Some Important Properties of Multiple G-Itô Integral in the G-Expectation Space

In the G-expectation space, we propose the multiple Ito integral, which is driven by multi-dimensional G-Brownian motion. We prove the recursive relationship of multiple G-Ito integrals by G-Ito formula and mathematical induction, and we obtain some computational formulas for a kind of multiple G-Ito integrals.


Introduction
With the rapid development of financial markets, traditional linear expectations cannot explain its uncertainty sometimes. In 2007, Peng [1] introduced a new sublinear expectation-G-expectation, and he introduced G-normal distribution and G-Brownian motion under the Gexpectation framework. In 2008, Peng [2] proved the law of large numbers and the central limit theorem under the sublinear expectation, and he defined the Itô integral about G-Brownian motion. Later, Peng [3] obtained the G-Itô formula and proved the existence and uniqueness of solution for the stochastic differential equations driven by G-Brownian motion (G-SDEs for short) and the backward stochastic differential equations driven by G-Brownian motion (G-BSDEs for short).
Since then, G-expectation space and the applications of G-Itô integral have been extensively studied by many researchers. In 2014, Hu, Ji, Peng and Song [4] studied the comparison theorem, nonlinear Feynman-Kac formula and Girsanov transformation of G-BSDE. In 2016, Hu, Wang and Zheng [5] obtained the Ito-Krylov formula under the G-expectation framework. Then they proved the reflection principle of G-Brownian motion, and they got the reflection principle of G-Brownian motion by Krylov's estimate in [6]. [7] studied rough path properties of stochastic integrals of Itô's type and Stratonovich's type with respect to G-Brownian motion. Then, Hu, Ji and Liu [8] studied the strong Markov property for G-SDEs in 2017. Wu [9] introduced the multiple Itô integrals driven by one-dimensional Brownian motion in G-expectation space. He also obtained the relationship between Hermite polynomials and multiple G-Itô integral. In 2012, Yin [10] introduced the Stratonovich integral with respect to G-Brownian motion, and she also researched the properties of G-Stratonovich integrals. In 2014, Sun [11] studied multiple stochastic integrals under one-dimensional G-Brownian motion and developed the L p estimation of maximal inequalities for n iterated integrals by the property of Hermite polynomials. The more contents about multiple random integrals can be found in the literature [12].
A nature question is how to define and calculate the multiple G-Itô integral of multi-dimensional G-Brownian motion. This problem will be solved in this paper. We define multiple Itô integrals driven by multi-dimensional G-Brownian motion under G-expectation space. And we prove the recursive relationship between multiple G-Itô integrals strictly by using G-Itô formula and mathematical induction method. Then we obtain some important formulas for calculating multiple G-Itô integrals and make some preparations for further study on scientific calculation of G-SDEs.
The remainder of this paper is organized as follows: In Section 2, we introduce some concepts and lemmas such as G-Brownian motion, G-Itô formula and so on. In Section 3, we define multiple Itô integrals driven by multi-dimensional G-Brownian motion, and prove the recursive relationship between multiple G-Itô integrals. Then we give some important formulas for calculating multiple G-Itô integrals. Finally, several concluding remarks are given in Section 4.

Preliminaries and Notation
In this section, we will give some basic theories about G-Brownian motion and multi-indices. Some more details can be found in literatures [1][2][3] and [12]. Let Ω be a given set, and let H be a linear space of real valued functions defined on Ω. For each c we suppose that c ∈ H, and |X| ∈ H if X ∈ H. The space H can be considered as the space of random variables.

G-Brownian Motion and G-Itô Formula
Firstly, we introduce some notations about G-Brownian motion.
Let G(·) : S(d) → R be a given monotonic and sublinear function. We denote by S(d) the collection of all d × d symmetric matrices.There exists a bounded, convex and closed subset In the following sections, we denote by We recall some important notions about G-Itô formula, product rule and so on (see [3]).

Definition 2. [3]
We denote the set of simple process , we define the Itô integral of G-Brownian motion is as follows: As µ(π N t ) → 0, the first term of the right side converges to 2 t 0 B s dB s in L 2 G (Ω). The second term must be convergent. We denote its limit by B t , i.e., By the above construction, ( B t ) t≥0 is an increasing process with B 0 = 0. We call it the quadratic variation process of the G-Brownian motion B. We denote B t be a m-dimensional G-Brownina motion. Let Φ ∈ C 2 (R n ) be bounded with bounded derivatives and ∂ 2 x i x j Φ are uniformly Lipschitz. Let s ∈ [0, T ] be fixed and let X i t be the i (i = 1, . . . , d)-th component of X t = (X 1 t , . . . , X d t ) satisfying where a i be the i-th of a = (a 1 , . . . , a d ) , η i,j and σ i,j is the lines i-th and j-th of η = (η i,j ) d×m and σ = (σ i,j ) d×m , and they are bounded process on M 2 G (0, T ). For t, s ≥ 0, then we have Lemma 2. [1,3] In G-expectation space, the following product rule is established: where v is the multi-index of length zero.

Multi-Indices
We write n(α) for the number of components of a multi-index α that are equal to 0 and s(α) for the number of components of a multi-index α that are equal to −1. Moreover, we write α− for the multi-index obtained by deleting the first component of α and −α for the multi-index obtained by deleting the first component of α. α − (j) for the multiindex obtained by deleting the last component of α = (j 1 , j 2 , . . . , j k , j) so we can get the multi-index (j 1 , j 2 , . . . , j k ). Additionally, given two multi-indices α 1 = (j 1 , j 2 , . . . , j k ) and α 2 = (i 1 , i 2 , . . . , i l ), we introduce the concatenation operator * on M defined by

Main Results
In this section, by a component j ∈ {1, 2, . . . , m} of a multi-index we will denote in a multiple stochastic integral the integration with respect to the j-th Wiener process. A component j = 0 will denote integration with respect to time. Lastly, a component j ∈ {−m, −(m − 1), . . . , −1} refer to an integration with respect to quadratic variation process. We shall define three sets of adapted right continuous stochastic processes g = {g(t, ω), t ∈ [0, T ]} with left hand limits.
The proof is completed.

Concluding Remarks and Future Work
In this work, we define G-Itô integral driven by multi-dimensional G-Brownian motion in G-expectation space. And we use G-Itô formula and mathematical induction to obtain a kind of multiple G-Itô integrals. As discussed in Section 1, this effort focuses on multiple G-Itô integrals driven by multi-dimensional G-Brownian motion rather than one-dimensional G-Brownian motion. Our future efforts will focus on introducing the properties of Stratonovich integral driven by multi-dimensional G-Brownian motion, and exploring the relationship between Stratonovich integral and G-Itô integral under the G-expectation framework.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.