Discretization Error of Stochastic Iterated Integrals

TL;DR

This paper investigates the weak convergence rate of discretization errors in stochastic iterated integrals involving jump processes, providing theoretical insights and applications to exponential processes.

Contribution

It establishes the convergence rate of discretization errors for stochastic iterated integrals with jumps, extending previous results to more general semimartingales.

Findings

Convergence rate is 1/n for semimartingales with continuous martingale parts.

Derived asymptotic behavior of normalized discretization errors.

Applied results to discretization of Doléans-Dade exponential.

Abstract

In this paper, the weak convergence about the discretization error of stochastic iterated integrals in the Skorohod sense are studied, while the integrands and integrators of iterated integrals are supposed to be semimartingales with jumps. We explored the rate of convergence of its approximation based on the asymptotic behaviors of the associated normalized error and obtained that the rate is $1/n$ when the driving process is semimartingale with a nonvanishing continuous martingale component. As an application, we also studied the discretization of the Dol\'eans-Dade exponential.