Uncertainty orders on the sublinear expectation space

(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

1. (i)Monotonicity: 𝔼[X] ≥ 𝔼[Y] if X ≥Y.
2. (ii)Constant preserving: 𝔼[c] = c for c ∈ ℝ.
3. (iii)Subadditivity: 𝔼[X + Y] ≤ 𝔼[X] + 𝔼[Y].
4. (iv)Positive homogeneity: 𝔼[λX] = λ𝔼[X] for λ ≥ 0.

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

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 $X\stackrel{d}{=}Y$, if E[φ(X)] = E[φ(Y)] for all φ ∈ C_l.Lip (ℝⁿ)

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 $X\stackrel{d}{=}\overline{X}$ and X̅ independent copy of X

where η̅ is an independent copy of η. In particular, for n = 1, we denote$\eta \stackrel{d}{=}N\left(\right[\underset{\_}{\mu },\overline{\mu }]\times \{0\left\}\right)$where μ = - 𝔼[-η] and μ̅ = 𝔼[η].

where X̅ is an independent copy of X. In particular, for n = 1, we denote$X\stackrel{d}{=}N\left(\right\{0\}\times [{\underset{\_}{\sigma }}^2,{\overline{\sigma }}^2\left]\right)$whereσ₂ = − 𝔼[− X²] and σ̅² = E[X²] and σ̅ ≥ σ ≥ 0.

where (X̅ η̅) is an independent copy of (X η) For n = 1, we denote$(X,\eta )\stackrel{d}{=}N\left(\right[\underset{\_}{\mu },\overline{\mu }]\times [{\underset{\_}{\sigma }}^2,{\overline{\sigma }}^2\left]\right)$where μ, = μ̅ σ² and σ̅² are defined as above

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

1. (i)X is said to precede Y in the monotonic order sense under 𝔼, denoted by $X{\le }_{mon}^{E}Y$, if for all increasing functions φ ∈ C_l.Lip (ℝⁿ), we have
   $\begin{array}{cc}\begin{array}{cc}E\left[\phi \right(X\left)\right]\le E\left[\phi \right(Y\left)\right]& and\end{array}& -\end{array}E[-\phi (X\left)\right]\le -E[-\phi (Y\left)\right].$(1)
2. (ii)X is said to precede Y in the convex order sense under 𝔼, denoted by $X{\le }_{con}^{E}Y$, if (1) holds for all convex functions φ ∈ C_l.Lip (ℝⁿ)
3. (iii)X is said to precede Y in the increasing convex order sense under 𝔼, denoted by $X{\le }_{icon}^{E}Y$, if (1) holds for all increasing convex functions φ ∈ C_l.Lip (ℝⁿ).

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.

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.

where Θ is a family of probability measures on (ℝⁿ, 𝓑(ℝⁿ)). In this sense, we say$X{\le }_{mon}^{E}Y$, 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).

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],

1. –if $X{\le }_{mon}^{E}Y$ or $X{\le }_{icon}^{E}Y$, then μ̅ ≤ v̅ and μ ≤ v̅.
2. –if $X{\le }_{con}^{E}Y$then 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.

Let$\eta \stackrel{d}{=}N\left(\right[\underset{\_}{\mu },\overline{\mu }]\times \{0\left\}\right)$ and $\xi \stackrel{d}{=}N\left(\right[\underset{\_}{v},\overline{v}]\times \{0\left\}\right)$be two maximal distributions on a sublinear expectation space (Ω, 𝓗, 𝔼). Then we have

1. (i)$\eta {\le }_{mon}^E\xi \iff \eta {\le }_{icon}^E\xi \iff \overline{\mu }\le \overline{v}$ and μ ≤ v.
2. (ii)$\eta {\le }_{con}^E\xi \iff \overline{\mu }=\overline{v}$ and μ = v, i.e., $\eta \stackrel{d}{=}\xi$.

(i) From the definitions of the monotonic order and the increasing convex order, $\eta {\le }_{mon}^E\xi$ implies $\eta {\le }_{icon}^E\xi$ is obvious.

Thus we have $\eta {\le }_{mon}^E\xi$

(ii) It is obvious that $\eta \stackrel{d}{=}\xi$ implies $\eta {\le }_{con}^E\xi$. 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 ${\le }_{con}^E$ and maximal distributions.

Let$X\stackrel{d}{=}N\left(\right\{0\}\times [{\underset{\_}{\sigma }}^2,{\overline{\sigma }}^2\left]\right)$ and $Y\stackrel{d}{=}N\left(\right\{0\}\times [{\underset{\_}{\rho }}^2,{\overline{\rho }}^2\left]\right)$be two G-normal distributions on a sublinear expectation space (Ω, 𝓗, 𝔼). Then we have

1. (i)$X{\le }_{mon}^{E}Y\iff {\overline{\sigma }}^2={\overline{\rho }}^2$ and σ ≤ ρ²i.e., $X\stackrel{d}{=}Y$.
2. (ii)$X{\le }_{con}^{E}Y\iff X{\le }_{icon}^{\text{E}}Y\iff {\overline{\sigma }}^2\le {\overline{\rho }}^2$ and σ² ≤ ρ².

(i) $X\stackrel{d}{=}Y$ implies $X{\le }_{mon}^{E}Y$ is obvious.

Similarly we can get $-E[-\phi (Y\left)\right]=\frac{1}{\sqrt{2\pi }}\underset{\_}{\rho }$. Since σ, ρ ≥ 0, we have σ² ≤ ρ⁻²from the definition of${\le }_{mon}^{\text{E}}$.

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., $X\stackrel{d}{=}Y$

(ii) Clearly we have $X{\le }_{con}^{E}Y\Rightarrow X{\le }_{icon}^{E}Y$. Repeating the arguments in the part (i) we have $X{\le }_{con}^{E}Y\Rightarrow {\overline{\sigma }}^2\le {\overline{\rho }}^{2}and{\underset{\_}{\sigma }}^2\le {\underset{\_}{\rho }}^2$ and σ² ≤ ρ² by choosing the increasing convex function φ(x) = x⁺.

It only needs to show that σ̅² ≤ ρ̅² and σ² ≤ ρ² ⟹ ${\underset{\_}{\sigma }}^2\le {\underset{\_}{\rho }}^2\Rightarrow X{\le }_{con}^{E}Y$.

By combining (6) with (8), we get $X{\le }_{con}^{E}Y$. The proof is complete. □

Let$(\eta ,X)\stackrel{d}{=}N\left(\right[\underset{\_}{\mu },\overline{\mu }]\times [{\underset{\_}{\sigma }}^2,{\overline{\sigma }}^2\left]\right)$ and $(\xi ,Y)\stackrel{d}{=}N\left(\right[\underset{\_}{v},\overline{v}]\times [{\underset{\_}{\rho }}^2,{\overline{\rho }}^2\left]\right)$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

1. (i)$(\eta ,X){\le }_{mon}^E(\xi ,Y)$, if and only ifμ̅ ≤ v̅, μ ≤ v, σ̅² = ρ̅²andσ² ≤ ρ².
2. (ii)$(\eta ,X){\le }_{con}^E(\xi ,Y)$, if and only ifμ̅ = v̅, μ ≤ v, σ̅² ≤ ρ̅²andσ² ≤ ρ².
3. (iii)$(\eta ,X){\le }_{icon}^E(\xi ,Y)$, if and only ifμ̅ ≤ v̅, μ ≤ v, σ̅² ≤ ρ̅²andσ² ≤ ρ².

The “only if” parts are the combinations of the results of Theorem 2.7 and Theorem 2.8. In fact, for example, if $(\eta ,X){\le }_{mon}^E(\xi ,Y)$, then we can derive that $\eta {\le }_{mon}^E\xi$ and $X{\le }_{mon}^{E}Y$. 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.

where $G(p,a)=E[\frac{1}{2}a{X}^2+p\eta ]$.

By combining (12) with (14), we obtain $(\eta ,X){\le }_{icon}^E(\xi ,Y)$. The proof is complete. □

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. Let$X\stackrel{d}{=}N(\mu ,{\sigma }^2)$ and $Y\stackrel{d}{=}N(v,{\rho }^2)$be the two normal distributions on the probability space (Ω, ℱ, P), we have

1. –$X{\le }_{mon}^{P}Y$, if and only ifμ ≤ v, σ² = ρ²,
2. –$X{\le }_{con}^{P}Y$, if and only ifμ = v, σ² ≤ ρ²,
3. –$X{\le }_{icon}^{P}Y$, if and only ifμ ≤ v, σ² ≤ ρ²,

Hence, our results generalize the classical results.

Theorem 2.9 looks like just combining stochastic orders’ results of two normal distributions $X\stackrel{d}{=}N(\overline{\mu },{\overline{\sigma }}^2)$ and $Y\stackrel{d}{=}N(\overline{v},{\overline{\rho }}^2)$ with $X^{\prime }\stackrel{d}{=}N(\underset{\_}{\mu },{\underset{\_}{\sigma }}^2)$ and $Y^{\prime }\stackrel{d}{=}N(\underset{\_}{v},{\underset{\_}{\rho }}^2)$ 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].

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

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

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].

And ${\breve{G}}_{\mu ,X}^{+}\left(\lambda \right)$ and ${\breve{G}}_{\mu ,X}^{-}\left(\lambda \right)$ are called the upper and lower (1 - λ)-quantile of X with respect to μ.

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].

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

1. (i)Ğ_{μ, X} (·) is a decreasing function;
2. (ii)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;
3. (iii)$\mu \left[X\right]={\int }_0^1{\breve{G}}_{\mu ,X}\left(t\right)dt;$
4. (iv)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.

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

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

1. (i)X is said to precede Y in the increasing convex ordering under μ, denoted by$X{\le }_{icon}^{\mu }Y$, if for all increasing and convex function ϕ : ℝ → ℝ
   $\mu \left(\varphi \right(X\left)\right)\le \mu \left(\varphi \right(Y\left)\right).$
2. (ii)X is said to precede Y in monotone ordering under μ, denoted by$X{\le }_{mon}^{\mu }Y$, if for all increasing function ϕ : ℝ → ℝ
   $\mu \left(\varphi \right(X\left)\right)\le \mu \left(\varphi \right(Y\left)\right).$

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

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

1. (i)$Y{\le }_{icon}^{\mu }X\iff \mu \left({(X-b)}^{+}\right)\ge \mu \left({(Y-b)}^{+}\right)$for any b ∈ ℝ.
2. (ii)$Y{\le }_{icon}^{\mu }X\iff AVa{R}_{\mu ,\text{λ}}\left(X\right)\ge AVa{R}_{\mu ,\text{λ}}\left(Y\right)$for any λ ∈ (0, 1)
3. (iii)$Y{\le }_{mon}^{\mu }X\iff G_{\mu ,X}\left(x\right)\ge G_{\mu ,Y}\left(x\right),\forall x\in ℝ\iff Va{R}_{\mu ,\text{λ}}\left(X\right)\ge Va{R}_{\mu ,\text{λ}}\left(Y\right),\forall \text{λ}\in (0,1).$

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 ${\le }_{mon}^{\mu }$ and ${\le }_{icon}^{\mu }$ 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 ${\le }_{mon}^{\mu }$ and ${\le }_{icon}^{\mu }$ in terms of distortion functions. We first list the definition of distortion functions, which can be found in any literature referred above.

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

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

1. (i)$Y{\le }_{mon}^{\mu }X\iff g\circ \mu \left(X\right)\ge g\circ \mu \left(Y\right)$for all distortion functions g.
2. (i)$Y{\le }_{icon}^{\mu }X\iff g\circ \mu \left(X\right)\ge g\circ \mu \left(Y\right)$for all concave distortion functions g.

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

Hence, g ∘ μ(X) = VaR_{μ, λ}(X). Then by Proposition 3.10(iii), we have $Y{\le }_{mon}^{\mu }X$.

Then by Proposition 3.10, we have $Y{\le }_{icon}^{\mu }X$.

“⟹” 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.

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. □

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.

is a sub modular capacity.

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. □

hold.

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

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.

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

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.

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