Stochastic integrals
• S-transform
• Chaos decomposition
• Invariance principle

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.

• 出版日期2018-8