Multiple G-Stratonovich integral in G-expectation space

Shaojin Fei

doi:10.1515/math-2025-0195

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Multiple G-Stratonovich integral in G-expectation space

Shaojin Fei

Published/Copyright: September 25, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

The Stratonovich integral represents a fundamental concept in stochastic calculus. In this article, we first propose a novel approach inspired by the multidimensional G-Itô formula, establishing a multiple G-Stratonovich integral within the G-expectation space. We then establish a relationship between Hermite polynomials and multiple G-Stratonovich integrals, which extends the findings reported in work (Li et al., Multiple G-Stratonovich integral driven by G-Brownian motion, J. Appl. Math. Phys. 6 (2018), no. 11, 2295–2301, DOI: https://doi.org/10.4236/jamp.2018.611190).

Keywords: G-Stratonovich integral; G-expectation; hermite polynomials; Itô formula; G-Brownian motion

MSC 2010: 60G20; 60H05

1 Introduction

Let B c = { B t c , t ≥ 0 } be a classical Brownian motion. A multiple stochastic integral with respect to B c was defined in [1], which constitutes a linear combination of iterated integrals. However, it should be noted that a more general scenario has its origins in the Itô integral [2]. This gives rise to the fact that the theories and applications of multiple ltooo stochastic integrals are rather rich. For example, Engel [3] laid out the background and framework for multiple integrals; Cheridito et al. [4] explored applications in finance; and Soner and Touzi [5] addressed applications in stochastic target problems.

To handle risk measures and stochastic volatility in financial problems, a new concept of sublinear expectation, known as G-expectation, was introduced in [6,7]. Subsequently, the G-normal distribution was defined within the framework of the G-expectation space. On the basis of this, Peng also defined a new canonical process B t ( ω ) = ω t , t > 0 , ω ∈ Ω as G-Brownian motion and constructed the G-Itô integral with respect to G-Brownian motion. Following this, Peng and Zhang [8] introduced the G-Itô process, providing a solid theoretical foundation for establishing the G-Stratonovich integral. In 2012, Yin [9] investigated a weighted G-Stratonovich integral with respect to G-Brownian motion. On the basis of the multiple G-Itô integral [10] Li et al. [11] constructed the multiple G-Stratonovich integral driven by G-Brownian motion. They also considered a special case where the integrand is 1, in which there exists a recursive relationship between Hermite polynomials and multiple Stratonovich integrals.

The G-expectation offers an early-stage risk measurement that is consistent with traditional risk measurements. In modern research, the Stratonovich approach is widely utilized in the analysis and modeling of various stochastic processes. For instance, in financial mathematics, stochastic differential equations are employed to describe the fluctuations of asset prices, and they can determine stock prices with precision. This holds significant implications for applying theoretical research to practical issues.

The objective of this article is to study the multiple G-Stratonovich integrals with symmetric functions in L 2 ( ( [ 0 , T ] ) n ) under the G-expectation. Next, we establish the relationship between Hermite polynomials and multiple G-Stratonovich integrals. The article is structured as follows: in Section 2, some concepts and notations related to the G-expectation space are reviewed (see, e.g., [1,2,4] and so forth). In Section 3, we then construct the multiple G-Stratonovich integrals with respect to G-Brownian motion. In Section 4, we establish a relation between Hermite polynomials and multiple G-Stratonovich integrals.

2 Preliminaries

In this section, from [7,12], we review some notions and results of G-expectation spaces. For more details, we can see [13–15] and the references therein.

Let Ω be a nonempty set and ℋ be a linear space of real valued functions defined on Ω . We assume that ℋ satisfies c ∈ ℋ for constant c and ∣ X ∣ ∈ ℋ if X ∈ ℋ . The space ℋ can be looked on as the space of random variables.

Let

(2.1) G ( α ) = 1 2 ( σ ¯ 2 α + − σ ̲ 2 α − )

be a real valued function defined on ℜ with 0 ≤ σ ̲ ≤ σ ¯ < ∞ . Then G ( ⋅ ) is a monotonic and sublinear function.

Definition 2.1

(See [7]). A functional sub-linear expectation is defined by E G : ℋ → ℜ , for any X , Y ∈ ℋ , satisfying

Monotonicity: E G [ X ] ≥ E G [ Y ] if X ≥ Y .
Constant preserving: E G [ c ] = c for c ∈ ℜ .
Sub-additivity: E G [ X + Y ] ≤ E G [ X ] + E G [ Y ] , X , Y ∈ ℋ .
Positive homogeneity: E G [ λ X ] = λ E G [ X ] for λ ≥ 0 .

The triple ( Ω , ℋ , E G ) is called a sublinear expectation space. In particular, if (1) and (2) are satisfied, E G is called a nonlinear expectation and the triple ( Ω , ℋ , E G ) is called a nonlinear expectation space.

Definition 2.2

(see [4]). We call a d -dimensional random vector X = ( X 1 , … , X d ) T on a sublinear expectation space ( Ω , ℋ , E G ) is G-normal distributed, denoted by X ∼ N ( 0 , [ σ ̲ 2 , σ ¯ 2 ] ) , if

(2.2) a X + b X ¯ ∼ a 2 + b 2 X , ∀ a , b ≥ 0 ,

where X ¯ is an independent copy of X and

(2.3) E G [ X 2 ] = σ ¯ 2 , − E G [ − X 2 ] = σ ̲ 2 .

For given T ∈ [ 0 , ∞ ) , N = 1 , 2 , … , ℜ n is a n -dimensional field of real numbers, C l , l i p ( ℜ n ) denotes the linear space of functions ψ satisfying

∣ ψ ( x ) − ψ ( y ) ∣ ≤ C ( 1 + ∣ x ∣ m + ∣ y ∣ m ) ∣ x − y ∣ for x , y ∈ ℜ n , C > 0 , m ∈ N depending on ψ .

Following the works in [10,14], we next introduce some spaces as outlined below:

Ω T = { ω ⋅ ∧ T : ω ∈ Ω } , L i p ( Ω T ) ≔ ψ ( ω t 1 ∧ T , … , ω t n ∧ T ) , n ∈ N , t 1 , … , t n ∈ [ 0 , T ] , ψ ∈ C l , l i p ( ℜ n ) , L i p ( Ω ) ≔ ⋃ n = 1 ∞ L i p ( Ω n ) ,

where B t is the canonical process, i.e., B t ( ω ) = ω t .

Following [7,11], we can obtain a sublinear expectation E G [ ⋅ ] on L i p ( Ω ) with a monotonic and sublinear function G ( ⋅ ) : ℜ → ℜ . The canonical process ( B t ) t ≥ 0 is called a G-Brownian motion under G-expectation E G ( ⋅ ) . We denote the completion of L i p ( Ω ) under the norm

(2.4) ‖ X ‖ p ≔ [ E ( ∣ X ∣ p ) ] 1 ⁄ p

by L G p for p ≥ 1 , and we have

L G p ( Ω ) ⊆ L G q ( Ω ) , p ≥ q .

Let M G p , 0 ( 0 , T ) be the collection of processes in the following form:

ξ t ( ω ) = ∑ i = 0 N − 1 η i ( ω ) I [ t i , t i + 1 ) ( t ) ,

where 0 = t 0 < t 1 < … < t N = T is any patition of [ 0 , T ] , η i ∈ L G p ( Ω t i ) , i = 0 , … , N − 1 . For any ξ ∈ M G p , 0 ( 0 , T ) , let

‖ ξ ‖ M G p = E ∫ 0 T ∣ ξ s ∣ p d s 1 ⁄ p ,

and we denote M G p ( 0 , T ) by the completion of M G p , 0 ( 0 , T ) under norm ‖ ⋅ ‖ M G p . We come to give the following definition of G-Brownian motion, G-quadratic variation process, and multiple G-Itô formula.

Definition 2.3

(see [4]). A d -dimensional process ( B t ) t ≥ 0 defined on a sublinear space ( Ω , ℋ , E G ) is called a G-Brownian motion if the following conditions are satisfied:

B 0 ( ω ) = 0 ;
for each t , s ≥ 0 , the increment B t + s − B t follows a distribution N ( 0 , [ s σ ̲ 2 , s σ ¯ 2 ] ) and is independent to ( B t 1 , B t 2 , … , B t n ) for all n ∈ N and t 1 ≤ t 2 ≤ … ≤ t n ≤ t .

Definition 2.4

(see [4]). Let ( B t ) t ≥ 0 be a d -dimensional G-Brownian motion. For a fixed a ∈ ℜ d , define ( B t a ) t ≥ 0 = ⟨ a , B t ⟩ . Then, ( B t a ) t ≥ 0 constitutes a 1-dimensional G a -Brownian motion. The quadratic variation process of B a can be defined as follows:

⟨ B a ⟩ t = lim μ ( π t N ) → 0 ∑ j = 0 N − 1 ( B t j + 1 N a − B t j N a ) 2 = ( B t a ) 2 − 2 ∫ 0 t B s a d B s a in L G 2 ( Ω ) .

Definition 2.5

(see [10], multidimensional G-Itô formula). Let Φ ∈ C 2 ( ℜ n ) with ∂ x i x j 2 Φ satisfy polynomial growth condition for i , j = 1 , … , n , and a n -dimensional Itô process X t = ( X t 1 , X t 2 , … , X t n ) be the form of

(2.5) X t i = X a i + ∫ 0 t α s i d s + ∑ j = 1 m ∫ 0 t β s i , j d ⟨ B j ⟩ s + ∑ j = 1 m ∫ 0 t γ s i , j d B s j ,

where α i is the i th component of α = ( α 1 , … , α d ) T , β i , j and γ i , j are the elements at the i th row and j th column of β = ( β i , j ) d × m and γ = ( γ i , j ) d × m respectively. Moreover, α i , β i , j , γ i , j ∈ M G 2 ( 0 , T ) are bounded processes. Then for each t , s > 0 , within L G 2 ( Ω t ) , we have

Φ ( X t ) − Φ ( X s ) = ∑ i = 1 d ∫ s t ∂ x i Φ ( X u ) α u i d u + ∑ j = 1 m ∫ s t ∂ x j Φ ( X u ) γ u i , j d B u j + ∫ s t ∑ i = 1 d ∑ j = 1 m ∂ x i Φ ( X u ) β u i , j + 1 2 ∑ i = 1 d ∑ j = 1 m ∂ x i x j 2 Φ ( X u ) γ u i , j γ u i , j d ⟨ B j ⟩ u .

Definition 2.6

(see [6,11]). Let ( Ω , ℋ , E G ) be a G-expectation space and B = ( B 1 , … , B d ) be a d -dimensional G-Brownian motion. The related product rule is as follows:

d B t i d B t j = δ i j d t = d ⟨ B i ⟩ t , i = j , 0 , i ≠ j , d t d t = 0 , d t d B t = 0 , d B t d t = 0 , d t d ⟨ B ⟩ t = 0 , d ⟨ B ⟩ t d t = 0 , d ⟨ B ⟩ t d ⟨ B ⟩ t = 0 , d ⟨ B ⟩ t d B t = 0 , d B t d ⟨ B ⟩ t = 0 , d B t d B t = d ⟨ B ⟩ t .

Definition 2.7

(see [9], G-Stratonovich integral). Let X t , Y t be two G-Itô processes defined in ( 2.5 ) for t ∈ [ a , b ] . Then the G-Stratonovich integral of X t with respect to Y t is defined by

(2.6) ∫ a b X t ∘ d Y t = ∫ a b X t d Y t + 1 2 ( d X t ) ( d Y t ) .

For a ≤ s ≤ t ≤ b , let X t , Y t be the forms of

X t = X a + ∫ a b f ( s ) d B ( s ) + ∫ a t ξ s d s , Y t = Y a + ∫ a b g ( s ) d B ( s ) + ∫ a t η s d s ,

where X a , Y a are ℱ a -measurable, f , g ∈ L G ( Ω , L 2 [ a , b ] ) , and ξ , η ∈ L G ( Ω , L 1 [ a , b ] ) . Then we have ( d X t ) ( d Y t ) = f ( t ) g ( t ) d t , and

(2.7) ∫ a b X t ∘ d Y t = ∫ a b X t g ( t ) d B ( t ) + ∫ a b ( X t ) η ( t ) + 1 2 f ( t ) g ( t ) d t .

Let’s recall related notations of multiple G-Itô integral as follows, writing

L 2 ( [ 0 , T ] n ) ≔ { f ∣ f : [ 0 , T ] n → ℜ , ‖ f ‖ L 2 ( [ 0 , T ] n ) 2 < ∞ } , L ˜ 2 ( [ 0 , T ] n ) ≔ { f ∣ f is a symmetric function in L 2 ( [ 0 , T ] n ) } ,

where

‖ f ‖ L 2 ( [ 0 , T ] n ) 2 = ∫ [ 0 , T ] n f 2 ( x 1 , … , x n ) d x 1 … d x n .

Put

Q n = { ( x 1 , … , x n ) ∈ [ 0 , T ] n : 0 ≤ x 1 ≤ … ≤ x n ≤ T } , n ∈ N .

Define

‖ f ‖ L 2 ( Q n ) 2 = ∫ Q n f 2 ( x 1 , … , x n ) d x 1 … d x n .

It is easy to see that ‖ f ‖ L Q n 2 2 < ∞ , and we can define the ( n -fold) iterated G-Itô integral by

(2.8) J n T ( f ) ≔ ∫ 0 T ∫ 0 t n … ∫ 0 t 3 ∫ 0 t 2 f ( t 1 , … , t n ) d B t 1 … d B t n .

Definition 2.8

(see [10], multiple G-Itô integral). For any f ∈ L ˜ 2 ( [ 0 , T ] n ) , we define

I n T ( f ) = ∫ [ 0 , T ] n f ( t 1 , … , t n ) d B t 1 … d B t n ≔ n ! J n T ( f ) .

For any c ∈ ℜ , we define

I 0 ( c ) ≔ 0 ! J 0 T ( c ) = c .

Note that for all f ∈ L ˜ 2 ( [ 0 , T ] n ) , due to [10], we have I n T ( f ) ∈ L G 2 ( Ω T ) and

E G [ ( I n T ) 2 ] = E G ( n ! ) 2 ( J n T ( f ) ) 2 ≤ σ ¯ 2 n ( n ! ) 2 ‖ f ‖ L 2 ( Q n ) 2 = σ ¯ 2 n n ! ‖ f ‖ L 2 ( [ 0 , T ] n ) 2 < ∞ .

3 Multiple G-Stratonovich integral

In this section, we introduce some definitions associated with G-Stratonovich integral following [8,15].

Definition 3.1

A set S ⊆ Ω is polar if c ( S ) = 0 . A property holds “quasi-surely” (q.s., for short) if it holds outside a polar set.

Definition 3.2

Let X = ( X 1 , … , X m ) ∈ M G 2 ( 0 , T ) be an m -dimensional Itô process, and suppose that ⟨ X i , B i ⟩ exists for all i ∈ N , 0 ≤ s < t . The G-Stratonovich integral of X against d -dimensional G-Brownian motion, B = ( B 1 , … , B m ) , taking values in L G 1 , is defined as follows:

∫ 0 t X s ( i ) ∘ d B s ( i ) = ∫ 0 t X s i d B s i + 1 2 ⟨ X i , B i ⟩ c ˆ -q.s. = ∫ 0 t X s i d B s i + 1 2 ∫ 0 t β s i d ⟨ B i , B l ⟩ s c ˆ -q.s. ,

where B 1 , … , B m are m independent G-Brownian motions on ℜ , and X t ≔ ∫ 0 t β s d B s l with β = ( β 1 , … , β m ) ∈ M G 2 .

We know that for a multiindex ν , the components equal to 0 correspond to integrations with respect to time; the components equal to i ∈ { 1 , 2 , … , l } correspond to integrations with respect to the G-Stratonovich integral. If ν = { j 1 , … , j l } for l ∈ N , then ν − = { j 1 , … , j l − 1 } denotes the multiindex set. We denote by ℳ the set of all multiindex, ℋ v and ℋ 0 the sets of functions h : ℜ + × ℜ m → ℜ such that h ( ⋅ , X ⋅ ) ∈ ℋ v and h ( ⋅ , X ⋅ ) ∈ ℋ 0 , respectively, where X = { X t , t ≥ 0 } is an m -dimensional Itô process satisfying the Stratonovich stochastic differential form ( 2.5 ) .

Definition 3.3

(see [11]). Let ρ and τ be two stopping times such that 0 ≤ ρ ( ω ) ≤ τ ( ω ) ≤ T holds with probability 1. Then for a multiindex ν = { i 1 , … , i l } ∈ ℳ and a function f ∈ ℋ ν , we define the multiple G-Stratonovich integral J ν [ f ( ⋅ ) ] ρ , τ recursively, as follows:

J ν [ ( ⋅ ) ] ρ , τ = f ( τ ) , l = 0 , ∫ ρ τ J ν − [ f ( ⋅ ) ] ρ , τ d z , l ≥ 1 , i l = 0 , ∫ ρ τ J ν − [ f ( ⋅ ) ] ρ , τ ∘ d B z i l , l ≥ 1 , i l ∈ { 1 , 2 , … , m } .

Following [10], we try to propose a multiple G-Stratonovich integral, denoted by I n S t r ( f ) , which extends of 1-dimensional G-Stratonovich integral [9]. Given a map l : [ 0 , T ] n → ℜ , for any l ∈ L 2 ( [ 0 , T ] n ) , in what follows we provide the definition of the multiple G-Stratonovich integral, by analogue with the multiple G-Itô integral.

Definition 3.4

Let l ∈ L 2 ( [ 0 , T ] n ) and B t = ( B t 1 , … , B t n ) be n -dimensional G-Brownian motions. For ‖ l ‖ L 2 ( Q n ) 2 < ∞ , we can define ( n -fold) iterated G-Stratonovich integral by

J n S t r ( l ) ≔ ∫ 0 T ∫ 0 t n … ∫ 0 t 3 ∫ 0 t 2 l ( t 1 , … , t n ) ∘ d B t 1 ∘ d B t 2 … ∘ d B t n ,

where B t 1 , … , B t n are n independent G-Brownian motions on ℜ .

Definition 3.5

For all l ∈ L ˜ 2 ( [ 0 , T ] n ) , we give the definition of multiple G-Stratonovich integral by

I n S t r ( l ) ≔ ∫ [ 0 , T ] n ( l ( t 1 , … , t n ) ∘ d B t 1 ) ∘ d B t 2 … ∘ d B t n ≔ n ! J n S t r ( l ) .

Let 0 = t 0 < t 1 < … < t m = t be a partition of [ 0 , t ] , m ∈ N . We summarize the definition given earlier as the following theorem.

Theorem 3.6

Assume that X t = ( X t 1 , … , X t m ) ∈ M G 2 ( 0 , T ) is an m-dimensional G-Itô process and B t = ( B t 1 … B t m ) is an m-dimensional independent G-Brownian motion on ℜ , if ⟨ X i , B i ⟩ exists for i ∈ N , a ≤ t ≤ b . Then the G-stochastic process

(3.1) L t i = ∫ a t X s i ∘ d B s i ,

are G-Itô processes. Let X t be a G-Itô process with the form of (2.5). Then

(3.2) L t = ∫ a t X s ∘ d B s , a ≤ s ≤ t ≤ b

is also a G-Itô process.

Proof

We can prove ( 3.1 ) is true from [9]. By Definition 3.2, we have

L t i = ∫ 0 t X s ( i ) ∘ d B s ( i ) = ∫ 0 t X s i d B s i + 1 2 ⟨ X i , B i ⟩ c ˆ -q.s. = ∫ 0 t X s i d B s i + 1 2 ∫ 0 t β s i d ⟨ B i , B l ⟩ s c ˆ -q.s. ,

where X t ≔ ∫ 0 t β s d B s l with β = ( β 1 , … , β m ) ∈ M G 2 .

Following [9, 11], we have

L t i = ∫ 0 t X s ( i ) ∘ d B s ( i ) = ∫ 0 t X s i α s i d s + ∫ 0 t X s i γ s i + 1 2 β s i γ s i d ⟨ B ⟩ s i + ∫ 0 t X s i γ s i d B s i .

By the continuity of X t and Definition 2.7, we have

( E G [ ∣ X t i α t ∣ p ] ) 1 ⁄ p ≤ ( E G [ sup t ∈ [ a , b ] ∣ X t i ∣ p ∣ α t ∣ p ] ) 1 ⁄ p ≤ ( sup t ∈ [ a , b ] ∣ X t i ∣ p ∣ ) 1 ⁄ p ( E G [ ∣ α t ∣ p ] ) 1 ⁄ p < ∞ .

Similarly, E G [ ∣ X t i γ s i ∣ ] < ∞ , E G [ ∣ β s i γ s i ∣ ] < ∞ , ( E G [ ∣ X t i γ s i ∣ 2 ] ) 1 ⁄ 2 < ∞ . This implies that L t i are also G-Itô processes for i ∈ N . Next we prove ( 3.2 ) is true. By the same discussion as the aforementioned case, for a ≤ s ≤ t ≤ b , we can obtain

L t = ∫ a t X s ∘ d B s = ∫ a t X s α s d s + ∫ a t X s γ s + 1 2 β s γ s d ⟨ B ⟩ s + X s γ s d B s .

Therefore, L t = ∫ a t X s ∘ d B s is also a G-Itô process.□

Example 3.7

In the G-expectation space, considering the Ho-Lee model, the short-term interest rate r ( t ) satisfies the following G-Stratonovich model:

(3.3) d r ( t ) = a ( t ) d t + δ 0 d B t ,

where a ( t ) is a time-dependent function, characterizing the trend of r ( t ) ’s variation; δ 0 is a constant, representing the amplitude of interest rate fluctuations; and B ( t ) is a G-Brownian motion.

4 Relation between Hermite polynomials and multiple G-Stratonovich integrals

In this section, we show the relation between Hermite polynomials and multiple G-Stratonovich integrals. Let Q n ( x ) denote the n th Hermite polynomial, which is defined by

Q n ( x ) = ( − 1 ) n e x 2 ⁄ 2 d n d x n e − x 2 ⁄ 2 , x ∈ ℜ , n ∈ N .

It is clear that the first three Hermite polynomials are given as follows:

Q 0 ( x ) = 1 , Q 1 ( x ) = x , Q 2 ( x ) = x 2 − 1 .

For n ∈ N , ( x , y ) ∈ ℜ 2 , some bivariate polynomial functions q n ( x , y ) are as follows:

q n ( x , y ) = y n Q n x y , y ≠ 0 , x n , y = 0 .

Theorem 4.1

For any l ∈ L 2 ( [ 0 , T ] ) , n ∈ N , put

r 0 = 1 , r n ( t 1 , t 2 , … , t n ) = l ( t 1 ) ∘ … ∘ l ( t n ) .

Then r n ∈ L ˜ 2 ( [ 0 , T ] n ) for n ∈ N , we have

(4.1) I n S t r ( r n ) = q n ( η T , ‖ l ‖ T ) , q.s. ,

where

‖ l ‖ t = ∫ 0 T l 2 ( s ) d ⟨ B ⟩ s 1 ⁄ 2 , t ∈ [ 0 , T ]

is a nonnegative random variable and

η t = ∫ 0 t l ( s ) ∘ d B s , t ∈ [ 0 , T ] .

Proof

We shall prove the theorem in two steps.

Step 1. We first prove ( 4.1 ) holds if and only if the following equality holds true for n ≥ 2 :

(4.2) I n S t r ( r n ) = η T I n − 1 S t r ( r n − 1 ) − ( n − 1 ) ‖ l ‖ T 2 I n − 2 S t r ( r n − 2 ) , q . s .

With the help of the recursion formula of Hermite polynomials

(4.3) Q n ( y ) = y Q n − 1 ( y ) − ( n − 1 ) Q n − 2 ( y ) , n ≥ 2 ,

we have

(4.4) q n ( x , y ) = x q n − 1 ( x , y ) − ( n − 1 ) y 2 q n − 2 ( x , y ) , n ≥ 2 .

If ( 4.1 ) holds for n ≥ 2 , then by (4.4), we have

I n S t r ( r n ) = q n ( η T , ‖ l ‖ T ) = η T q n − 1 ( η T , ‖ l ‖ T ) − ( n − 1 ) ‖ l ‖ T 2 q n − 2 ( η T , ‖ l ‖ T ) = η T I n − 1 T ( r n − 1 ) − I n − 2 T ( r n − 2 ) .

We find (4.2) holds for n ≥ 2 .

On the other hand, assume that ( 4.2 ) is true, then

(4.5) q 0 ( η T , ‖ l ‖ T ) = 1 = I 0 S t r ( m 0 ) ,

(4.6) q 1 ( η T , ‖ l ‖ T ) = η T = I 1 S t r ( r 1 ) .

If n = 2 , applying G-Stratonovich integral formula to η t 2 , we obtain

d η t 2 = ∫ 0 t l ( s ) ∘ d B s l ( t ) ∘ d B t + l 2 ( t ) d ⟨ B ⟩ t ,

by using “quasi-surely,” we have

η T 2 = 2 ∫ 0 T ∫ 0 t l ( t ) ( l ( s ) ∘ d B s ) ∘ d B t + l 2 ( t ) ∘ d ⟨ B ⟩ t .

Thus,

q 2 ( η T , ‖ l ‖ T ) = η T 2 − ‖ l ‖ T 2 = 2 ∫ 0 T ∫ 0 t l ( t ) ( l ( s ) ∘ d B s ) ∘ d B t , q . s .

Hence,

(4.7) q 2 ( η T , ‖ l ‖ T ) = I 2 S t r ( r 2 ) , q . s .

It shows that (4.1) holds for n ∈ N by (4.5)–(4.7). Furthermore, for n ≥ 2 , (4.1) can be proved by mathematical induction with (4.2) and (4.4), we omit it here.

Step 2. We know that (4.2) holds in the case n = 2 . Next, we need to use mathematical induction on n to prove that (4.2) holds. We may assume that m ≥ 3 is an arbitrary integer and that (4.2) holds when n ≤ m − 1 . Let

X t = ∫ 0 t ∫ 0 t m − 1 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 1 ) ∘ d B t 1 ∘ … ∘ d B t m − 1 .

Then by G-Stratonovich integral formula, we obtain

d η t X t = ∫ 0 t ∫ 0 t m − 1 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 1 ) ∘ d B t 1 ∘ … ∘ d B t m − 1 l ( t ) ∘ d B t + η t ∫ 0 t ∫ 0 t m − 2 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 2 ) ∘ d B t 1 ∘ … ∘ d B t m − 2 l ( t ) ∘ d B t + ∫ 0 t ∫ 0 t m − 2 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 2 ) ∘ d B t 1 ∘ … ∘ d B t m − 2 l 2 ( t ) ∘ d ⟨ B ⟩ t .

Therefore, by using “quasi-surely,” we have

η T I m − 1 S t r ( m m − 1 ) = ( m − 1 ) ! η T X T = ( m − 1 ) ! ∫ 0 T ∫ 0 t ∫ 0 t m − 1 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 1 ) ∘ d B t 1 ∘ … ∘ d B t m − 1 ∘ d B t + ∫ 0 T ∫ 0 t ∫ 0 t m − 2 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 2 ) ∘ d B t 1 ∘ … ∘ d B t m − 2 l 2 ( t ) ∘ d ⟨ B ⟩ t + ∫ 0 T η t ∫ 0 t ∫ 0 t m − 2 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 2 ) ∘ d B t 1 ∘ … ∘ d B t m − 2 l ( t ) ∘ d B t .

Hence, we obtain

η T I m − 1 S t r ( r m − 1 ) = I m S t r ( r m ) + ( m − 1 ) ! ∫ 0 T ∫ 0 t ∫ 0 t m − 2 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 3 ) l ( t m − 2 ) ∘ d B t 1 ∘ … ∘ d B t m − 2 l 2 ( t ) ∘ d ⟨ B ⟩ t + ∫ 0 T η t ∫ 0 t ∫ 0 t m − 2 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 2 ) ∘ d B t 1 ∘ … ∘ d B t m − 2 l ( t ) ∘ d B t − ( m − 1 ) ∫ 0 T ∫ 0 t ∫ 0 t m − 1 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 1 ) l ( t ) ∘ d B t 1 ∘ … ∘ d B t m − 1 ∘ d B t = I m S t r ( r m ) + ( m − 1 ) ! ∫ 0 T ∫ 0 t ∫ 0 t m − 2 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 3 ) l ( t m − 2 ) ∘ d B t 1 ∘ … ∘ d B t m − 2 l 2 ( t ) ∘ d ⟨ B ⟩ t + ( m − 1 ) ϕ m ,

where

ϕ m = ∫ 0 T ( m − 2 ) ! η t ∫ 0 t ∫ 0 t m − 2 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 2 ) ∘ d B t 1 ∘ … ∘ d B t m − 2 − ( m − 1 ) ! ∫ 0 t ∫ 0 t m − 1 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 1 ) ∘ d B t 1 ∘ … ∘ d B t m − 1 l ( t ) ∘ d B t = ∫ 0 T [ η t I m − 2 S t r ( r m − 2 ) − I m − 1 S t r ( r m − 1 ) ] l ( t ) ∘ d B t .

From (4.2), by using “quasi-surely,” we have

ϕ m = ∫ 0 T [ ( m − 2 ) ‖ l ‖ t 2 I m − 3 S t r ( r m − 3 ) ] l ( t ) ∘ d B t = ( m − 2 ) ! ∫ 0 T ∫ 0 t l 2 ( s ) ∘ d ⟨ B ⟩ s ⋅ ∫ 0 t ∫ 0 t m − 3 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 3 ) ∘ d B t 1 ∘ … ∘ d B t m − 3 l ( t ) ∘ d B t .

If we let

X t = ∫ 0 t ∫ 0 t m − 2 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 2 ) ∘ d B t 1 ∘ … ∘ d B t m − 2 .

Then repeated “quasi-surely,” by using G-Stratonovich formula to ‖ l ‖ t 2 X t , we have

( m − 1 ) ‖ f ‖ T 2 I m − 2 S t r ( r m − 2 ) = ( m − 1 ) ! ‖ l ‖ T 2 X T = ( m − 1 ) ! ∫ 0 T ∫ 0 t ∫ 0 t m − 2 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 2 ) ∘ d B t 1 ∘ … ∘ d B t m − 2 l 2 ( t ) d ⟨ B ⟩ t + ( m − 1 ) ! ∫ 0 T ∫ 0 t l 2 ( s ) d ⟨ B ⟩ s ⋅ ∫ 0 t ∫ 0 t m − 3 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 3 ) ∘ d B t 1 ∘ … ∘ d B t m − 3 l ( t ) ∘ d B t .

Hence,

( m − 1 ) ‖ l ‖ T 2 I m − 2 S t r ( r m − 2 ) = ( m − 1 ) ! ∫ 0 T ∫ 0 t ∫ 0 t m − 2 … ∫ 0 t 2 l ( t 1 ) … l ( t m − 2 ) ∘ d B t 1 ∘ … ∘ d B t m − 2 l 2 ( t ) d ⟨ B ⟩ t + ( m − 1 ) ϕ m .

Thus, “quasi-surely” implies that

η T I m − 1 S t r ( r m − 1 ) = I m S t r ( r m ) + ( m − 1 ) ‖ l ‖ T 2 I m − 2 S t r ( r m − 2 ) ,

and ( 4.2 ) holds for n = m . The proof is complete.□

Remark 4.2

If σ ¯ 2 = σ ̲ 2 , G-Brownian motion degenerates to the classical case. In the situation, ( 4.1 ) also becomes the relation between the classical multiple Stratonovich integrals and Hermite polynomials.

Based on the aforementioned theorem, we would like to consider a special case in the following.

Corollary 4.3

Let X t = 1 and B t = ( B t 1 , … , B t n ) be n-dimensional G-Brownian motion. Then we have

∫ 0 T ∫ 0 t ∫ 0 t n … ∫ 0 t 2 d B t 1 ∘ … ∘ d B t n = ∑ m = 0 [ n ⁄ 2 ] ( − 1 ) m ⟨ B ⟩ T m ∘ B T n − 2 m 2 m m ! ( n − 2 m ) ! , q . s . ,

where [ x ] denotes the large integer not greater than x.

Proof

From the aforementioned theorem, we have

∫ 0 T ∫ 0 t ∫ 0 t n … ∫ 0 t 2 d B t 1 ∘ … ∘ d B t n = 1 n ! q n ( B T , ⟨ B ⟩ T 1 ⁄ 2 ) = 1 n ! ( B t i i ) n ,

where i ∈ N , t ∈ [ 0 , T ] .

It is very interesting in the corollary that Hermite can be written explicitly as follows:

□ Q n ( x ) = n ! ∑ m = 0 [ n ⁄ 2 ] ( − 1 ) m 2 m m ! ( n − 2 m ) ! x n − 2 m , x ∈ ℜ , n ∈ N .

5 Conclusion

In this article, we address the problem of “Multiple G-Stratonovich Integrals in G-Expectation Space.” This problem emerges, for instance, when constructing the framework for G-Stratonovich-type stochastic analysis within the sublinear expectation space. Motivated by the G-Itô formula, we then develop a multiple G-Stratonovich calculus in the G-expectation space, which is a sublinear expectation space. Building upon the aforementioned research work, we ultimately establish a relationship between Hermite polynomials and multiple G-Stratonovich integrals within the framework of the G-expectation space.

We must emphasize that the G-Stratonovich method holds potential application value in fields such as financial risk measurement and stochastic control in uncertain environments. Future research directions may encompass the analysis of weak convergence and the extension of parameter estimation methods.

Acknowledgments

The author would like to express their appreciation to the reviewers and the editor for their time and comments.

Funding information: This study was supported by Suqian Sci&Tech Program (Grant No. M202206).
Author contributions: The author conceived the study, designed the methodology, collected and analyzed all data, and wrote the manuscript.
Conflict of interest: The author declares that there is no conflict of interest in this study.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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Received: 2025-01-04

Revised: 2025-06-13

Accepted: 2025-08-08

Published Online: 2025-09-25

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2025-0195

Keywords for this article

G-Stratonovich integral; G-expectation; hermite polynomials; Itô formula; G-Brownian motion

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