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In the G-expectation framework, Wang [1] first obtained the Jensen inequality of one-dimensional function. In this paper, under some stronger conditions, we obtain the Jensen inequality of bivariate function based on Wang’s proof method. And we give some examples to illustrate the application of Jensen inequality of bivariate function.

As we know, expected utility theory has been widely used in the field of mathematical finance, especially in measuring the preference and aversion of risk. However, because the classical mathematical expectation is linear, the Von-Neumann expected utility cannot accurately measure risk aversion. Hence, economists hope to find a tool that can have certain properties of the classical expectation and accurately measure risk aversion. Driven by this problem, Peng [

Since Peng’s pioneering work, many scholars devoted themselves to the study of related problems and obtained a wealth of scientific achievements. Bai and Buckdahn [

When expected utility function denotes the aversion and preference of risk, the Jensen’s inequality of mathematical expectation takes an important role. Because of the importance of the Jensen inequality, many scholars studied the Jensen inequality in different cases. In the g-expectation framework, Li [

This paper is organized as follows. In Section 2, we present a brief review of the primary concepts under the G-framework, including the definition and some useful properties of G-expectation. Then, we give the basic concept about the G-Browian and the computation of E ^ [ | B t | n ] . In Section 3, we demonstrate the G-Jensen inequality of bivariate function under the stronger conditions and give some examples of Jensen inequality of binary function.

In this section, we will give some basic theories about G-expectation and G-Brownian motion. Some more details can be found in literatures [

| φ ( x ) − φ ( y ) | ≤ C ( 1 + | x | m + | y | m ) | x − y | , ∀ x , y ∈ ℝ n , (1)

for C > 0 , m ∈ ℕ rely on φ . For each T ∈ [ 0, ∞ ) , let

L i p ( F T ) : = { φ ( B t 1 , ⋯ , B t n ) : 0 ≤ t 1 , ⋯ , t n ≤ T , ∀ φ ∈ C l . L i p ( ℝ n ) , n ∈ ℕ }

where B t is a canonical process. Let L G p ( F ) : = ∪ n = 1 ∞ L i p ( F n ) . For a given p ≥ 1 , we also denote L G p ( F ) the completion of L i p ( F ) under norm ‖ X ‖ p : = ( E ^ [ | X | p ] ) 1 p . Then let G ( x ) be a monotonic and sublinear function:

G ( x ) = 1 2 ( x + − σ _ 2 x − ) , x ∈ ℝ ,

where x + = max { 0 , x } , x − = ( − x ) + .

Firstly, we introduce some notations about G-expectations.

Definition 1. [

a) Monotonicity: If X ≥ Y , then E ^ [ X ] ≥ E ^ [ Y ] .

b) Preserving of constants: E ^ [ c ] = c , ∀ c ∈ ℝ .

c) Sub-additivity: E ^ [ X ] − E [ Y ] ≤ E ^ [ X − Y ] .

d) Positive homogeneity: E ^ [ λ X ] = λ E ^ [ X ] , ∀ λ ≥ 0 .

e) Constant translatability: E ^ [ X + c ] = E ^ [ X ] + c , ∀ c ∈ ℝ .

The triple ( Ω , H , E ^ ) is called a sublinear expectation spaces. If only c) and d) are satisfied, E ^ is called a sublinear functional.

Remark 1. If the inequality in c) is equality, then E ^ is a linear expectation on H . Moreover, the sublinear expectation E ^ can be represented as the upper expectation of a subset of linear expectation { E θ : θ ∈ Θ } , i.e., E ^ [ X ] = sup θ ∈ Θ E Θ [ X ] . In most cases, this subset is often treated as an uncertain model of probabilities { P θ : θ ∈ Θ } and the notion of sublinear expectation provides a robust way to measure a risk loss X.

The following simple properties is very useful in sublinear analysis.

Lemma 1. [

E ^ [ X + Y ] = E ^ [ X ] + E ^ [ Y ] .

In particular, if E ^ [ Y ] = E ^ [ − Y ] = 0 , then E ^ [ X + Y ] = E ^ [ X ] .

2) According to the property d) of G-expectation, it is easy to deduce that

E ^ [ λ X ] = λ + E ^ [ X ] + λ − E ^ [ − X ] , λ ∈ R .

3) For arbitrary X , Y ∈ H , we have

| E ^ [ X ] − E ^ [ Y ] | ≤ E ^ [ X − Y ] ∨ E ^ [ Y − X ] ≤ E ^ [ | X − Y | ] .

For this properties of G-expectation will often be used in this article. They can can simplify our calculations.

Now let us introduce the notation about G-Brownian.

Definition 2. [

1) B 0 ( ω ) = 0 ;

2) For each t , s ≥ 0 , the increment B t + s − B t and B s have the identically distribution. For arbitrary n ∈ ℕ and { t 1 , t 2 , ⋯ , t n } ∈ [ 0, t ] , B t + s − B t is independent from { B t 1 , B t 2 , ⋯ , B t n } .

Just Like the classical expectation situation, the increments of G-Brownian motion ( B t + s − B s ) t ≥ 0 is independent of F s . In fact it is a new G-Brownian motion since, just like the classical situation, the increments of B are identically distributed. Then we introduce some computation formula of standard G-Browian motion.

Lemma 2. [

E ^ [ B t − B s | F s ] = 0 E ^ [ | B t − B s | n | F s ] = E ^ [ | B t − B s | n ] = 1 2 π ( t − s ) ∫ − ∞ + ∞ | x | n e − x 2 2 ( t − s ) d x

Exactly as in classical cases, we have

E ^ [ ( B t − B s ) 2 ] = t − s , E ^ [ | B t − B s | 3 ] = 2 2 ( t − s ) 3 / 2 π E ^ [ ( B t − B s ) 4 ] = 3 ( t − s ) 2 , E ^ [ | B t − B s | 5 ] = 8 2 ( t − s ) 5 / 2 π . (2)

Definition 3. [

f ( x , y ) − 1 2 [ f ( x + Δ x , y + Δ y ) + f ( x − Δ x , y − Δ y ) ] ≤ 0

We can call the the bivariate function f ( x , y ) is convex function in the region D.

Lemma 3. [

f ( x 2 , y 2 ) ≥ f ( x 1 , y 1 ) + f ′ x ( x 1 , y 1 ) ( x 2 − x 1 ) + f ′ y ( x 1 , y 1 ) ( y 2 − y 1 )

Lemma 4. [

Using Wang’s proof method, we can easily obtain the following theorem.

Theorem 1. Assuming that function h ( x , y ) : ℝ × ℝ ↦ ℝ has the second partial derivatives and satisfies the inequation:

h ( E ^ [ X ] , E ^ [ Y ] ) ≤ E ^ [ h ( X , Y ) ] , (3)

where X , Y ∈ L G 1 ( F ) , h ( x , y ) ∈ L G 1 ( F ) , then h ( x , y ) is the viscosity subsolution of the following equation:

− G ( ∂ 2 h ∂ x 2 ( x , y ) b 2 + ∂ 2 h ∂ y 2 ( x , y ) d 2 + 2 ∂ 2 h ∂ x ∂ y ( x , y ) b d ) = 0 , ( b , d ) ∈ ℝ 2 . (4)

Based on the Theorem 1, we obtain the following Jensen inequality of bivariate function.

Theorem 2. Assuming that function h ( x , y ) : ℝ × ℝ ↦ ℝ has the second partial derivatives and the bivariate function is non-increasing w.r.t. one variable at least. The following two conditions are equivalent:

1) Function h is convex function;

2) The Jensen inequality based on G-expectation can hold:

h ( E ^ [ X ] , E ^ [ Y ] ) ≤ E ^ [ h ( X , Y ) ] , (5)

where X , Y ∈ L G 1 ( F ) , h ( X , Y ) ∈ L G 1 ( F ) .

Proof: ( i ) ⇒ ( ii ) Suppose for a moment that convex function h ( x , y ) is non-increasing w.r.t. independent variable y. For each ( X , Y ) and ( E ^ [ X ] , E ^ [ Y ] ) , we have

h ( X , Y ) ≥ h ( E ^ [ X ] , E ^ [ Y ] ) + h ′ x ( E ^ [ X ] , E ^ [ Y ] ) ( X − E ^ [ X ] ) + h ′ y ( E ^ [ X ] , E ^ [ Y ] ) ( Y − E ^ [ Y ] ) .

Then let l : = h ′ x ( E ^ [ X ] , E ^ [ Y ] ) , k : = h ′ y ( E ^ [ X ] , E ^ [ Y ] ) . Apparently, k ≤ 0 .

Then we have

E ^ [ h ( X , Y ) ] − h ( E ^ [ X ] , E ^ [ Y ] ) ≥ E ^ [ l ( X − E ^ [ X ] ) + k ( Y − E ^ [ Y ] ) ] = E ^ [ l ( X − E ^ [ X ] ) − k − ( Y − E ^ [ Y ] ) ] ≥ E ^ [ l ( X − E ^ [ X ] ) ] − E ^ [ k − ( Y − E ^ [ Y ] ) ] = E ^ [ l ( X − E ^ [ X ] ) ] .

Now we only consider E ^ [ l ( X − E ^ ( X ) ) ] .

E ^ [ l ( X − E ^ [ X ] ) ] = E ^ [ l + ( X − E ^ [ X ] ) + l − ( X − E ^ [ X ] ) ] = l + E ^ [ X − E ^ [ X ] ] + l − E ^ [ − ( X − E ^ [ X ] ) ] .

Because of l + E ^ [ X − E ^ [ X ] ] = 0 and l − E ^ [ − ( X − E ^ [ X ] ) ] ≥ 0 , we can know

E ^ [ h ( X , Y ) ] − h ( E ^ [ X ] , E ^ [ Y ] ) ≥ 0.

( ii ) ⇒ ( i ) The proof by contradiction.Suppose the function h ( x , y ) is not a convex function. And exist constants α , β , p , q such that the inequality ρ ( x , y ) ≥ h ( x , y ) can’t hold true in the domain of definition D = { ( x , y ) | α ≤ x ≤ β , p ≤ y ≤ q } , where

ρ ( x , y ) = h ( β , p ) − h ( α , p ) β − α ( x − α ) + h ( α , q ) − h ( α , p ) q − p ( y − p ) + h ( α , p ) .

Define a new function

ρ δ ( x , y ) = ρ ( x , y ) − δ [ ( x − α ) ( x − β ) + ( y − p ) ( y − q ) 2 ] , ( x , y ) ∈ D .

Exist a constant δ 0 > 0 and a point ( x * , y * ) ∈ D , let h ( x * , y * ) > ρ δ 0 ( x * , y * ) . For this fixed constant δ 0 , we assume the maximum of function h ( x , y ) − ρ δ 0 is achieved at a point ( x ¯ , y ¯ ) . Thenet let l = h ( x ¯ , y ¯ ) − ρ δ 0 ( x ¯ , y ¯ ) . We can obtain the following function

h δ 0 ( x , y ) = δ 0 ( x , y ) + l , ∀ ( x , y ) ∈ D .

Obviously, h ≤ h δ 0 . According to theorem 1, for ∀ ( b , d ) ∈ ℝ 2 , h ( x , y ) is the viscosity subsolution of the Equation (1), which yields

− G ( ∂ 2 h δ 0 ∂ x 2 ( x , y ) b 2 + ∂ 2 h δ 0 ∂ y 2 ( x , y ) d 2 + 2 ∂ 2 h δ 0 ∂ x ∂ y ( x , y ) b d ) ≤ 0. (6)

The Inequation (6) can be rewritten as follows

− G ( − δ 0 ( b 2 + d 2 ) ) ≤ 0.

According to definition of G ( x ) , we can know − ( − 1 2 δ 0 ( b 2 + d 2 ) σ 0 2 ) ≤ 0 . This conflicts with δ 0 ≥ 0 . Therefore, function h ( x , y ) is convex function.

Remark 2. Similarly, for the n-variables function, if we suppose that the n-variables function has the second partial derivatives and it is non-increasing w.r.t. n − 1 variables at least, we can also obtain the corresponding Jensen inequality In the G-expectation framework. Then, we give some examples to illustrate the application of Jensen inequality of bivariate function.

Example 1. Assume B t and Z s is the standard G-Borwnian motion. Then E ^ [ B t ] = E ^ [ B s ] = 0 . The function h ( x , y ) = x 2 m + e − y , ( x , y ) ∈ ℝ 2 , m ∈ ℕ .

Obviously, the function h ( x , y ) is convex function and satisfies ∂ h ∂ y ≤ 0 in the region D. According to Theorem 2, we can obtain

E ^ [ B t 2 m + e − Z t ] ≥ ( E ^ [ B t ] ) 2 m + e − E ^ [ Z s ] = t 2 m + 1

From this example, we can find that the Jensen inequality of bivariate function can be used to proof the inequality or estimate the G-expectation. We can also use the bivariate expected utility function to define uncertain preference based on this Jensen inequality of the bivariate function.

In this work, we suppose that the bivariate function is non-increasing w.r.t. one variable at least and has the second partial derivatives. Then we obtain the Jensen inequality of bivariate function in the G-expectation framework. Moreover, we give some examples to illustrate the application of Jensen inequality of bivariate function. As discussed in Section 1, this effort focuses on the Jensen inequality of bivariate function in the G-expectation. Our future efforts will focus on demonstrating the Jensen inequality of multivariate function and exploring the condition for this inequality.

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

Feng, L.Y. (2020) Jensen Inequality of Bivariate Function in the G-Expectation Framework. Journal of Mathematical Finance, 10, 35-41. https://doi.org/10.4236/jmf.2020.101004