Uncertainty orders on the sublinear expectation space

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.

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.

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].

Sublinear expectation spaces
Let be a given set and let H be a linear space of real valued functions defined on such that c 2 H for all constants c and jX j 2 H if X 2 H. The space H can be considered as the space of random variables. We further suppose that if X 1 ; ; X n 2 H, then '.X 1 ; ; X n / 2 H for each ' 2 C l:Lip .R n /, where C l:Lip .R n / denotes the linear space of functions ' satisfying j'.x/ '.y/j Ä C.1 C jxj m C jyj m /jx yj; 8x; y 2 R n ; where m depends only on '.
Definition 2.1 (see [8] ; X n /, X i 2 H, denoted by X 2 H n , be a given n-dimensional random vector on a sublinear expectation space . ; H; E/. Define a function on C l:Lip .R n / by F X OE' WD EOE'.X /; 8' 2 C l:Lip .R n /: The triple .R n ; C l:Lip .R n /; F X / forms a sublinear expectation space. F X is called the distribution of X. Let Y be another n-dimensional random vector on . ; H; E/, we denote X d D Y , if EOE'.X / D EOE'.Y / for all ' 2 C l:Lip .R n /. Definition 2.2 (see [6,8]). Let . ; H; E/ be a sublinear expectation space. A random vector Y 2 H n is said to be independent from another random vector X 2 H m under E, if for each test function ' 2 C l:Lip .R mCn / we have And Y is said to be weakly independent from X under E, if the above test functions ' are taken only among, instead of C l:Lip .R mCn /, '.x; y/ D 0 .y/ C 1 .y/jxj C 2 .y/jxj 2 ; i 2 C l:Lip .R n /; i D 0; 1; 2: Let X, N X be two n-dimensional random vectors on a sublinear expectation space . ; H; E/. N X is called an independent copy of X if X d D N X and N X is independent from X . Definition 2.3 (see [8]). Let . ; H; E/ be a sublinear expectation space. A n-dimensional random vector Á on where N Á is an independent copy of Á. In particular, for n D 1, where N X is an independent copy of X. In particular, for n D 1, we denote X where . N X; N Á/ is an independent copy of .X; Á/. For n D 1, we denote .X; Á/ d D N.OE ; N OE 2 ; N 2 / where , N , 2 and N 2 are defined as above.

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. (1) holds for all increasing convex functions ' 2 C l:Lip .R n /.
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 EOE '.X/ Ä EOE '.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 where ‚ is a family of probability measures on .R n ; B.R n //. In this sense, we say X Ä E mon Y , i.e., the best and worst expectations of a subset of linear expectations fE Â W Â 2 ‚g for loss random variables '.X / are both less than those of '.Y / for all increasing functions ' 2 C l:Lip .R n /.
For any loss random variable X 2 H of a sublinear expectation space . ; H; E/, we have the following four typical parameters: The internals OE ; N and OE 2 ; N 2 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 . ; H; E/, with the mean-uncertainty interval OE ; N and the variance-uncertainty interval 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. If N Ä N and Ä , for any increasing function ' 2 C l:Lip .R/, from the equivalent definition of the maximal distribution Definition 1.1 in Chapter II of [8], we have As for the other direction, choosing the convex functions '.x/ D x and '.x/ D x respectively, we can easily obtain N D N and D by the definitions of Ä E con and maximal distributions.
Recall the fact that EOE'. / can be explicitly calculated for G-normal distributions such that ' 2 C l:Lip .R/ is a convex or concave function in [8], for an increasing convex function '.x/ D x C satisfying ' 2 C l:Lip .R/, thus we have Similarly we obtain EOE'.Y / D 1 p 2 N . Since N ; N 0, we have N 2 Ä N 2 by the definition of Ä E mon . On the other hand, we have Similarly we can get EOE '.Y / D 1 p 2 . Since ; 0, we have 2 Ä N 2 from the definition of Ä E mon . Taking an increasing concave function '.x/ D x , we can derive N 2 N 2 and 2 2 using the arguments We conclude from the above that N 2 D N 2 and 2 D 2 , i.e., X It only needs to show that N 2 Ä N 2 and 2 Ä 2 H) X Ä E con Y . For any convex function ' 2 C l:Lip .R/. Consider the following G-heat equation for X ( @u @t We have that u.t; x/ WD EOE'.x C p t X /; .t; x/ 2 OE0; C1/ 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 Similarly we obtain v.t; x/ WD EOE'.x C p t Y /; .t; x/ 2 OE0; C1/ R, is the unique viscosity solution of the following G-heat equation ( @v @t Since N 2 Ä N 2 , 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 In particular, taking .t; x/ D .1; 0/, we have Since ' is a concave function, we can similarly show that m.t; x/ WD EOE '.x C p t X / and n.t; x/ WD EOE '.x C p tY /, .t; x/ 2 OE0; C1/ R, are the unique viscosity solutions of the following G-heat equations respectively ( @m @t C 1 2 2 . @ 2 m @x 2 / D 0; mj t D0 D ': and ( @n @t C 1 2 2 . @ 2 n @x 2 / D 0; nj t D0 D ': Due to the facts that 2 Ä 2 and the comparison theorem for the viscosity solutions of (7), setting .t; x/ D .1; 0/, we have By combining (6) with (8), we get X Ä E con Y . The proof is complete. Proof. The "only if" parts are the combinations of the results of Theorem 2.7 and Theorem 2.8. In fact, for example, if .Á; X / Ä E mon . ; Y /, then we can derive that Á Ä E mon and X Ä E mon 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.
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. Let X d D N. ; 2 / and Y d D N. ; 2 / be the two normal distributions on the probability space . ; F; P /, we have -X Ä P mon Y if and only if Ä , 2 D 2 , -X Ä P con Y if and only if D , 2 Ä 2 -X Ä P icon Y if and only if Ä , 2 Ä 2 . 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 ; 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] .

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.

Quantile functions and risk measures
Let . ; F/ be a measurable space, and for simplicity we only consider the situation bounded random variables. Let -Continuous from below: if A n ; A 2 F, and A n " A, then .A n / " .A/; -Continuous from above: if A n ; A 2 F, and A n # A, then .A n / # .A/. Now we introduce the definitions of capacity and Choquet integral (see, for instance, [19], [3]). Definition 3.1. A set function W F ! OE0; 1 is called a capacity if it is monotonic, normalized and continuous from below and continuous from above.
Definition 3.2. Let be a capacity, X 2 L 1 , and denote OEX by We call .X/ the Choquet integral of X with respective to the capacity .
Let be a capacity and X 2 L 1 . Put G ;X .x/ WD .X > x/: 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 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 , denoted by X Ä icon Y , if for all increasing and convex function W R ! R . .X // Ä . .Y //: (ii) X is said to precede Y in monotone ordering under , denoted by X Ä mon Y , if for all increasing function Some characterizations about these uncertainty orders were considered in [14,15], see Propositions 2.6-2.8 and Proposition 3.1 in [15].
Here, we give an another characterization of the uncertainty orders Ä mon and Ä icon 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 Ä mon and Ä icon in terms of distortion functions. We first list the definition of distortion functions, which can be found in any literature referred above.
The distortion risk measure associated with distortion function g and capacity is denoted by g ı . / and is defined as Proposition 3.13. Let be a capacity. For two losses X; Y 2 L 1 , (i) Y Ä mon X " g ı .X / g ı .Y / for all distortion functions g.
Proof. (i) The "H)" implication follows immediately from G ;X .x/ G ;Y .x/ and the non-decreasing property of any distortion function. "(H" For 2 .0; 1/, let distortion function g be defined by g.x/ D I x> ; x 2 OE0; 1. By the translation invariance of VaR ; . / and g ı . /, we may assume without loss of generality that X 0, then by (16), we have Hence, g ı .X/ D VaR ; .X /. Then by Proposition 3.10(iii), we have Y Ä mon X .
we see that g is a continuous concave distortion function. Furthermore, we find that Then by Proposition 3.10, we have Y Ä icon X . "H)" 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 ı .X / g ı .Y / for all continuous concave 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!1 g n .x/ D g.x/ for all x 2 OE0; 1. Since g n ı .X/ g n ı .Y / for all n, by monotone convergence theorem, we have g ı .X / g ı .Y /. The proof is complete. Proposition 3.14. If is a sub modular capacity, then for any concave distortion function g, g ı . / is a coherent risk measure on L 1 . In particular, for 2 .0; 1, AVaR ; . / is a coherent risk measure.
Proof. It is known that, for any sub modular capacity , . / is a coherent risk measure on L 1 (see [12], Theorem 4.88 (17), we then find that AVaR ; . / D g ı . / for a continuous concave function. Hence AVaR ; . / is a coherent risk measure. The proof is complete. And we assume that the induced set function

Characterizations from the probability measures viewpoint
is a capacity. Thus . ; B. /; / 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 2 H; 2 .0; 1/, we define VaR P; .X / WD sup Q2P VaR Q; .X /; AVaR P; .X / WD sup Q2P AVaR Q; .X /; where VaR Q; .X/ D inffxjQ.X > x/ Ä g and AVaR Q; .X / D 1 R 0 VaR Q;t .X /dt are the classical definitions (see Pages 177-179 in [12]). Remark 3.15. It can be verified that if is the supremum of a family of probability measures, then for all X 2 H and 2 .0; 1/ the following VaR P; .X / D VaR ; .X /; and AVaR P; .X / D AVaR ; .X / hold.
In the case of AVaR under a given probability measure P , we have known that AVaR P; . / 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 ; . / is coherent risk measure. And from Theorem 4.47 in [12], we can obtain that for any 2 .0; 1; X 2 H, where Q D fR 2 MjR Q and dR dQ Ä 1 ; Q a:s:g: Finally, from the characterizations results of uncertainty orders in Propositions 3.10-3.13, we conclude the following theorems.
Theorem 3.16. Let be a capacity induced by a family of probability measures P, which is determined by the sublunar expectation E. Let X; Y 2 H. The following statements are equivalent.
Theorem 3.17. Let be a capacity induced by a family of probability measures P, which is determined by the sublunar expectation E. Let X; Y 2 H. The following statements are equivalent.

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.