Multiple G-It\^{o} integral in the G-expectation space

TL;DR

This paper introduces the multiple G-Itô integral within the G-expectation space, explores its calculation methods, and extends classical Itô results by establishing a relationship with Hermite polynomials, relevant to mathematical finance.

Contribution

It presents the first definition of multiple G-Itô integrals in the G-expectation space and extends classical Itô-Hermite polynomial relations to this new setting.

Findings

Established the relationship between Hermite polynomials and multiple G-Itô integrals.

Extended classical Itô results to the G-expectation framework.

Provided methods for calculating multiple G-Itô integrals.

Abstract

In this paper, motivated by mathematic finance we introduce the multiple G-It\^{o} integral in the G-expectation space, then investigate how to calculate. We get the the relationship between Hermite polynomials and multiple G-It\^{o} integrals which is a natural extension of the classical result obtained by It\^{o} in 1951.