Higher-order error estimates of the discrete-time Clark--Ocone formula

TL;DR

This paper studies the convergence rates of higher-order errors in the discrete-time Clark--Ocone formula, extending previous work to include more general Wiener functionals with various differentiability indices.

Contribution

It introduces estimates for higher-order errors and arbitrary differentiability indices in the discrete-time Clark--Ocone formula, broadening the scope of convergence analysis.

Findings

Derived higher-order error estimates for Wiener functionals.

Extended convergence analysis to arbitrary differentiability indices.

Enhanced understanding of error behavior in discrete-time stochastic calculus.

Abstract

In this article, we investigate the convergence rate of the discrete-time Clark--Ocone formula provided by Akahori--Amaba--Okuma [1]. In that paper, they mainly focus on the $L_{2}$-convergence rate of the first-order error estimate related to the tracking error of the delta hedge in mathematical finance. Here, as two extensions, we estimate "the higher order error" for Wiener functionals with an integrability index $2$ and "an arbitrary differentiability index."