Multiple G-Stratonovich Integral Driven by G-Brownian Motion

In this paper, we propose the multiple Stratonovich integral driven by G-Brownian motion under the G-expectation framework. Then based on G-Itô formula, we obtain the relationship between Hermite polynomials and multiple G-Stratonovich integrals by using mathematical induction method.


Introduction
With the rapid development of the internet, computer science and data information technology, we are facing a real world with more and more dynamic characteristics, often dealing with a large number of high-dimensional random data, and the uncertainty is becoming more and more large. The Choquet expectation theory cannot satisfy the dynamic economic model in the risk study, such as financial risk with highly dynamic and complex characteristics. By introducing a backward stochastic differential equation (BSDE) in typical probability space, in 1997, Peng [1] constructs a new class of nonlinear expectations which are uniquely determined by the generating function g of BSDE, which is named g-expectation. In a sense, the discovery establishes the theoretical basis of dynamic nonlinear mathematical expectation. With more and more scholars studying, g-expectation has become a powerful tool for studying recursive utility theory and financial risk measurement [2] [3] [4]. The concept of g-expectation can be applied to handle a set { } P θ θ∈Θ of uncertain probabilities by reference probability P. However, especially for ( ) 0 P A = and ( ) 0 a consistent risk measure, the theory has an important application in financial theory [6] [7] [8] [9]. In G-expectation theory, G-normal distribution theory is a sublinear expectation defined by Peng in the space of global continuous orbit.
Next, there are concepts, which are introduced, such as a new stochastic process called G-Brownian motion, G-Itô integral and so on. Subsequently, the law of large numbers and central limit theorems under G-expectation are also proved [10]. Now based on the multiple G-Itô integral, scholars get the relationship between Hermite polynomials and multiple G-Itô integrals. Stratonovish [11] introduced the Brown movement. The problems related to the Stratonovish integral are not easy to solve. In 2012, Yin [12] introduced one weight G-Stratonovish integral of Brownian motion.
In this paper, according to the definition of Stratonovish integral of Brownian motion in G-expectation space, we not only introduce the multiple G-Stratonovish integral of Brownian motion but also obtain the relationship between Hermite polynomials and multiple G-Stratonovish integrals.
The structure of this paper is as follows: in Section 2, we first introduce the basic theoretical framework of nonlinear expectation related to the main concepts. In Section 3, two related theorems which are the relationships between Hermite polynomials and multiple G-Stratonovish integrals are given by mathematical induction for the G-Stratonovish integral of Brownian motion.

Preliminaries and Notation
Let Ω be a given set and let  be a linear space of real valued functions defined on Ω . In this paper, we suppose that  satisfies c ∈  for each constant c and X ∈  . The space  can be considered as the space of random variables. Peng [13] gave the nonlinear G-mathematical expectation and G-normal distribution as follows.
Definition 1 [13] We define a functional sublinear expectation : → We call a sublinear expectation space, which is the triple ( ) , , Ω   . If 1) and 2) are satisfied,  is called a nonlinear expectation and the triple ( ) where X is an independent copy of X. Now we give the definition of G-Brownian motion, G-quadratic variation Journal of Applied Mathematics and Physics process, and multi-dimensional G-Itô formula.
is called a G-Brownian motion if the following properties are satisfied: is called the quadratic variation process of a B . Definition 5 [13] Let  [13] In the G-expectation space, the following product rule is established: The definition of G-Stratonovich integral for G-Brownnian motion is as below.
Definition 6 [13] Let , t t X Y is G-Itô process, then the G-Stratonovich integral of t X is

Main Result
In this section, in a multi-index α the components that equal 0 refer to an integration with respect to time; the components that equal to an integration with respect to Stratonovich integral. We shall denote by v  and 0  the sets of functions : There is a recursive relationship for multiple Stratonovich integrals analogous to that for multiple Itô integrals when the integrand is identically equal to 1. In order to state it succinctly we shall use the abbreviation [ ] We can derive The proof is completed.
The next Theorem, gives a clear indication of the same structure offered by multiple Stratonovich integrals when compared with its counterpart for multiple G-Itô integrals. Similarly, we will give the proof process. In fact, we only need to prove that Taking integral about the above equation, and combined with formula (18), the proof is completed.