Nonlinear expectations of random sets

If \(C=\{0\}\), then \(\operatorname{co}\mathcal{F}(\mathbb{R}^{d},\{0\})\) is the family \(\operatorname{co}\mathcal{F}\) of all convex closed sets in \(\mathbb{R}^{d}\). If \(C=\mathbb{R}_{-}^{d}\), then \(\operatorname{co}\mathcal{F}(\mathbb{R}^{d},\mathbb{R}_{-}^{d})\) is the family of lower convex closed sets, and a random closed convex set with realisations in this family is called a random lower set.

Let \(C\) be a convex closed cone in \(\mathbb{R}^{d}\) which does not coincide with the whole space. If \(X=\xi +C\) for \(\xi \in L^{p}(\mathbb{R}^{d})\), then \(X\) belongs to the space \(L^{p}(\operatorname{co}\mathcal{F}(\mathbb{R}^{d},C))\). For each \(\zeta \in L^{q}( C^{o} )\), we have \(h(X,\zeta )=\langle \xi ,\zeta \rangle \).

Let \(X=H_{\eta }(0)\) be the random half-space with the normal vector \(\eta \) having a non-atomic distribution. Then \(\mathbb{E}[X]\) is the whole space. The support function of \(X\) is finite only on the random ray \(\{c\eta : c\geq 0\}\).

Let \(X,Y\in L^{p}(\operatorname{co}\mathcal{F}(\mathbb{R}^{d},C))\). If we have \(\mathbb{E}[h(Y,\zeta )]\leq \mathbb{E}[h(X,\zeta )]\) for all \(\zeta \in L^{q}( C^{o} )\), then \(Y\subseteq X\) a.s.

For each \(A\in \mathfrak{F}\), replacing \(\zeta \) with \(\zeta \mathbf{1}_{A}\) yields whence \(h(Y,\zeta )\leq h(X,\zeta )\) a.s. The same holds for a general \(\zeta \in L^{q}(\mathbb{R}^{d})\) by splitting it into the cases when \(\zeta \in C^{o} \) and \(\zeta \notin C^{o} \). For a general \(\zeta \in L^{0}(\mathbb{R}^{d})\), we have \(h(Y,\zeta _{n})\leq h(X,\zeta _{n})\) a.s. with \(\zeta _{n}=\zeta \mathbf{1}_{\{|\zeta |\leq n\}}\) for \(n\in \mathbb{N}\). Thus \(h(Y,\zeta ) \leq h(X,\zeta )\) almost surely for all \(\zeta \in L^{0}(\mathbb{R}^{d})\), and the statement follows from [22, Corollary 3.6]. □

The distribution of \(X\in L^{p}(\operatorname{co}\mathcal{F}(\mathbb{R}^{d},C))\) is uniquely determined by \(\mathbb{E}[h(X,\zeta )]\) for \(\zeta \in L^{q}( C^{o} )\).

Apply Lemma 2.4 with \(Y=\{\xi \}\), so that the values of \(\mathbb{E}[h(X,\zeta )]\) identify all \(p\)-integrable selections of \(X\), and note that \(X\) equals the closure of the family of its \(p\)-integrable selections; see [25, Proposition 2.1.4]. □

Recall that \(C\) denotes a generic convex cone in \(\mathbb{R}^{d}\) which differs from the whole space. If \(X\) is a \(p\)-integrable random \(C\)-closed convex set, then \(L^{p}(X)\) is a nonempty convex \(\sigma (L^{p},L^{q})\)-closed and \(L^{p}(C)\)-closed subset of \(L^{p}(\mathbb{R}^{d})\).

If \(\xi _{n}\in L^{p}(X)\) and \(\xi _{n}\to \xi \in L^{p}(\mathbb{R}^{d})\) in \(\sigma (L^{p},L^{q})\), then for all \(\zeta \in L^{q}(\mathbb{R}^{d})\). Thus \(\xi \) is a selection of \(X\) by Lemma 2.4. The statement concerning \(C\)-closedness is obvious. □

A sublinear set-valued expectation is a function such that

i) for each deterministic \(a\in \mathbb{R}^{d}\) (additivity on deterministic singletons);

ii) for all deterministic \(F\in \operatorname{co}\mathcal{F}(\mathbb{R}^{d},C)\);

iii) if \(X\subseteq Y\) almost surely (monotonicity);

iv) for all \(c>0\) (homogeneity);

v) is subadditive, that is, for all \(p\)-integrable random closed convex sets \(X\) and \(Y\). A superlinear set-valued expectation satisfies the same properties with the exception of ii) replaced by and (3.1) replaced by the superadditivity property The nonlinear expectations and are said to be law-invariant if they retain their values on identically distributed random closed convex sets.

All nonlinear expectations on \(L^{p}(\operatorname{co}\mathcal{F}(\mathbb{R}^{d},C))\) take their values in \(\operatorname{co}\mathcal{F}(\mathbb{R}^{d},C)\).

If \(a\in C\), then \(X+a\subseteq X\) a.s., whence . Therefore, . □

If \(X+X'=\mathbb{R}^{d}\) a.s. with \(X'\) being an independent copy of \(X\), then for each law-invariant sublinear expectation .

By subadditivity and law-invariance,  □

Let \(C=\mathbb{R}_{-}^{d}\). If for a vector-valued function \(\overline{\mathtt{e}}:L^{p}(\mathbb{R}^{d})\to \mathbb{R}^{d}\), then \(\overline{\mathtt{e}}(\xi )\) splits into the vector of superlinear expectations applied to the components of \(\xi =(\xi _{1},\dots ,\xi _{d})\); see Theorem A.1.

It is possible to consider nonlinear expectations defined only on some special random sets, e.g. singletons or half-spaces. It is then only required that the family of such sets is closed under translations, under dilations by positive reals and for Minkowski sums.

Utility functions of random variables are usually assumed to be superadditive. Risk measures of random variables are defined by inverting the sign and so become subadditive. In order to resemble the terminology common for risk measures, the family \(\operatorname{co}\mathcal{F}\) could be ordered by the reverse inclusion ordering; then the terminology is correspondingly adjusted, e.g. a superlinear expectation becomes sublinear and monotonically decreasing. The use of the reverse inclusion order promoted by Hamel et al. [13, 14] is largely motivated by financial terminology, where risk measures are traditionally assumed to be antimonotonic and subadditive; see e.g. Föllmer and Schied [10, Chap. 4]. In the reverse inclusion order, set-valued risk measures become subadditive, exactly as conventional risk measures of random variables are. We, however, systematically consider the conventional inclusion order, and so our set-valued setting extends the setup advocated by Delbaen [7] in the numerical case. He considers utility functions instead of risk measures: utility functions are superlinear and increasing, corresponding to the properties of the superlinear set-valued expectation . Thus up to a change of terminology, our superlinear expectation corresponds to the sublinear set-valued risk measure of Hamel et al. [13, 14]. On the other hand, our sublinear expectation is a different object, which requires a separate treatment. Indeed, in the set-valued framework, a change of sign (that is, the central symmetry) does not alter the direction of the inclusion, and so it is not possible to convert a superlinear function to a sublinear one.

Motivated by financial applications, it is possible to replace the homogeneity and sub-(super-)additivity properties with convexity or concavity, e.g. But then can be turned into a superlinear expectation for random sets in the space \(\mathbb{R}^{d+1}\) by letting The arguments of are random closed convex sets \(Y=\{t\}\times X\); they form a family closed for dilations, Minkowski sums and translations by singletons from \(\mathbb{R}_{+}\times \mathbb{R}^{d}\). Note that selections of \(\{t\}\times X\) are given by \((t,\xi )\) with \(\xi \) being a selection of \(X\). In view of this, all results in the homogeneous case apply to the convex case if the dimension is increased by one.

Let denote the set of fixed points of a random closed set \(X\). If \(X\) is almost surely convex, then \(F_{X}\) is also almost surely convex, and if \(X\) is compact with positive probability, then \(F_{X}\) is compact. It is easy to see that \(F_{X+Y}\) contains \(F_{X}+F_{Y}\), whence is a law-invariant superlinear expectation. With a similar idea, it is possible to define the sublinear expectation as the support of \(X\), which is the set of points \(x \in \mathbb{R}^{d}\) such that \(X\) hits any open neighbourhood of \(x\) with positive probability. By the monotonicity property, for any \(x\in F_{X}\), whence is a subset of any other normalised superlinear expectation of \(X\). By a similar argument, dominates any other constant-preserving sublinear expectation.

Fix \(C=\{0\}\) and let \(X=[\beta ,\infty )\subseteq \mathbb{R}\) be a half-line. The functional is superlinear if and only if \(\beta \mapsto \mathtt{e}(\beta )\) is sublinear in the usual sense of (1.2). For random sets of the type \(Y=(-\infty ,\beta ]\), the superlinearity of corresponds to the univariate superlinearity of \(\beta \mapsto \mathtt{u}(\beta )\). This example shows that numerical sublinear expectations may be converted to both sublinear and superlinear set-valued ones depending on the choice of relevant random sets.

Let \(X=[\beta ',\beta ]\) be a random interval on the line with \(\beta ,\beta '\in L^{p}(\mathbb{R})\) and let \(C=\{0\}\). Then is the interval formed by a numerical superlinear expectation of \(\beta '\) and a numerical sublinear expectation of \(\beta \) such that \(\mathtt{u}\) is dominated by \(\mathtt{e}\), e.g. if \(\mathtt{u}\) and \(\mathtt{e}\) form an exact dual pair. The superlinear expectation may be empty.

We have , and the same holds for the superlinear expectation.

Let \(C=\{0\}\). For each \(\xi \in L^{p}(\mathbb{R}^{d})\) and any normalised superlinear expectation , the set is either empty or a singleton, and is additive on .

By (3.2) applied to \(X=\{\xi \}\) and \(Y=\{-\xi \}\), we have whence is either empty or a singleton, and then . If and are singletons (and so are nonempty) for \(\xi ,\xi '\in L^{p}(\mathbb{R}^{d})\), then whence the inclusion turns into equality. □

If is a sublinear expectation defined on random sets \(\xi +C\) for \(\xi \in L^{p}(\mathbb{R}^{d})\), then its minimal extension (4.1) is a sublinear expectation.

The additivity of on deterministic singletons follows from this property of  . For a deterministic \(F\in \operatorname{co}\mathcal{F}(\mathbb{R}^{d},C)\), The homogeneity and monotonicity properties of are obvious. The subadditivity follows from the fact that \(L^{p}(X+Y)\) is the \(L^{p}\)-closure of the sum \(L^{p}(X)+L^{p}(Y)\); see [25, Proposition 2.1.6]. □

If is superlinear on half-spaces with the same normal, that is, for all \(\beta ,\beta '\in L^{p}(\mathbb{R})\) and \(\eta \in L^{0}(\mathbb{S}^{d-1}\cap C^{o} )\), and is scalarly upper semicontinuous on half-spaces with the same normal, that is, if \(\beta _{n}\to \beta \) in \(\sigma (L^{p},L^{q})\), then its maximal extension given by (4.3) is superlinear and upper semicontinuous with respect to the scalar convergence of random closed convex sets. If is law-invariant on half-spaces, then is law-invariant.

The additivity on deterministic singletons follows from the fact that we have \(H_{\eta }(X+a)=H_{\eta }(X)+a\) for all \(a\in \mathbb{R}^{d}\). If \(F\in \operatorname{co}\mathcal{F}(\mathbb{R}^{d},C)\) is deterministic, then The homogeneity and monotonicity properties of the extension are obvious. For two \(p\)-integrable random closed convex sets \(X\) and \(Y\), (4.4) yields that Assume that \((X_{n})\) scalarly converges to \(X\). Let and let \((x_{n_{k}})\) converge to \(x\). Then for all \(\eta \in L^{0}(\mathbb{S}^{d-1}\cap C^{o} )\). Since \(h(X_{n_{k}}, \eta )\to h(X,\eta )\) in \(\sigma (L^{p},L^{q})\), scalar upper semicontinuity of on half-spaces yields that we have , whence for all \(\eta \). Therefore , confirming the upper semicontinuity of the maximal extension. The law-invariance property is straightforward. □

A map defined for \(\xi \in L^{p}(\mathbb{R}^{d})\) is a \(\sigma (L^{p},L^{q})\)-lower semicontinuous normalised sublinear expectation if and only if for \(u\notin C^{o}\) and where \(\mathcal{Z}_{u}\), \(u\in C^{o} \), are convex \(\sigma (L^{q},L^{p})\)-closed cones in \(L^{q}( C^{o} )\) such that for all \(u\neq 0\), \(\mathcal{Z}_{cu}=\mathcal{Z}_{u}\) for all \(c>0\), \(\mathcal{Z}_{0}=\{0\}\) and

(Sufficiency) For linearly independent \(u\) and \(v\) in \(\mathbb{R}^{d}\), each \(\zeta \in \mathcal{Z}_{u+v}\) satisfies \(\zeta =\zeta _{1}+\zeta _{2}\) with \(\mathbb{E}[\zeta _{1}]=t_{1}u\) and \(\mathbb{E}[\zeta _{2}]=t_{2}v\). Thus \(\mathbb{E}[\zeta ]=t(u+v)\) only if \(t_{1}=t_{2}=t\). Therefore, Since \(\mathcal{Z}_{cu}=\mathcal{Z}_{u}=c\mathcal{Z}_{u}\) for any \(c>0\), so the function is sublinear in \(u\) and hence a support function. The additivity property on singletons follows from the construction since for each deterministic \(a\in \mathbb{R}^{d}\). Furthermore, which implies that . The homogeneity property is obvious. The function is subadditive since Finally, for \(u\in C^{o} \), the set \(\{\zeta \in \mathcal{Z}_{u} : \mathbb{E}[\zeta ]=u\}\) is closed in \(\sigma (L^{q},L^{p})\). Since is the support function of the closed set \(\{\zeta \in \mathcal{Z}_{u} : \mathbb{E}[\zeta ]=u\}\) in the direction \(\xi \), it is lower semicontinuous as a function of \(\xi \) so that (3.3) holds.

(Necessity) By Proposition 3.2, the support function is infinite for \(u\notin C^{o} \). For \(u\in C^{o} \), let The map is sublinear from \(L^{p}(\mathbb{R}^{d})\) to \((-\infty ,\infty ]\). By sublinearity, \(\mathcal{A}_{u}\) is a convex cone in \(L^{p}(\mathbb{R}^{d})\), and \(\mathcal{A}_{cu}=\mathcal{A}_{u}\) for all \(c>0\). Furthermore, \(\mathcal{A}_{u}\) is closed with respect to the scalar convergence \(\xi _{n}+C\to \xi +C\) by the assumed lower semicontinuity of . Hence it is closed with respect to the convergence \(\xi _{n}\to \xi \) in \(\sigma (L^{p},L^{q})\).

Note that \(0\in \mathcal{A}_{u}\) and let be the polar cone of \(\mathcal{A}_{u}\). For \(u=0\), we have \(\mathcal{A}_{0}=L^{p}(\mathbb{R}^{d})\) and \(\mathcal{Z}_{0}=\{0\}\). Consider \(u\neq 0\). Letting \(\xi =a\mathbf{1}_{A}\) for an event \(A\) and a deterministic \(a\) such that \(\langle a,u\rangle \leq 0\), we obtain a member of \(\mathcal{A}_{u}\), whence each \(\zeta \in \mathcal{Z}_{u}\) satisfies \(\langle \mathbb{E}[\zeta ],a\mathbf{1}_{A}\rangle \leq 0\) whenever \(\langle a,u\rangle \leq 0\). Thus \(\zeta \in C^{o} \) a.s., and letting \(A=\Omega \) yields that \(\mathbb{E}[\zeta ]=tu\) for some \(t\geq 0\) and all \(\zeta \in \mathcal{Z}_{u}\). The subadditivity property of the support function of yields that \(\mathcal{A}_{u+v}\supseteq (\mathcal{A}_{u}\cap \mathcal{A}_{v})\) for \(u,v\in C^{o} \). By a Banach space analogue of Schneider [33, Theorem 1.6.9], the polar of \(\mathcal{A}_{u}\cap \mathcal{A}_{v}\) is the closed sum \(\mathcal{Z}_{u}+\mathcal{Z}_{v}\) of the polars, whence (5.1) holds.

By the definition of \(\mathcal{A}_{u}\), Since \(\mathcal{A}_{u}\) is convex and \(\sigma (L^{p},L^{q})\)-closed, the bipolar theorem yields that  □

A function is a scalarly lower semicontinuous minimal normalised sublinear expectation if and only if admits the representation and for \(u\notin C^{o} \), where \(C^{o} \) and the sets \(\mathcal{Z}_{u}\), \(u\in \mathbb{R}^{d}\), satisfy the conditions of Lemma 5.1.

(Necessity) Lemma 5.1 applies to the restriction of onto random sets \(\xi +C\). By the minimality assumption, coincides with its minimal extension (4.2). By Lemma 5.1, (4.2) and (2.4), for \(u\in C^{o} \),

(Sufficiency) The right-hand side of (5.2) is sublinear in \(u\) and so is a support function. The additivity on singletons, monotonicity, subadditivity and homogeneity properties of are obvious. For a deterministic \(F\in \operatorname{co}\mathcal{F}(\mathbb{R}^{d},C)\), the sublinearity of the support function yields that whence . The minimality of follows from Since the support function of given by (5.2) is the supremum of scalarly continuous functions of \(X\), the minimal sublinear expectation is scalarly lower semicontinuous. □

The sets \(\mathcal{Z}_{u}\), \(u\in \mathbb{R}^{d}\), constructed in the proof of necessity in Lemma 5.1 are maximal sets representing the sublinear expectation.

If \(u\in \mathcal{Z}_{u}\) for all \(u\in \mathbb{R}^{d}\), then for all \(p\)-integrable \(X\) and any scalarly lower semicontinuous normalised minimal sublinear expectation .

By (5.2), for all \(u\in C^{o} \). □

The sublinear expectation given by (5.2) is law-invariant if and only if the sets \(\mathcal{Z}_{u}\) are law-complete, that is, with each \(\zeta \in \mathcal{Z}_{u}\), the set \(\mathcal{Z}_{u}\) contains all random vectors that have the same distribution as \(\zeta \). If \(p=\infty \), then the elements of \(\mathcal{Z}_{u}\) can be represented as vectors composed of probability measures absolutely continuous with respect to ℙ. This is also possible for \(p\in [1,\infty )\) using measures with \(p\)-integrable densities.

Let \(Z\) be a random matrix with \(\mathbb{E}[Z]\) being the identity matrix, and let \(\mathcal{Z}_{u}=\{tZu^{\top }: t\geq 0\}\) for \(u\in C^{o} =\mathbb{R}^{d}\), where \(C=\{0\}\). Then (5.2) turns into the condition , whence . In this example, is not solely determined by \(h(X,u)\). This sublinear expectation is not necessarily constant-preserving.

Let \(X=H_{\eta }(\beta )\) with \(\beta \in L^{p}(\mathbb{R})\) and \(\eta \in L^{0}(\mathbb{S}^{d-1}\cap C^{o} )\) be a random half-space. By (5.2), is finite for \(u\in \mathbb{S}^{d-1}\cap C^{o} \) only if each \(\zeta \in \mathcal{Z}_{u}\) with \(\mathbb{E}[\zeta ]=u\) satisfies \(\zeta =\gamma \eta \) a.s. with \(\gamma \in L^{q}(\mathbb{R}_{+})\). For such \(u\), If the normal \(\eta =u\) is deterministic and then with Otherwise, . Thus the sublinear expectation of a random half-space with a deterministic normal is either a half-space with the same normal or the whole space.

A mapping is a scalarly lower semicontinuous constant-preserving minimal sublinear expectation if and only if for \(u\notin C^{o} \) and where \(\mathcal{M}_{u}\), \(u\in C^{o} \), are convex \(\sigma (L^{q},L^{p})\)-closed cones in \(L^{q}(\mathbb{R}_{+})\) with \(\mathcal{M}_{cu}=\mathcal{M}_{u}\) for all \(c>0\) and \(\mathcal{M}_{u+v}\subseteq \mathcal{M}_{u}\cap \mathcal{M}_{v}\) for all \(u,v\in C^{o} \).

(Sufficiency) If \(\mathcal{M}_{u}\), \(u\in C^{o} \), satisfy the imposed conditions, then \(\mathcal{Z}_{u}\) given by \(\{\gamma u : \gamma \in \mathcal{M}_{u}\}\) for \(u\in C^{o} \) satisfy the conditions of Lemma 5.1. Indeed, \(\mathcal{Z}_{cu}=\mathcal{Z}_{u}\) for all \(c>0\) and for all \(u,v\in C^{o} \). If \(F\in \operatorname{co}\mathcal{F}(\mathbb{R}^{d},C)\) is deterministic, then whence is constant-preserving.

(Necessity) Since is minimal, the support function of is given by (5.2). The constant-preserving property yields that for all half-spaces \(H_{u}(t)\) with \(u\in C^{o} \). By the argument from Example 5.7, the minimal sublinear expectation of a half-space \(H_{u}(t)\) is distinct from the whole space only if (5.3) holds.

The properties of \(\mathcal{Z}_{u}\) imply those of \(\mathcal{M}_{u}=\{\gamma : \gamma u\in \mathcal{Z}_{u}\}\). Indeed, assume that \(\gamma \in \mathcal{M}_{u+v}\) so that \(\gamma (u+v)\in \mathcal{Z}_{u+v}\). Hence \(\gamma (u+v)\in (\mathcal{Z}_{u}+\mathcal{Z}_{v})\), meaning that \(\gamma (u+v)\) is the limit of \(\gamma _{1n}u+\gamma _{2n}v\) for \(\gamma _{1n}u\in \mathcal{Z}_{u}\) and \(\gamma _{2n}v\in \mathcal{Z}_{v}\), \(n \in \mathbb{N}\). The linear independence of \(u\) and \(v\) yields \(\gamma _{1n}\to \gamma \) and \(\gamma _{2n}\to \gamma \), whence \(\gamma \in (\mathcal{M}_{u}\cap \mathcal{M}_{v})\). □

Each scalarly lower semicontinuous constant-preserving minimal sublinear expectation is exact.

Since (5.4) yields that if \(\eta \) is random, the maximal extension of by an analogue of (4.3) reduces to deterministic \(\eta \), and so is the reduced maximal extension. For \(u\in \mathbb{S}^{d-1}\cap C^{o} \) and \(\beta \in L^{p}(\mathbb{R})\), we have ; cf. Example 5.7. Thus the reduced maximal extension of is given by Comparing with (5.5), we see that . The opposite inclusion is obvious, whence . □

If is a scalarly lower semicontinuous constant-preserving minimal normalised sublinear expectation, then for each deterministic \(F\in \operatorname{co}\mathcal{F}(\mathbb{R}^{d},C)\).

Let be a scalarly lower semicontinuous constant-preserving minimal law-invariant sublinear expectation. Then we have for all \(X\in L^{p}(\operatorname{co}\mathcal{F}(\mathbb{R}^{d},C))\) and any \(\sigma \)-algebra \(\mathfrak{H}\subseteq \mathfrak{F}\). In particular, .

The law-invariance of implies that \(\mathtt{e}_{u}\) is law-invariant. The sublinear expectation \(\mathtt{e}_{u}\) is dilatation-monotonic, meaning that \(\mathtt{e}_{u}(\mathbb{E}[\beta |\mathfrak{H}])\leq \mathtt{e}_{u}( \beta )\) for all \(\beta \in L^{p}(\mathbb{R})\); see Föllmer and Schied [10, Corollary 4.59] for this fact derived for risk measures. The statement follows from (5.5). □

A scalarly lower semicontinuous constant-preserving minimal superlinear expectation satisfies for all \(\beta \in L^{p}(\mathbb{R}_{+})\) and some \(r\geq 0\) if and only if (5.4) holds with \(\mathcal{M}_{u}=\mathcal{M}\) for all \(u\neq 0\). Then where \(\mathtt{e}\) admits the representation (1.5). Furthermore,

Assume that the \(\mathcal{M}_{u}\) are constructed as in the proof of Theorem 5.8 so that \(\mathcal{M}_{u}\) is maximal for each \(u\in C^{o} \). The right-hand side of does not depend on \(u\in \mathbb{S}^{d-1}\cap C^{o} \) if and only if \(\mathcal{M}_{u}=\mathcal{M}\) for all \(u\in C^{o} \). The representation (5.6) follows from (5.5) with \(\mathcal{M}_{u}=\mathcal{M}\). In view of (1.5), By (5.6), the support functions of both sides of (5.7) are identical. □

Equality (5.6) can be viewed as a scalarisation of the sublinear expectation. Indeed, it represents the convex set as an intersection of half-spaces \(H_{u}(\mathtt{e}(h(X,u)))\) and so provides a dual representation of . Such scalarisations have been considered by Hamel and Heyde [13] and Hamel et al. [14] for set-valued risk measures, which are sublinear for the reverse inclusion. In that case, the exact equality may be violated, see (6.5) below, and the scalarisation is defined as the support function of the superlinear expectation.

For an integrable \(X\) and \(n\in \mathbb{N}\), consider the sublinear expectation where \(X_{1},\dots ,X_{n}\) are independent copies of \(X\). It is easy to see that is a minimal constant-preserving sublinear expectation; it is given by (5.6) with the corresponding numerical sublinear expectation \(\mathtt{e}(\beta )\) being the expected maximum of \(n\) i.i.d. copies of \(\beta \in L^{1}(\mathbb{R})\). By Corollary 5.9, this sublinear expectation is exact.

For \(\alpha \in (0,1)\), let \(\mathcal{P}_{\alpha }\) be the family of random variables \(\gamma \) with values in \([0,\alpha ^{-1}]\) and such that \(\mathbb{E}[\gamma ]=1\). Furthermore, let ℳ be the cone generated by \(\mathcal{P}_{\alpha }\), that is, \(\mathcal{M}=\{t\gamma :\gamma \in \mathcal{P}_{\alpha }, t\geq 0\}\). In finance, the set \(\mathcal{P}_{\alpha }\) generates the average value-at-risk, which is the risk measure obtained as the average quantile; see Föllmer and Schied [10, Definition 4.43]. Similarly, the numerical sublinear \(\mathtt{e}\) and superlinear \(\mathtt{u}\) generated by this set ℳ are represented as average quantiles. Namely, \(\mathtt{e}(\beta )\) is the average of the quantiles of \(\beta \) at levels \(t\in (1-\alpha ,1)\), and \(\mathtt{u}(\beta )\) is the average of the quantiles at levels \(t\in (0,\alpha )\).

A map is a scalarly upper semicontinuous normalised maximal superlinear expectation if and only if for a collection of convex \(\sigma (L^{q},L^{p})\)-closed cones \(\mathcal{M}_{\eta }\subseteq L^{q}(\mathbb{R}_{+})\) parametrised by \(\eta \in L^{0}(\mathbb{S}^{d-1}\cap C^{o} )\) and such that \(\mathcal{M}_{u}\) is strictly larger than \(\{0\}\) for each deterministic \(\eta =u\in \mathbb{S}^{d-1}\cap C^{o} \).

(Necessity) Fix \(\eta \in L^{0}(\mathbb{S}^{d-1}\cap C^{o} )\) and let \(\mathcal{A}_{\eta }\) be the set of all \(\beta \in L^{p}(\mathbb{R})\) such that contains the origin. Since , we have \(0\in \mathcal{A}_{\eta }\). Since , the family \(\mathcal{A}_{u}\) does not contain \(\beta =t\) for \(t<0\) and \(u\in \mathbb{S}^{d-1}\cap C^{o} \).

If \(\beta _{n}\to \beta \) in \(\sigma (L^{p},L^{q})\), then \(\mathbb{E}[h(H_{\eta }(\beta _{n}),\gamma \eta )]\to \mathbb{E}[h(H_{\eta }(\beta ),\gamma \eta )]\) for all \(\gamma \in L^{q}(\mathbb{R})\), whence \(H_{\eta }(\beta _{n})\to H_{\eta }(\beta )\) scalarly in \(\sigma (L^{p},L^{q})\). Therefore, by the assumed upper semicontinuity of . Thus \(\mathcal{A}_{\eta }\) is a convex \(\sigma (L^{p},L^{q})\)-closed cone in \(L^{p}(\mathbb{R})\). Consider its positive dual cone Since , we have whenever \(C\subseteq X\) a.s. In view of this, if \(\beta \) is a.s. nonnegative, then \(H_{\eta }(\beta )\) a.s. contains zero and so \(\beta \in \mathcal{A}_{\eta }\). Thus each \(\gamma \) from \(\mathcal{M}_{\eta }\) is a.s. nonnegative. The bipolar theorem yields that Since \((-t)\notin \mathcal{A}_{u}\), (6.2) implies that the cone \(\mathcal{M}_{u}\) is strictly larger than \(\{0\}\). Since is assumed to be maximal, (4.3) implies that

(Sufficiency) It is easy to check that given by (6.1) is additive on deterministic singletons, homogeneous and monotonic. If \(F\in \operatorname{co}\mathcal{F}(\mathbb{R}^{d},C)\) is deterministic, then letting \(\eta =u\) in (6.1) be deterministic and using the nontriviality of \(\mathcal{M}_{u}\) yields that . Furthermore, , since contains the origin and so is not empty. The superadditivity of follows from the fact that It is easy to see that coincides with its maximal extension.

Note that (6.1) is equivalently written as If \((X_{n})\) scalarly converges to \(X\) and \(x_{n_{k}}\to x\) for , \(k\in \mathbb{N}\), then \(\mathbb{E}[h(X_{n}-x_{n},\gamma \eta )]\) converges to \(\mathbb{E}[h(X-x,\gamma \eta )]\) for all \(\gamma \in L^{q}(\mathbb{R}_{+})\) and \(\eta \) from \(L^{0}(\mathbb{S}^{d-1}\cap C^{o} )\). Thus \(\mathbb{E}[h(X-x,\gamma \eta )]\geq 0\), whence , and the upper semicontinuity of follows. □

If \(1\in \mathcal{M}_{\eta }\) for all \(\eta \), then for all \(p\)-integrable \(X\) and any scalarly upper semicontinuous maximal normalised superlinear expectation .

Restrict the intersection in (6.1) to deterministic \(\eta =u\) and \(\gamma =1\), so that the right-hand side of (6.1) becomes \(\mathbb{E}[X]\). □

Let \(X=H_{\eta }(\beta )\) be the half-space with normal \(\eta \in L^{0}(\mathbb{S}^{d-1})\) and \(\beta \in L^{p}(\mathbb{R})\). If \(C=\{0\}\), the maximal superlinear expectation of \(X\) is given by Assume that \(d=2\) and let \(\eta =(1,\pi )/\sqrt{1+\pi ^{2}}\) with \(\pi \) being an almost surely positive random variable. This example represents the case of two currencies exchangeable at rate \(\pi \) without transaction costs. We then have where \(\mathtt{u}\) is the numerical superlinear expectation with the representing set In particular, if \(\beta =0\) a.s., then the random set \(H_{\eta }(0)\) describes all portfolios available at price zero for two currencies with the exchange rate \(\pi \), and Hence with \(w'=(1,\mathtt{e}(\pi ))\) and \(w''=(1,\mathtt{u}(\pi ))\) for the exact dual pair \(\mathtt{e}\) and \(\mathtt{u}\) of nonlinear expectations with the representing set ℳ.

A map is a scalarly upper semicontinuous normalised reduced maximal superlinear expectation if and only if for a collection of nontrivial convex \(\sigma (L^{q},L^{p})\)-closed cones \(\mathcal{M}_{v}\subseteq L^{q}(\mathbb{R}_{+})\) parametrised by \(v\in \mathbb{S}^{d-1}\cap C^{o} \).

Let be a scalarly upper semicontinuous law-invariant normalised reduced maximal superlinear expectation, and let the probability space be non-atomic. Then is dilatation-monotonic, meaning that for each sub-\(\sigma \)-algebra \(\mathfrak{H}\subseteq \mathfrak{F}\) and all \(X\in L^{p}(\operatorname{co}\mathcal{F}(\mathbb{R}^{d},C))\), In particular, .

Since \(\mathtt{u}_{v}(\beta )\) given by (6.4) is a law-invariant concave function of \(\beta \in L^{p}(\mathbb{R})\) and the probability space is non-atomic, it is dilatation-monotonic, meaning that \(\mathtt{u}_{v}(\mathbb{E}[\xi |\mathfrak{H}])\geq \mathtt{u}_{v}(\xi )\); see Föllmer and Schied [10, Corollary 4.59]. Hence, Thus the infimum on the right-hand side of (6.3) written for is dominated by the infimum corresponding to . This implies the inclusion of the two sets. □

If \(\mathcal{M}_{v}=\mathcal{M}\) in (6.3) is nontrivial and does not depend on \(v\), then (6.3) turns into where \(\mathtt{u}\) given by (6.4) is the numerical superlinear expectation with the representing set ℳ. In this case, is the largest convex set whose support function is dominated by \(\mathtt{u}(h(X,v))\), that is, Note that \(\mathtt{u}(h(X,\cdot ))\) may fail to be a support function. The left-hand side of (6.5) is the scalarisation of the superlinear expectation ; cf. Hamel et al. [13, 14]. Since for \(X\in L^{p}(\operatorname{co}\mathcal{F}(\mathbb{R}^{d},C))\), this reduced maximal superlinear expectation admits an equivalent representation as

Let \(X=\xi +C\) for a \(\xi \in L^{p}(\mathbb{R}^{d})\) and a deterministic convex closed cone \(C\) that is different from the whole space. Then If \(\mathcal{M}_{v}=\mathcal{M}\) for all \(v\in \mathbb{S}^{d-1}\cap C^{o} \), then \(\mathtt{u}_{v}=\mathtt{u}\) and A cone \(C\) is said to be a Riesz cone (or lattice cone) if \(\mathbb{R}^{d}\) with the partial order generated by \(C\) is a Riesz space (or a vector lattice), that is, the maximum of any two points from \(\mathbb{R}^{d}\) is well defined. If this is the case, then for some \(x\), since an intersection of translations of \(C\) is again a translation of \(C\); see Aliprantis and Tourky [1, Theorem 1.16].

Let for \(n\) independent copies of \(X\), noticing that the expectation is empty if the intersection \(X_{1}\cap \cdots \cap X_{n}\) is empty with positive probability. This superlinear expectation is not a reduced maximal one. For instance, so that the reduced maximal extension is the largest convex set whose support function is dominated by , \(v\in \mathbb{S}^{d-1}\). However, the support function of \(\mathbb{E}[X_{1}\cap \cdots \cap X_{n}]\) is the expectation of the largest sublinear function dominated by \(\min (h(X_{i},v), i=1,\dots ,N)\), and so may be a strict subset of .

For instance, let \(X=\xi +\mathbb{R}_{-}^{d}\) for \(\xi \in L^{p}(\mathbb{R}^{d})\). Then where the minimum is applied coordinatewise to independent copies of \(\xi \), while is the largest convex set whose support function is dominated by the function \(v \mapsto \mathbb{E}[\min (\langle \xi _{i},v\rangle ,i=1,\dots ,n)]\) for \(v\in \mathbb{R}_{+}^{d}\). Obviously, with a possibly strict inequality.

Let be a normalised superlinear expectation satisfying the conditions of Proposition 4.2. Then for \(\xi \in L^{p}(\mathbb{R}^{d})\) only if for all \(v\in \mathbb{S}^{d-1}\).

By a variant of Proposition 4.2 for the reduced maximal extension, this extension satisfies the conditions of Theorem 6.4 and hence admits the representation (6.3). If \(\xi \in L^{p}(\mathbb{R}^{d})\), then (6.3) yields that which is not empty only if (6.7) holds. □

Let be a scalarly upper semicontinuous law-invariant normalised reduced maximal superlinear expectation, and let the probability space be non-atomic. Then the minimal extension given by (6.8) is a law-invariant superlinear expectation.

Let \(x\) and \(x'\) belong to the union on the right-hand side of (6.8) (without closure). Then and for \(\xi ,\xi '\in L^{p}(X)\), and the superlinearity of yields that for each \(t\in [0,1]\). Since \(t\xi +(1-t)\xi '\) is a selection of \(X\), the convexity of easily follows. Additivity on deterministic singletons, monotonicity and homogeneity are evident from (6.8). If \(F\in \operatorname{co}\mathcal{F}(\mathbb{R}^{d},C)\) is deterministic, the dilatation-monotonicity of (see Corollary 6.5) yields that For the superadditivity property, consider \(x\) and \(y\) from the nonclosed right-hand side of (6.8) for \(X\) and \(Y\), respectively. Then for some \(\xi \in L^{p}(X)\) and for some \(\xi '\in L^{p}(Y)\). Hence, Finally, let \(\mathfrak{F}_{X}\) be the \(\sigma \)-algebra generated by \(X\), that is, \(\mathfrak{F}_{X}\) is generated by the events \(\{X\cap K\neq \varnothing \}\) for all compact sets \(K\) in \(\mathbb{R}^{d}\). The convexity of \(X\) implies that \(\mathbb{E}[\xi |\mathfrak{F}_{X}]\) is a selection of \(X\) for any \(\xi \in L^{p}(X)\). By the dilatation-monotonicity from Corollary 6.5, it is possible to replace \(\xi \in L^{p}(X)\) in (6.8) with an \(\mathfrak{F}_{X}\)-measurable \(p\)-integrable selection of \(X\). The families of \(\mathfrak{F}_{X}\)-measurable selections of \(X\) and \(\mathfrak{F}_{Y}\)-measurable selections of \(Y\) coincide for two identically distributed random sets \(X\) and \(Y\); see Molchanov [25, Proposition 1.4.5]. □

Assume that \(p\in (1,\infty ]\), satisfies the conditions imposed in Theorem 6.10, and that for all nontrivial \(\xi \in L^{p}(C)\). Then the minimal extension is scalarly upper semicontinuous.

It suffices to omit the closure in (6.8) and consider with \(x_{n}\to x\) and \(X_{n}\to X\) scalarly in \(\sigma (L^{p},L^{q})\). For each \(n\in \mathbb{N}\), there exists a \(\xi _{n}\in L^{p}(X_{n})\) such that .

Assume first that \(p\in (1,\infty )\) and \(\sup _{n\in \mathbb{N}}\mathbb{E}[|\xi _{n}|^{p}]<\infty \). Then \((\xi _{n})_{n\in \mathbb{N}}\) is relatively compact in \(\sigma (L^{p},L^{q})\). Without loss of generality (if necessary, passing to subsequences), assume that \((\xi _{n})\) converges to \(\xi \) in \(\sigma (L^{p},L^{q})\). Since \(\langle \xi _{n},\zeta \rangle \leq h(X_{n},\zeta )\) for all \(\zeta \in L^{q}( C^{o} )\), taking expectations, letting \(n\to \infty \) and using the convergence \(\xi _{n}\to \xi \) and \(X_{n}\to X\) yields that \(\mathbb{E}[h(\xi ,\zeta )]\leq \mathbb{E}[h(X,\zeta )]\). By Lemma 2.4, \(\xi \) is a selection of \(X\). By the upper semicontinuity of , the upper limit of is a subset of . Hence for some \(\xi \in L^{p}(X)\) so that .

Assume now that \(\|\xi _{n}\|_{p}^{p}=\mathbb{E}[| \xi _{n} |^{p}]\to \infty \). Let \(\xi '_{n}=\xi _{n}/\|\xi _{n}\|_{p}\). This sequence is bounded in the \(L^{p}\)-norm, and so we can assume without loss of generality that \(\xi '_{n}\to \xi '\) in \(\sigma (L^{p},L^{q})\). Since the upper semicontinuity of yields that . For each \(\zeta \in L^{q}( C^{o} )\), we have \(\langle \xi _{n},\zeta \rangle \leq h(X_{n},\zeta )\). Dividing by \(\|\xi _{n}\|_{p}\), taking expectations and letting \(n\to \infty \) yields that \(\mathbb{E}[\langle \xi ',\zeta \rangle ]\leq 0\). Thus \(\xi '\in C\) almost surely. Given that \(\mathbb{E}[\|\xi '\|]=1\), this contradicts the fact that contains the origin.

The proof for \(p=\infty \) follows the exact same steps, splitting the cases when \(\sup _{n\in \mathbb{N}} | \xi _{n} |\) is essentially bounded (in which case the sequence is relatively compact in \(\sigma (L^{\infty },L^{1})\)) and when the essential supremum of \((\xi _{n})\) converges to infinity. □

Consider \(\xi \in L^{p}(\mathbb{R}^{d})\) and a deterministic \(F\in \operatorname{co}\mathcal{F}(\mathbb{R}^{d},C)\). Assume that in (6.8) satisfies the conditions of Corollary 6.5. Then where \(L^{p}(F,\sigma (\xi ))\) is the family of selections of \(F\) which are measurable with respect to the \(\sigma \)-algebra \(\sigma (\xi )\) generated by \(\xi \). Indeed, for each \(\xi '\in L^{p}(F)\), the dilatation-monotonicity of yields that . It remains to note that \(\mathbb{E}[\xi '|\sigma (\xi )]\in L^{p}(F,\sigma (\xi ))\).

Assume that is defined by (6.8), where is a scalarly upper semicontinuous reduced maximal superlinear expectation with representation (6.6). Then for all \(v\in \mathbb{S}^{d-1}\cap C^{o} \) and \(\beta \in L^{p}(\mathbb{R})\), and the reduced maximal extension of coincides with .

By (6.3), . In view of (6.9), it suffices to show that each \(x\in H_{v}(\mathtt{u}(\beta ))\) also belongs to . Let \(y\) be the projection of \(x\) onto the subspace orthogonal to \(v\). It suffices to show that . Noticing that \(H_{v}(\beta )-y=H_{v}(\beta )\), it is possible to assume that \(x=tv\) for \(t\leq \mathtt{u}(\beta )\). Consider \(\xi =\beta v\). Then Since \(\langle tv,w\rangle \leq \langle v,w\rangle \mathtt{u}(\beta )\), we deduce that . Since and coincide on half-spaces, the reduced maximal extension of is  □

Consider \(\xi \in \mathbb{R}^{2}\) which takes with equal probabilities two possible values: the origin and \(a=(a_{1},a_{2})\). Let \(X=\xi +C\), where \(C\) is the cone containing \(\mathbb{R}_{-}^{2}\) and with points \((1,-\pi )\) and \((-\pi ',1)\) on its boundary such that \(\pi ,\pi '>1\).

Let \(\mathcal{M}_{v}=\mathcal{M}\) be the family from Example 5.15 and let \(\mathtt{u}\) be the superlinear expectation with the representing set ℳ. For each \(\beta \in L^{1}(\mathbb{R})\), \(\mathtt{u}(\beta )\) equals the average of the \(t\)-quantiles of \(\beta \) over \(t\in (0,\alpha )\). If \(\alpha \in (0,1/2]\) and \(\beta \) takes two values with equal probabilities, then \(\mathtt{u}(\beta )\) is the smaller value of \(\beta \). Then so that coincides with in this case.

Now assume that \(\alpha \in (1/2,1)\). If \(\beta \) with equal probabilities takes two values \(t\) and \(s\), then \(\mathtt{u}(\beta )=\max (t,s)-|t-s|/(2\alpha )\) and for all \(v\) from \(C^{o}\). Since \(C\) is a Riesz cone, for some \(x\); see Example 6.7. For \(v\in C^{o} \), the linear function \(x \mapsto \langle x,v\rangle \) is dominated by \(\frac{1}{2\alpha }\langle a,v\rangle \) if \(\langle a,v\rangle <0\) and by \((1-\frac{1}{2\alpha })\langle a,v\rangle \) otherwise. By an elementary calculation, In view of Example 6.12, for the minimal extension, it suffices to consider selections of \(C\) which are measurable with respect to the \(\sigma \)-algebra \(\sigma (\xi )\) generated by \(\xi \); these selections take two values from the boundary of \(C\) with equal probabilities. The minimal extension can be found via (6.10), letting \(\xi '\) with equal probabilities take two values \(y=(y_{1},y_{2})\) and \(z=(z_{1},z_{2})\) on the boundary \(\partial C\) of \(C\). Then Figure 1 shows and for \(\pi =\pi '=2\), \(a=(1,-1)\) and \(\alpha =0.7\). It shows that the minimal extension may indeed be a strict subset of the underlying reduced maximal superlinear expectation.

Fix \(\alpha \in (0,1)\). For \(\beta \in L^{1}(\mathbb{R})\), define where \(q_{\beta }(s)\) is an \(s\)-quantile of \(\beta \) (in case of nonuniqueness, the choice of a particular quantile does not matter because of integration). The risk measure \(r(\beta )=\mathtt{e}_{\alpha }(-\beta )\) is called the average value-at-risk. Denote by the corresponding minimal sublinear expectation constructed by (5.6), so that for all \(u\). The set is the zonoid-trimmed region of \(\xi \) at level \(\alpha \); see Cascos [5] and Mosler [28, Sect. 3.1]. This set can be obtained as where \(\mathcal{P}_{\alpha }\subseteq L^{1}(\mathbb{R}_{+})\) consists of all random variables with values in \([0,\alpha ^{-1}]\) and expectation 1; see Example 5.15. This setting is a special case of Theorem 5.12 with \(\mathcal{M}=\{t\gamma : \gamma \in \mathcal{P}_{\alpha },t\geq 0\}\). The value of \(\alpha \) controls the size of the zonoid-trimmed region; \(\alpha =1\) yields a single point, being the expectation of \(\xi \). The subadditivity property of zonoid-trimmed regions was first noticed by Cascos and Molchanov [6].

Let \(X\) be an integrable random closed convex set. Consider the random set \(Y\) in \(\mathbb{R}^{d+1}\) given by the convex hull of the origin and \(\{1\}\times X\). The selection expectation \(Z_{X}=\mathbb{E}[Y]\) is called the lift expectation of \(X\); see Diaye et al. [8]. If \(X=\{\xi \}\) is a singleton, then \(Z_{X}\) is the lift zonoid of \(\xi \); see Mosler [28, Sect. 2.2]. By the definition of the selection expectation, \(Z_{X}\) is the closure of the set of \((\mathbb{E}[\beta ],\mathbb{E}[\beta \xi ])\), where \(\beta \) runs through the family of random variables with values in \([0,1]\). Equivalently, \((\alpha ,x)\) belongs to \(Z_{X}\) if and only if \(x=\alpha \mathbb{E}[\gamma \xi ]\) for \(\gamma \) from the family \(\mathcal{P}_{\alpha }\); see Example 7.1. Thus the minimal extension of from Example 7.1 is .

Choosing in (7.1) and (7.2) yields a family of nonlinear expectations depending on the parameter \(n\), which are also easy to compute.