Discretizing Malliavin calculus

TL;DR

This paper establishes necessary and sufficient conditions for the convergence of discretized Malliavin calculus operators and chaos decompositions, bridging continuous and discrete stochastic analysis.

Contribution

It provides rigorous $L^2$-convergence criteria for discretized Malliavin derivatives, Skorokhod integrals, and chaos coefficients, extending the theory to binary noise.

Findings

Conditions for weak and strong $L^2$-convergence of discretized operators

Convergence criteria for chaos coefficients in discrete approximations

Support for analogies between Wiener and Bernoulli Malliavin calculus

Abstract

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 of $X$. In the special case of binary noise, our results support the known formal analogies between Malliavin calculus on the Wiener space and Malliavin calculus on the Bernoulli space by rigorous $L^2$-convergence results.