Uncertainty orders on the sublinear expectation space

Dejian Tian; Long Jiang

doi:10.1515/math-2016-0023

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Uncertainty orders on the sublinear expectation space

Dejian Tian und Long Jiang

Veröffentlicht/Copyright: 23. April 2016

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$Open Mathematics$

Aus der Zeitschrift Open Mathematics Band 14 Heft 1

Abstract

In this paper, we introduce some definitions of uncertainty orders for random vectors in a sublinear expectation space. We all know that, under some continuity conditions, each sublinear expectation 𝔼 has a robust representation as the supremum of a family of probability measures. We describe uncertainty orders from two different viewpoints. One is from sublinear operator viewpoint. After giving definitions such as monotonic orders, convex orders and increasing convex orders, we use these uncertainty orders to derive characterizations for maximal distributions, G-normal distributions and G-distributions, which are the most important random vectors in the sublinear expectation space theory. On the other hand, we also establish some uncertainty orders’ characterizations from the viewpoint of probability measures and build some connections with the theory of risk measures.

Keywords: Uncertainty order; Sublinear expectation; Choquet integral; Quantile function; Risk measure

MSC 2010: 62C86; 91B06; 90B50

1 Introduction

Nonlinear expectations have been a prominent topic in mathematical economics ever since the famous Allais paradox. Typical examples are monetary risk measures introduced in [1, 2], Choquet expectations developed in [3], g-expectations and G-expectations established and studied by [4–10] respectively.

The theory of sublinear expectation, an important and a special nonlinear expectation, is not based on a given (linear) probability space. We all know that, under some mild continuity conditions, each sublinear expectation 𝔼 has a robust representation as the supremum of a subset of probability measures. It thus provides a new way to describe the uncertainties.

In a financial context, when facing risks one often uses the stochastic orders to compare the “good or bad” between the portfolios. In this case, a probability measure is given on the set of scenarios, so we can focus on the resulting payoff or loss distributions. The comparison of financial risks plays an important role for both regulators and agents in financial markets. Under the framework of a classical probability space, based on expected utility theory developed by [11], lots of elegant results about the stochastic orders have been obtained. For example, in the stochastic order theory it has been shown that the monotonic order is equivalent to saying that one risk position is preferred over the other by all decision makers who have increasing real functions for which the expectations exist. For more details and properties of stochastic orders, see [12, 13].

Motivated by the classical stochastic orders, the object of this paper is to explore the uncertainty orders for random vectors in a sublinear expectation space. These uncertainty orders will provide useful criterions for describing the comparisons of the uncertainty degrees between two random vectors on the sublinear expectation space.

We establish the uncertainty orders in the sublinear expectation space from two different viewpoints. One is from the sublinear operator viewpoint. We give some definitions of uncertainty orders such as monotonic orders, convex orders and increasing convex orders. Then we use these uncertainty orders to derive the characterizations for maximal distributions, G-normal distributions and G-distributions, which are the most important random vectors in the theory of sublinear expectation space.

On the other hand, we study the uncertainty orders in the sublinear expectation space by a family of probability measures 𝒫 induced by the sublinear expectation 𝔼. We can define the capacity space from 𝒫, then we get some characterizations with the help of the recent results about capacity orders obtained by [14, 15] in the capacity space. Besides, we also give the characterization of uncertainty orders by distortion functions.

The paper is organized as follows. In Section 2, we first introduce some preliminaries about the sublinear expectation space. Then we give the definitions of uncertainty orders from the sublinear operator viewpoint, and establish some characterizations for maximal distributions, G-normal distributions and G-distributions. In Section 3, we introduce the other viewpoint of characterizations for uncertainty orders in the sublinear expectation space by using some terminologies of a capacity space. We conclude the paper in Section 4.

2 Characterizations of uncertainty orders from the sublinear operator viewpoint

We first present some preliminaries of sublinear expectation space theory. Then we give the definitions of uncertainty orders in the sublinear expectation space. More details for the first subsection can be found in [6, 8].

2.1 Sublinear expectation spaces

Let Ω be a given set and let 𝓗 be a linear space of real valued functions defined on Ω such that c ∈ 𝓗 for all constants c and ∣X∣ ∈ 𝓗 if X ∈ 𝓗. The space 𝓗 can be considered as the space of random variables. We further suppose that if X₁, ... , X_n ∈ 𝓗, then φ(X₁, ... , X_n) ∈ 𝓗 for each φ ∈ C_l.Lip(ℝⁿ), where C_l.Lip (ℝⁿ) denotes the linear space of functions φ satisfying

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

where m depends only on φ.

Definition 2.1

(see [8]). A sublinear expectation 𝔼 on 𝓗 is a functional 𝔼 : 𝓗 → ℝ satisfying the following properties: for all X, Y ∈ 𝓗, we have

Monotonicity: 𝔼[X] ≥ 𝔼[Y] if X ≥Y.
Constant preserving: 𝔼[c] = c for c ∈ ℝ.
Subadditivity: 𝔼[X + Y] ≤ 𝔼[X] + 𝔼[Y].
Positive homogeneity: 𝔼[λX] = λ𝔼[X] for λ ≥ 0.

The triple (Ω, 𝓗, 𝔼) is called a sublinear expectation space.

Let X = (X₁, ... , X_n), X_i ∈ 𝓗, denoted by X ∈ 𝓗ⁿ, be a given n-dimensional random vector on a sublinear expectation space (Ω, 𝓗, 𝔼). Define a function on C_l.Lip (ℝⁿ) by

FX[φ]:=E[φ(X)],∀φ∈Cl.Lip(ℝn).

The triple (ℝⁿ, C_l.Lip (ℝⁿ), 𝔽_X) forms a sublinear expectation space. 𝔽_X is called the distribution of X. Let Y be another n-dimensional random vector on (Ω, 𝓗, 𝔼), we denote X=dY, if E[φ(X)] = E[φ(Y)] for all φ ∈ C_l.Lip (ℝⁿ)

Definition 2.2

(see [6, 8]). Let (Ω, 𝓗, 𝔼) be a sublinear expectation space. A random vector Y ∈ 𝓗ⁿis said to be independent from another random vector X ∈ 𝓗^munder 𝔼, if for each test function φ ∈ C_l.Lip (ℝ^{m + n}) we have

E[φ(X,Y)]=E[E[φ(x,Y)]x=X].

And Y is said to be weakly independent from X under 𝔼, if the above test functions φ are taken only among, instead of C_l.Lip (ℝ^{m + n}),

φ(x,y)=ψ0(y)+ψ1(y)|x|+ψ2(y)|x|2,ψi∈Cl.Lip(ℝn),i=0,1,2.

Let X, X̅ be two n-dimensional random vectors on a sublinear expectation space (Ω, 𝓗, 𝔼). X̅ is called an independent copy of X if X=dX¯ and X̅ independent copy of X

Definition 2.3

(see [8]). Let (Ω, 𝓗, 𝔼) be a sublinear expectation space. A n-dimensional random vector η on (Ω, 𝓗, 𝔼) is called maximal distributed if

aη+bη¯=d(a+b)η,fora,b≥0,

where η̅ is an independent copy of η. In particular, for n = 1, we denoteη=d⁡N([μ_,μ¯]×{0})where μ = - 𝔼[-η] and μ̅ = 𝔼[η].

A n-dimensional random vector X on (Ω, 𝓗, 𝔼) is called G-normal distributed if

aX+bX¯=da2+b2X,fora,b≥0,

where X̅ is an independent copy of X. In particular, for n = 1, we denoteX=d⁡N({0}×[σ_2,σ¯2])whereσ₂ = − 𝔼[− X²] and σ̅² = E[X²] and σ̅ ≥ σ ≥ 0.

A pair of n-dimensional random vectors (X, η) on (Ω, 𝓗, 𝔼) is called G-distributed if

(aX+bX¯,a2η+b2η¯)=d(a2+b2X,(a2+b2)η)fora,b≥0,

where (X̅ η̅) is an independent copy of (X η) For n = 1, we denote(X,η)=dN([μ_,μ¯]×[σ_2,σ¯2])where μ, = μ̅ σ² and σ̅² are defined as above

2.2 Characterizations from the sublinear operator viewpoint

Throughout the following paper, we interpret the risk positions as loss random vectors. Motivated by the definitions of stochastic orders in a probability space, we give the definitions of uncertainty orders in a sublinear expectation space. We then use these uncertainty orders to derive the characterizations for maximal distributions, G-normal distributions and G-distributions.

Definition 2.4

Let (Ω, 𝓗, 𝔼) be a sublinear expectation space. Let X, Y be two n-dimensional random vectors on (Ω, 𝓗, 𝔼).

X is said to precede Y in the monotonic order sense under 𝔼, denoted by X≤monEY, if for all increasing functions φ ∈ C_l.Lip (ℝⁿ), we have
(1)E[φ(X)]≤E[φ(Y)]and−E[−φ(X)]≤−E[−φ(Y)].
X is said to precede Y in the convex order sense under 𝔼, denoted by X≤conEY, if (1) holds for all convex functions φ ∈ C_l.Lip (ℝⁿ)
X is said to precede Y in the increasing convex order sense under 𝔼, denoted by X≤iconEY, if (1) holds for all increasing convex functions φ ∈ C_l.Lip (ℝⁿ).

Remark 2.5

From the definitions of uncertainty orders as above, we can see that the uncertainty orders only involve the distributions of random vectors X and Y, thus we can also consider the random vectors X and Y from the different sublinear expectation spaces.

Remark 2.6

Compared with the stochastic orders, here we impose an extra restriction condition - 𝔼[−φ(X)] ≤ 𝔼[−φ(Y)] on the uncertainty orders, which is a redundant condition in the linear probability space.

Recall Lemma 3.4 (the representation theorem) in Chapter I established in [8], we can see that it is reasonable to define the uncertainty orders as above. In fact, the condition (1) is equivalent to

(2)maxθ∈ΘEθ[φ(X)]≤maxθ∈ΘEθ[φ(Y)]andminθ∈ΘEθ[φ(X)]≤minθ∈ΘEθ[φ(Y)],

where Θ is a family of probability measures on (ℝⁿ, 𝓑(ℝⁿ)). In this sense, we sayX≤monEY, i.e., the best and worst expectations of a subset of linear expectations {E_θ : θ ∈ Θ} for loss random variables φ(X) are both less than those of φ(Y) for all increasing functions φ ∈ C_l.Lip (ℝⁿ).

For any loss random variable X ∈ 𝓗 of a sublinear expectation space (Ω, 𝓗, 𝔼), we have the following four typical parameters:

μ_=−E[−X],μ¯=E[X],σ_2=−E[−X2],σ¯2=E[X2].

The internals [μ, μ̅] and [σ²;σ̅²] characterize the mean-uncertainty and the variance-uncertainty of X respectively.

From Definition 2.4 of the uncertainty orders, we easily obtain that for another loss random variable Y in (Ω, 𝓗, 𝔼), with the mean-uncertainty interval [vv̅] and the variance-uncertainty interval [ρ², ρ̅²],

if X≤monEY or X≤iconEY, then μ̅ ≤ v̅ and μ ≤ v̅.
if X≤conEYthen μ̅ = v̅, μ = v, σ̅² ≤ ρ̅² and σ² ≤ ρ².

In the following three theorems we show that for some particular distributions the above necessary conditions are also the sufficient conditions. And these distributions are very important in the sublinear expectation space theory.

Theorem 2.7

Letη=dN([μ_,μ¯]×{0}) and ξ=dN([v_,v¯]×{0})be two maximal distributions on a sublinear expectation space (Ω, 𝓗, 𝔼). Then we have

η≤monEξ⇔η≤iconEξ⇔μ¯≤v¯ and μ ≤ v.
η≤conEξ⇔μ¯=v¯ and μ = v, i.e., η=dξ.

Proof

(i) From the definitions of the monotonic order and the increasing convex order, η≤monEξ implies η≤iconEξ is obvious.

If η≤iconEξ, choosing an increasing convex function φ(x) = x satisfying φ ∈ C_l.Lip (ℝ), from the definitions of the increasing convex orders and maximal distributions, we have

μ¯=E[η]≤E[ξ]=v¯andμ_=−E[−η]≤−E[−ξ]=v_.

If μ̅ ≤ v̅ and μ ≤ v, for any increasing function φ ∈ C_l.Lip (ℝ), from the equivalent definition of the maximal distribution Definition 1.1 in Chapter II of [8], we have

E[φ(η)]=maxμ_≤y≤μ¯φ(y)=φ(μ¯)≤φ(v¯)=maxv_≤y≤v¯φ(y)=E[φ(ξ)],

and

−E[−φ(η)]=minμ_≤y≤μ¯φ(y)=φ(μ_)≤φ(v_)=minv_≤y≤v¯φ(y)=−E[−φ(ξ)].

Thus we have η≤monEξ

(ii) It is obvious that η=dξ implies η≤conEξ. As for the other direction, choosing the convex functions φ(x) = x and φ(x) = -x respectively, we can easily obtain μ̅ = v̅ and μ = v by the definitions of ≤conE and maximal distributions.

Theorem 2.8

LetX=dN({0}×[σ_2,σ¯2]) and Y=dN({0}×[ρ_2,ρ¯2])be two G-normal distributions on a sublinear expectation space (Ω, 𝓗, 𝔼). Then we have

X≤monEY⇔σ¯2=ρ¯2 and σ ≤ ρ²i.e., X=dY.
X≤conEY⇔X≤iconEY⇔σ¯2≤ρ¯2 and σ² ≤ ρ².

Proof

(i) X=dY implies X≤monEY is obvious.

Suppose X≤monEY. Recall the fact that 𝔼[φ(·)] can be explicitly calculated for G-normal distributions such that φ ∈ C_l.Lip (ℝ) is a convex or concave function in [8], for an increasing convex function φ(x) = x⁺ satisfying φ ∈ C_l.Lip (ℝ), thus we have

E[φ(X)]=12π∫−∞+∞φ(σ¯y)exp(−y22)dy=σ¯2π∫0+∞yexp(−y22)dy=12πσ¯.

Similarly we obtain E[φ(Y)]=12πρ¯. Since σ̅, ρ̅ ≥ 0, we have σ̅² ≤ ρ̅²by the definition of≤monEOn the other hand, we have

−E[−φ(X)]=12π∫−∞+∞φ(σ_y)exp(−y22)dy=σ_2π∫0+∞yexp(−y22)dy=12πσ_.

Similarly we can get −E[−φ(Y)]=12πρ_. Since σ, ρ ≥ 0, we have σ² ≤ ρ⁻²from the definition of≤monE.

Taking an increasing concave function φ(x) = -x ̅, we can derive σ̅² ≥ ρ̅² and σ² ≥ ρ² using the arguments as φ(x) = x⁺.

We conclude from the above that σ̅² = ρ̅² and σ² = ρ², i.e., X=dY

(ii) Clearly we have X≤conEY⇒X≤iconEY. Repeating the arguments in the part (i) we have X≤conEY⇒σ¯2≤ρ¯2andσ_2≤ρ_2 and σ² ≤ ρ² by choosing the increasing convex function φ(x) = x⁺.

It only needs to show that σ̅² ≤ ρ̅² and σ² ≤ ρ² ⟹ σ_2≤ρ_2⇒X≤conEY.

For any convex function φ ∈ C_l.Lip (ℝ). Consider the following G-heat equation for X

(3){∂u∂t−12(σ¯2(∂2u∂x2)+−σ_2(∂2u∂x2)−)=0,u|t=0=φ.

We have that v(t,x):=E[φ(x+tX)], (t, x) ∈ [0, ∞) × ℝ, is the unique viscosity solution of (3). Furthermore, we can check that u(t, x) is convex in x. Thus the above G-heat equation (3) becomes

(4)∂u∂t−12σ¯2(∂2u∂x2)+=0,u|t=0=φ.

Similarly we obtain v(t, x):=E[φ(x+tY)], (t, x) ∈ [0, ∞) × ℝ, is the unique viscosity solution of the following G-heat equation

(5){∂v∂t−12ρ¯2(∂2v∂x2)+=0, v|t=0=φ.

Since σ̅² ≤ ρ̅², then by the comparison theorem for the viscosity solutions of (4) and (5) (For example, see [8], Theorem 2.6 in Appendix C), we derive that

u(t, x)≤v(t, x), (t, x)∈[0, +∞]×ℝ.

In particular, taking (t, x) = (1, 0), we have

(6)E[φ(X)]≤E[φ(Y)].

Since -φ is a concave function, we can similarly show that m(t, x):=E[−φ(x+tX)] and n(t, x):=E[−φ(x+tY)], (t, x) ∈ [0, ∞) × ℝ, are the unique viscosity solutions of the following G-heat equations respectively

(7){∂m∂t+12σ_2(∂2m∂x2)−=0, m|t=0=−φ.and{∂n∂t+12ρ_2(∂2n∂x2)−=0, n|t=0=−φ.

Due to the facts that σ² ≤ ρ² and the comparison theorem for the viscosity solutions of (7), setting (t, x) = (1, 0) we have

(8)−E[−φ(X)]≤−E[−φ(Y)].

By combining (6) with (8), we get X≤conEY. The proof is complete. □

Theorem 2.9

Let(η, X)=dN([μ_, μ¯]×[σ_2, σ¯2]) and (ξ, Y)=dN([v_, v¯]×[ρ_2, ρ¯2])be two G-distributions on a sublinear expectation space (Ω, 𝓗, 𝔼). Moreover, suppose η is weakly independent from X and Ȉ is weakly independent from Y respectively. Then we have

(η, X)≤monE(ξ, Y), if and only ifμ̅ ≤ v̅, μ ≤ v, σ̅² = ρ̅²andσ² ≤ ρ².
(η, X)≤conE(ξ, Y), if and only ifμ̅ = v̅, μ ≤ v, σ̅² ≤ ρ̅²andσ² ≤ ρ².
(η, X)≤iconE(ξ, Y), if and only ifμ̅ ≤ v̅, μ ≤ v, σ̅² ≤ ρ̅²andσ² ≤ ρ².

Proof

The “only if” parts are the combinations of the results of Theorem 2.7 and Theorem 2.8. In fact, for example, if (η, X)≤monE(ξ, Y), then we can derive that η≤monEξ and X≤monEY. Thus from Theorem 2.7 and Theorem 2.8 we get the results.

For the proof of the converse implications, the key ideas are both the applications of the comparison theorem of the viscosity solutions to G-equations. We only show the case (iii). Cases (i) and (ii) are verified by an analogous argument.

Assume μ̅ ≤ v̅, μ ≤ v, σ̅² ≤ ρ̅² and σ² ≤ ρ². For any increasing convex function φ ∈ C_l.Lip (ℝ²), by Proposition 1.10 in Chapter II of [8], we have that u(t, x, y):=E[φ(x+tX, y+tη)], (t, x, y) ∈ [0, ∞)×ℝ×ℝ, is the unique viscosity solution of the following G-equation for (η, X)

(9){∂u∂t−G(∂u∂y, ∂2u∂x2)=0, u|t=0=φ,

where G(p, a)=E[12aX2+pη].

Since η is weakly independent from X, we have

G(p, a)=E[12aX2+pη]=E[12aX2]+E[pη]=12(σ¯2a+−σ_2a−)+(μ¯p+−μ_p−).

On the other hand, since φ is an increasing convex function in C_l.Lip (ℝ²), we can get (t, x, y) is convex in x and increasing in y. Thus (9) becomes

(10){∂u∂t−(12σ¯2(∂2u∂x2)++μ¯(∂u∂y)+)=0, u|t=0=φ.

Similarly we obtain v(t,x,y):=E[φ(x+tY,y+tξ)],(t,x,y)∈[0,+∞)×R×R, is the unique viscosity solution of the following G-equation

(11){∂v∂t−(12ρ¯2(∂2v∂x2)++v¯(∂v∂y)+)=0, u|t=0=φ.

Since μ̅ ≤ v̅ and σ̅² ≤ ρ̅², then by the comparison theorem for the viscosity solutions of (10) and (11), setting (t, x, y) = (1, 0, 0), we have

(12)E[φ(X, η)]≤E[φ(Y, ξ)].

Since φ is an increasing convex function, we can similarly obtain that m(t, x, y):=E[−φ(x+tX, y+tη)] and u(t, x, y):=E[−φ(x+tX, y+tη)] (t, x, y) ∈ [0, ∞) × ℝ × ℝ, are the unique viscosity solutions of the following G-equations respectively

(13)∂m∂t+12σ_2(∂2m∂x2)−+μ_(∂m∂y)−=0, m|t=0=−φ.and∂n∂t+12ρ_2(∂2n∂x2)−+v_(∂n∂y)−=0, n|t=0=−φ.

Due to the facts that μ ≤ v and σ² ≤ ρ², by the comparison theorem for the viscosity solutions of (13), setting (t, x, y) = (1, 0, 0), we have

(14)−E[−φ(X, η)]≤−E[−φ(Y, ξ)].

By combining (12) with (14), we obtain (η, X)≤iconE(ξ, Y). The proof is complete. □

Remark 2.10

In the classical linear expectation space, for the stochastic orders’ results to the normal distributions, the reader can refer to [12] and [16]. We list the results as follows. LetX=dN(μ, σ2) and Y=dN(v, ρ2)be the two normal distributions on the probability space (Ω, ℱ, P), we have

X≤monPY, if and only ifμ ≤ v, σ² = ρ²,
X≤conPY, if and only ifμ = v, σ² ≤ ρ²,
X≤iconPY, if and only ifμ ≤ v, σ² ≤ ρ²,

Hence, our results generalize the classical results.

Remark 2.11

Theorem 2.9 looks like just combining stochastic orders’ results of two normal distributions X=dN(μ¯, σ¯2) and Y=dN(v¯, ρ¯2) with X′=dN(μ_, σ_2) and Y′=dN(v_, ρ_2) together. However, we can not understand it in this way, because G-distribution is not a simple collection of a family of normal distributions, see [8].

In this subsection, we introduce some uncertainty orders in the sublinear expectation space. Here we only consider some random variables with special distributions. It is not easy to characterize other distributions. For more properties or computations, the readers can refer to [17, 18].

3 Characterizations of uncertainty orders from the probability measures viewpoint

In this section, we first list some properties of a capacity and quantile functions. We then introduce the recent results of uncertainty orders in the capacity space introduced by [14, 15]. We also establish some characterizations by distortion functions. Finally, we derive the characterizations for uncertainty orders by capacity space theory, induced by sublinear expectation space.

3.1 Quantile functions and risk measures

Let (Ω, ℱ) be a measurable space, and for simplicity we only consider the situation bounded random variables. Let L^∞ = L^∞ (Ω, ℱ) be the space of bounded ℱ-measurable functions, endowed with the supremum norm ∥·∥.

Firstly, we introduce some properties of set functions μ : ℱ →[0, 1] by [3]:

Monotonicity: if A, B ∈ ℱ and A ⊆ B, then μ(A) ≤ μ(B);
Normalization: if μ (∅) = 0 and μ(Ω) = 1;
Continuous from below: if A_n, A ∈ ℱ, and A_n ↑ A, then μ(A_n) ↑ μ(A);
Continuous from above: if A_n, A ∈ ℱ, and A_n ↓ A, then μ(A_n) ↓ μ(A).

Now we introduce the definitions of capacity and Choquet integral (see, for instance, [19], [3]).

Definition 3.1

A set function μ : ℱ → [0, 1] is called a capacity if it is monotonic, normalized and continuous from below and continuous from above.

Definition 3.2

Let μ be a capacity, X ∈ L^∞, and denote μ[X] by

μ(X):=∫∞0(μ(X>t)−1)dt+∫0∞μ(X>t)dt.

We call μ(X) the Choquet integral of X with respective to the capacity μ.

Let μ be a capacity and X ∈ L^∞. Put

Gμ, X(x):=μ(X>x).

We call G_{μ, X} the decreasing distribution function of X with respective to μ. Taking into account the continuity property from below of the capacity μ, we derive that G_{μ, X} is right continuous. We introduce the definition of quantile functions of X with respect to μ by [3].

Definition 3.3

([3]). Let μ be a capacity and X ∈ L^∞. Then we say that Ğ_{μ, X} is a quantile function of X under the capacity μ if for all λ ∈ (0, 1),

G˘μ, X+(λ):=sup{x|Gμ, X(x)≥λ}≥G˘μ, X(λ)≥sup{x|Gμ, X(x)>λ}=:G˘μ, X−(λ).

And G˘μ, X+(λ) and G˘μ, X-(λ) are called the upper and lower (1 - λ)-quantile of X with respect to μ.

Remark 3.4

It is easy to verify that the lower and upper (1 - λ)-quantile of the distribution of X with respect to μ can also be represented as

(15)G˘μ, X−(λ)=inf{x|μ(X>x)≤λ}, G˘μ, X+(λ)=inf{x|μ(X>x)<λ}.

Any two quantile functions coincide for all levels λ, except on at most a countable set. We also have the following properties about the quantile functions of X under the capacity μ (see Chapter 1 and Chapter 4 of [3]). Note that (iv) of Lemma 3.5 holds here because the capacity is continuous from below and above, the reader can obtain a similar proof of Lemma A.23 in [12], see also Remark 2.4 in [15].

Lemma 3.5

Let μ be a capacity and X ∈L^∞, we have

Ğ_{μ, X} (·) is a decreasing function;
If v is an another capacity and Y ∈ L^∞, such that G_{v, Y} ≤ G_{μ, X}, thenĞ_{v, Y} ≤ Ğ_{μ, X}except on at most a countable set;
μ[X]=∫01G˘μ, X(t)dt;
If u is an increasing function, then Ğ_{μ, u(X)} = u ∘ Ğ_{μ, X} except on at most a countable set.

Note that VaR of a financial position X ∈ L^∞ under a given probability measure P on (Ω, ℱ) is the quantile function of the distribution of X. We give the following definitions, which generalize the definitions of VaR and AVaR under a priori probability measure. These definitions can also be found in [14, 15], where Grigorova used the different notations but the economic implications are the same.

Definition 3.6

Let μ be a capacity and a loss random variable X ∈ L^∞. For λ ∈ (0, 1), we define the Value at Risk with a confidence level λ ∈ (0, 1) of X under the capacity μ as

VaRμ, λ(X):=G˘μ, X−(λ)=inf{x|μ(X>x)≤λ}.

Definition 3.7

The Average Value at Risk under a capacity μ at λ ∈ (0, 1] of a loss position X ∈ L^∞is given by

AVaRμ, λ(X):=1λ∫0λVaRμ, t(X)dt.

Remark 3.8

For any X ∈ L^∞, Definition 3.6 can also be used to define VaR_{μ, 0}(X) and VaR_{μ, 1} (X) We set that VaR_{μ, 0}. (X) -∞ and VaR_{μ, 1} (X) = sup X. In the sequel, we will often use the following equivalence relation which holds for all x ∈ ℝ and λ ∈ [0, 1]:

(16)VaRμ, λ(X)≤x⇔Gμ, X(x)≤λ.

Since any two quantiles coincide for all levels λ, except on at most a countable set, then the AVaR under a capacity μ at λ ∈ (0, 1] of a financial position X ∈ L^∞can be defined by any quantile function of X under μ, i.e.,

AVaRμ, λ(X)=1λ∫0λG˘μ, X(t)dt.

We list some concepts of uncertainty orders under a capacity introduced by Grigorova. See Definition 2.7 and Definition 3.1 in [15].

Definition 3.9

Let μ be a capacity. Let X and Y be two losses positions in L^∞.

X is said to precede Y in the increasing convex ordering under μ, denoted byX≤iconμY, if for all increasing and convex function ϕ : ℝ → ℝ
μ(ϕ(X))≤μ(ϕ(Y)).
X is said to precede Y in monotone ordering under μ, denoted byX≤monμY, if for all increasing function ϕ : ℝ → ℝ
μ(ϕ(X))≤μ(ϕ(Y)).

Some characterizations about these uncertainty orders were considered in [14, 15], see Propositions 2.6–2.8 and Proposition 3.1 in [15].

Proposition 3.10

([15]) Let μ be a capacity, X, Y ∈ L^∞. The following statements hold.

Y≤iconμX⇔μ((X−b)+)≥μ((Y−b)+)for any b ∈ ℝ.
Y≤iconμX⇔AVaRμ, λ(X)≥AVaRμ, λ(Y)for any λ ∈ (0, 1)
Y≤monμX⇔Gμ, X(x)≥Gμ, Y(x), ∀x∈ℝ⇔VaRμ, λ(X)≥VaRμ, λ(Y), ∀λ∈(0, 1).

Remark 3.11

In Proposition 3.10, we use our notations. In fact, γ_X(t), in the Proposition 2.6 of [15], is equal to our Ğ_{μ, –X} (1 - t) ∀t ∈ (0, 1), thus (ii) and (iii) of our claims hold by simple transformation.

Here, we give an another characterization of the uncertainty orders ≤monμ and ≤iconμ by distortion functions. Distortion functions play an important role in the field of mathematical finance. We refer to [20] for decision choice under risk, [21] for insurance premiums and [12] for risk measures.

Motivated by [22], we give the characterizations of the uncertainty orders ≤monμ and ≤iconμ in terms of distortion functions. We first list the definition of distortion functions, which can be found in any literature referred above.

Definition 3.12

A distortion function is defined as a non-decreasing function g :[0, 1] → [0, 1] such that g(0) = 0 and g(1) = 1.

The distortion risk measure associated with distortion function g and capacity μ is denoted by g ∘μ(·) and is defined as

g∘μ(X)=∫−∞0(g[μ(X>x)]−1)dx+∫0+∞g[μ(X>x)]dx, ∀X∈L∞.

Proposition 3.13

Let μ be a capacity. For two losses X, Y ∈ L^∞

Y≤monμX⇔g∘μ(X)≥g∘μ(Y)for all distortion functions g.
Y≤iconμX⇔g∘μ(X)≥g∘μ(Y)for all concave distortion functions g.

Proof

(i) The “⟹” implication follows immediately from G_{μ, X}(x) ≥ G_{μ, Y} (x) and the non-decreasing property of any distortion function.

“⟸” For λ ∈ (0, 1), let distortion function g be defined by g(x) = I_x>λ, x ∈ [0, 1]. By the translation invariance of VaR_{μ, λ}(·) and g ∘ μ(·), we may assume without loss of generality that X ≥ 0, then by (16), we have

g[μ(X>x)]={1, if VaRμ, λ(X)>x, 0, if VaRμ, λ(X)≤x.

Hence, g ∘ μ(X) = VaR_{μ, λ}(X). Then by Proposition 3.10(iii), we have Y≤monμX.

(ii) “⟸” For any continuous distortion function g, by Fubini’s theorem, we get that

g∘μ(X)=∫0, 1VaRμ, t(X)dg(t).

For λ ∈ (0, 1), taking

(17)g(x)=min(xλ,1),x∈[0,1],

we see that g is a continuous concave distortion function. Furthermore, we find that

(18)g∘μ(x)=1λ∫0λVaRμ, t(X)dt=AVaRμ, λ(X).

Then by Proposition 3.10, we have Y≤iconμX.

“⟹” Any concave distortion functions (the concave distortion function may be not continuous at the point 0). Without loss of generality, we only need to show g ∘ μ(X) ≥ g ∘ μ (Y) for all continuous concave distortion functions g.

First we prove it holds for continuous concave piecewise linear distortion functions. As [22] constructed, any such function can be written as

g(x)=∑i=1nαi(βi−βi+1) min (x/αi, 1),

where 0 = α₀ < α₁ < ⋯ < α_n-1 < α_n = 1 and β₁ > β₂ > ⋯ > β_n > β_n+1 = 0. By (17) and (18), we have that g ∘ μ(·) can be represented by

g∘μ(⋅)=∑i=1nαi(βi−βi+1)AVaRμ, αi(⋅).

By Proposition 3.10, we thus obtain g ∘ μ(X) ≥ g ∘ μ(Y) for all continuous concave piecewise linear distortion functions g.

For any continuous concave distortion function g, there exists an increasing sequence of continuous concave piecewise linear distortion functions g_n such that lim_n→∞g_n(x) = g(x) for all x ∈ [0, 1]. Since g_n ∘ μ(X) ≥ g_n ∘ μ(Y) for all n, by monotone convergence theorem, we have g ∘ μ(X) ≥ g ∘ μ(Y). The proof is complete. □

Proposition 3.14

If μ is a sub modular capacity, then for any concave distortion function g, g ∘ μ(−·) is a coherent risk measure on L^∞. In particular, for λ ∈ (0, 1], AVaR_{μ, λ}(−·) is a coherent risk measure.

Proof

It is known that, for any sub modular capacity v, v(−·) is a coherent risk measure on L^∞ (see [12], Theorem 4.88). So the point is to show that

v(A):=g(μ(A)), ∀A∈F,

is a sub modular capacity.

Suppose that g is concave. Take A, B ∈ 𝓕 with μ(A) ≤ μ(B). We show that

g(μ(A))−g(μ(A∩B))≤g(μ(A∪B))−g(μ(B)).

It is trivial if r = 0, where

μ(A)−μ(A∩B)≥μ(A∪B)−μ(B):=r.

For r = 0, the concavity of g yields that

v(A)−v(A∩B)μ(A)−μ(A∩B)≥v(A∪B)−v(B)μ(A∪B)−μ(B).

Multiplying both sides with (μ(A)− μ(A ∩ B)) derives the result.

Let g be defined by (17), we then find that AVaR_{μ, λ}(·) = g ∘ μ(·) for a continuous concave function. Hence AVaR_{μ, λ}(−·) is a coherent risk measure. The proof is complete. □

3.2 Characterizations from the probability measures viewpoint

Given a sublinear expectation space (Ω, 𝓗, 𝔼). In this subsection, let Ω be a complete separable metric space equipped with the distance d, 𝓑(Ω) the Borel σ-algebra of Ω and 𝒨 the collection of all probability measures on (Ω, 𝓑(Ω)). Let B_b(Ω) be the all bounded functions of 𝓑(Ω)-measurable real functions, and we assume that 𝓗 = B_b(Ω). For the sublinear operator 𝔼, we assume that there exists a family of probability measures 𝒫 ⊂ 𝒨 such that

E[X]=supQ∈P⁡EQ[X], ∀X ∈H.

And we assume that the induced set function μ

μ(A):=E[IA]=supQ∈P⁡EQ[IA]=supQ∈P⁡Q(A), ∀A∈B(Ω),

is a capacity. Thus (Ω, 𝓑(Ω), μ) become a capacity space.

Kervarec defined the robust VaR and AVaR using a family of probability measures in [23]. [24] also introduced the similar notations under a family of absolutely continuous probability measures. For any X ∈ 𝓗, λ ∈ (0, 1), we define

(19)VaRP, λ(X):=supQ∈P⁡VaRQ, λ(X),

(20)AVaRP, λ(X):=supQ∈P⁡AVaRQ, λ(X),

where VaR_{Q, λ}(X) = inf{x|Q(X > x) ≤ λ} and AVaRQ, λ(X)=1λ∫0λVaRQ, t(X)dt are the classical definitions (see Pages 177-179 in [12]).

Remark 3.15

It can be verified that if μ is the supremum of a family of probability measures, then for all X ∈ 𝓗 and λ ∈ (0, 1) the following

VaRP, λ(X)=VaRμ, λ(X), andAVaRP, λ(X)=AVaRμ, λ(X)

hold.

In the case of AVaR under a given probability measure P, we have known that AVaR_{P, λ}(−·) is a coherent risk measure (see [12], Theorem 4.47). For coherent risk measures, we refer to [1] and [2]. From Remark 3.15, we can see that AVaR_{μ, λ}(−·) is coherent risk measure. And from Theorem 4.47 in [12], we can obtain that for any λ ∈ (0, 1], X ∈ 𝓗,

AVaRP, λ(X)=supQ∈PmaxR∈QλER[X],

where

Qλ={R∈M|R≪QanddRdQ≤1λ, Q−a.s.}.

Finally, from the characterizations results of uncertainty orders in Propositions 3.10–3.13, we conclude the following theorems.

Theorem 3.16

Let μ be a capacity induced by a family of probability measures 𝒫, which is determined by the sublunar expectation 𝔼. Let X; Y ∈ 𝓗. The following statements are equivalent.

μ(𝜙(X)) ≥ μ(𝜙(Y)) for all increasing and convex function 𝜙 : ℝ → ℝ.
μ((X − b⁺) ≥ μ((Y − b⁺) for any b ∈ ℝ.
AVaR_{𝒫, λ}(X) ≥ AVaR_{𝒫, λ}(Y) for any λ ∈ (0, 1).
g ∘ μ(X) ≥ g ∘ μ(Y) for all concave distortion functions g.

Theorem 3.17

Let μ be a capacity induced by a family of probability measures 𝒫, which is determined by the sublunar expectation 𝔼. Let X, Y ∈ 𝓗. The following statements are equivalent.

μ(𝜙(X)) ≥ μ(𝜙(Y)) for all increasing function 𝜙 : ℝ → ℝ.
G_{μ, X}(x) ≥ G_{μ, Y}(x), ∀x ∈ ℝ.
VaR_{𝒫, λ}(X) ≥ VaR_{𝒫, λ}(Y), for any λ ∈ (0, 1).
g ∘ μ(X) ≥ g ∘ μ (Y) for all distortion functions g.

4 Conclusions

This paper considers the uncertainty orders on the sublinear expectation space from two different viewpoints. It is worth noting that the sublinear expectation does not equal to the Choquet integral generally, where the capacity is induced by sublinear expectation. The readers can refer to [25] and [26] for more details. We only consider some special distributions in the first viewpoint, and other plausible formulation leaves to a future study.

Competing interests: The authors declare that they have no competing interests.
Authors’ contributions: All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgement

The authors are very grateful to the editor and an anonymous referee for several constructive and insightful comments on how to improve the paper. This work was supported by the National Natural Science Foundation of China (11371362), the Natural Science Foundation of Jiangsu Province (BK20150167).

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Received: 2104-5-8

Accepted: 2016-1-8

Published Online: 2016-4-23

Published in Print: 2016-1-1

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.