Abstract
We provide a characterization in terms of Fatou closedness for weakly closed monotone convex sets in the space of \({\mathcal P}\)quasisure bounded random variables, where \({\mathcal P}\) is a (possibly nondominated) class of probability measures. Applications of our results lie within robust versions the Fundamental Theorem of Asset Pricing or dual representation of convex risk measures.
Keywords
Capacities Fatou closedness/property Sequential order closedness Convex duality under model uncertainty Fundamental Theorem of Asset PricingMathematics Subject Classification
31A15 46A20 46E30 60A99 91B301 Introduction
A fundamental result attributed to Grothendieck ([22, p. 321, Exercise 1]) and based on the Krein–Smulian theorem characterizes weak*closedness of a convex subset of \(L_P^\infty :=L^\infty (\Omega , {\mathcal F},P)\), where \((\Omega ,{\mathcal F},P)\) is a probability space, by means of a property called Fatou closedness as follows:
Theorem 1.1
 (i)
\({\mathcal A}\) is weak*closed (i.e. closed in \(\sigma (L_P^\infty , L_P^1)\)).
 (ii)
\({\mathcal A}\) is Fatou closed, i.e. if \((X_n)_{n\in {\mathbb N}}\subset {\mathcal A}\) is a bounded sequence which converges Palmost surely to X, then \(X\in {\mathcal A}\).
Note that \(L_P^\infty \) is a Banach lattice (see Sect. 2) and that from this point of view property (ii) in Theorem 1.1 equals sequential order closedness of \({\mathcal A}\) which in fact implies order closedness since \(L_P^\infty \) has the countable sup property, i.e. every nonempty subset possessing a supremum contains a countable subset possessing the same supremum. Theorem 1.1 is very useful and often applied in the mathematical finance literature such as in the classic proof of the Fundamental Theorem of Asset Pricing, see e.g. [12] or [13], or in the dual representation of convex risk functions, see e.g. [19]. In all cases the problem is that the norm dual of \(L_P^\infty \) contains undesired singular elements, whereas in the weak*duality \((L_P^\infty ,\sigma (L_P^\infty ,L_P^1))\) the elements of the dual space are identified with \(\sigma \)additive measures. However, as the weak*topology is generally not firstcountable, verifying that some set is weak*closed is typically quite challenging. This is where Theorem 1.1 proves helpful.
The aim of this paper is to study the existence of a version of Theorem 1.1 for the case when the probability measure P is replaced by a class \({\mathcal P}\) of probability measures on \((\Omega ,{\mathcal F})\). In general this class \({\mathcal P}\) does not allow for a dominating probability. Applications of such a result lie for instance in the field of mathematical finance, where currently there is much attention paid to deriving versions of the Fundamental Theorem of Asset Pricing as well as dual representations of convex risk functions in socalled robust frameworks as studied in [4, 6, 7, 8, 26, 28]. These kind of frameworks have become increasingly popular to describe a decision maker who has to deal with the uncertainty which arises from model ambiguity. Here the class of probability models \({\mathcal P}\) the decision maker takes into account represents her degree of ambiguity about the right probabilistic model. If \({\mathcal P}=\{P\}\) there is no ambiguity. In many studies which account for model ambiguity \({\mathcal P}\) in fact turns out to be a nondominated class of probability measures, see [6, 7, 8, 26] and the reference therein.
 (WC)

\({\mathcal A}\subset L^\infty _c\) is \(\sigma (L^\infty _c,ca_c)\)closed,
 (FC)

\({\mathcal A}\subset L^\infty _c\) is Fatou closed: for any bounded sequence \(\{X_n\}\subset {\mathcal A}\) and \(X\in L^\infty _c\) such that \(X_n\rightarrow X\) \({\mathcal P}\)quasi surely we have that \(X\in {\mathcal A}\),
Another requirement which is crucial for our proof of (WC) \(\Leftrightarrow \) (FC) is that the dual space of \(ca_c\) may be identified with \(L^\infty _c\). This condition is in fact equivalent to the order completeness of the Banach lattice \(L^\infty _c\), i.e. the existence of a supremum for any bounded subset of \(L^\infty _c\), see Proposition 3.10, and it thus corresponds to aggregation type results as in [11, 27]. If \(L^\infty _c\) is order complete, then the property (FC) equals sequential order closedness of \({\mathcal A}\). However, order completeness does not imply that \(L^\infty _c\) possesses the countable sup property, see Examples 3.11 and 3.12, so even under this condition (FC) does in general not imply order closedness of \({\mathcal A}\).
We also provide a counter example showing that for nondominated \({\mathcal P}\) there is no proof of (WC) \(\Leftrightarrow \) (FC) without further requirements such as \({\mathcal P}\)sensitivity, see Example 3.4. Moreover, we illustrate that many conditions, in particular on \({\mathcal P}\), one would think of in the first place to ensure (WC) \(\Leftrightarrow \) (FC), indeed imply that \({\mathcal P}\) is dominated, so we are back to Theorem 1.1. Hence, a further contribution of this paper is to provide a deeper insight into the fallacies one might encounter when attempting to extend Theorem 1.1 to a robust case.
The paper is structured as follows: Sect. 2 provides a list of useful notations which will be used throughout the paper. Section 3 contains the main results of the paper, and in particular Theorem 3.9 is the robust version of Theorem 1.1. Finally, applications of Theorem 3.9 in the field of mathematical finance are collected in Sect. 4. Here we do not assume that the reader is familiar with mathematical finance. However, we try to keep the presentation concise, referring to the relevant literature for more background information.
2 Notation
For the sake of clarity we propose here a list of the basic notations and definitions that we shall use throughout this paper.
 (i)
\(ba:=\{\mu :{\mathcal F}\rightarrow {\mathbb R}\mid \mu \text{ is } \text{ finitely } \text{ additive }\}\) and \(ca:=\{\mu :{\mathcal F}\rightarrow {\mathbb R}\mid \mu { is}\sigma \text{additive }\}\). These are both Banach lattices once endowed with the total variation norm TV and \(\mu =\mu ^++\mu ^\) where \(\mu =\mu ^+\mu ^\) is the Jordan decomposition (see [1] for further details).
 (ii)
\(ba_+\) (resp. \(ca_+\)) is the set of all positive additive (resp. \(\sigma \)additive) set functions on \((\Omega ,{\mathcal F})\).
 (iii)In absence of any reference probability measure we have the following sets of random variablesIn particular \({\mathcal L}^\infty \) is a Banach space under the (pointwise) supremum norm \(\Vert \cdot \Vert _{\infty }\) with dual space ba.$$\begin{aligned} {\mathcal L}:= & {} \{f:\Omega \rightarrow {\mathbb R}\mid f\, \text{ is }\, {\mathcal F}\text{measurable }\}, \\ {\mathcal L}_+:= & {} \{f\in {\mathcal L}\mid f(\omega )\ge 0, \forall \, \omega \in \Omega \}, \\ {\mathcal L}^\infty:= & {} \{f\in {\mathcal L}\mid \text{ f } \text{ is } \text{ bounded }\}. \end{aligned}$$
 (iv)
\({\mathcal M}_1\subset ca_+\) is the set of all probability measures on \((\Omega ,{\mathcal F})\).
 (v)
Throughout this paper we fix a set of probability measures \({\mathcal P}\subset {\mathcal M}_1\).
 (vi)We introduce the sublinear expectationand by some abuse of notation we define the capacity \(c(A):=c(1_A)\) for \(A\in {\mathcal F}\).$$\begin{aligned} c(f):=\sup _{Q\in {\mathcal P}}E_Q[f], \quad f\in {\mathcal L}_+ \end{aligned}$$
 (vii)Let \({\widehat{{\mathcal P}}},{\widetilde{{\mathcal P}}}\subset {\mathcal M}_1\). \({\widehat{{\mathcal P}}}\) dominates \({\widetilde{{\mathcal P}}}\), denoted by \({\widetilde{{\mathcal P}}} \ll {\widehat{{\mathcal P}}} \), if for all \(A\in {\mathcal F}\):We say that two classes \({\widehat{{\mathcal P}}}\) and \({\widetilde{{\mathcal P}}}\) are equivalent, denoted by \({\widehat{{\mathcal P}}} \thickapprox {\widetilde{{\mathcal P}}}\), if \({\widetilde{{\mathcal P}}}\ll {\widehat{{\mathcal P}}} \) and \({\widehat{{\mathcal P}}}\ll {\widetilde{{\mathcal P}}}\).$$\begin{aligned} \sup _{P\in {\widehat{{\mathcal P}}}} P(A)=0 \quad \Rightarrow \quad \sup _{P\in {\widetilde{{\mathcal P}}}} P(A)=0 . \end{aligned}$$
 (viii)
A statement holds \({\mathcal P}\)quasi surely (q.s.) if the statement holds Qalmost surely (a.s.) for any \(Q\in {\mathcal P}\).
 (ix)
The space of finitely additive (resp. countably additive) set functions dominated by c is given by \(ba_c=\{\mu \in ba \mid \mu \ll c \}\) (resp. \(ca_c=\{\mu \in ca \mid \mu \ll c \}\)). Here \(\mu \ll c\) means: \(c(A)=0\) for some \(A\in {\mathcal F}\) implies \(\mu (A)=0\). When \({\mathcal P}=\{Q\}\) we shall write \(ba_Q\) or \(ca_Q\) for the sake of simplicity.
 (x)We consider the quotient space \(L_c:={\mathcal L}_{/\sim }\) where the equivalence is given byWe shall use capital letters to distinguish equivalence classes of random variables \(X\in L_c\) from a representative \(f\in X\), with \(f\in {\mathcal L}\). In case \({\mathcal P}=\{Q\}\) we shall write \(L_Q^1\) instead of \(L_c\). It is a wellknown consequence of the RadonNikodym theorem ([1, Theorem 13.18]) that \(ca_Q\) may be identified with \(L_Q^1\).$$\begin{aligned} f\sim g\quad \Leftrightarrow \quad \forall P\in {\mathcal P}: P(f=g)=1 . \end{aligned}$$
 (xi)
For any \(f,g\in {\mathcal L}\) and \(P\in {\mathcal M}_1 \), we write \(f\le g\) Pa.s. if and only if \(P(f\le g)=1\). Similarly \(f\le g\) \({\mathcal P}\)q.s. if and only if \(f\le g\) Pa.s. for all \(P\in {\mathcal P}\). This relation is a partial order on \({\mathcal L}\) and it also induces a partial order on \(L_c\) where \(X\le Y\) for \(X,Y\in L_c\) if and only if \(f\le g\) \({\mathcal P}\)q.s. for any \(f\in X\) and \(g\in Y\).
 (xii)We define \(L_c^\infty :={\mathcal L}^\infty _{/\sim }\) and endow this space with the norm \((L_c^\infty , \Vert \cdot \Vert _{c,\infty })\) is a Banach lattice with the same partial order \(\le \) as on \(L_c\). Its norm dual is \(ba_c\). In case \({\mathcal P}=\{Q\}\) we shall write \(L_Q^\infty \) and \(\Vert \cdot \Vert _{Q,\infty }\) for the sake of simplicity. Note that \(\Vert \cdot \Vert _{c,\infty }\) is never order continuous for any choice of \({\mathcal P}\).
3 Towards a robust version of Theorem 1.1
Theorem 3.1
 (i)
\({\mathcal Z}=\emptyset \).
 (ii)
\(M_c^*\subset ca\).
 (iii)
\(H_c^*\subset ca\).
In particular, if \({\mathcal Z}=\emptyset \), then there exists a countable subset \({\widetilde{{\mathcal P}}}\subset {\mathcal P}\) such that \({\widetilde{{\mathcal P}}}\thickapprox {\mathcal P}\), and thus there is a probability measure \(Q\in {\mathcal M}_1\) such that \(\{Q\}\thickapprox {\mathcal P}\).
Proof
(ii) \(\Rightarrow \) (i) and (iii) \(\Rightarrow \) (i) follow directly from Proposition A.2
The last statement of this theorem is Proposition A.1. \(\square \)
Remark 3.2
Note that \({\mathcal Z}=\emptyset \) is equivalent to sequential order continuity of \(\Vert \cdot \Vert _c\). According to Theorem 3.1, if \({\mathcal P}\) is not dominated, then \({\mathcal Z}\ne \emptyset \) and hence the norm \(\Vert \cdot \Vert _c\) on \(M_c\) or \(H_c\) is not order continuous.
Also note that the converse of the last statement of Theorem 3.1 is not true, i.e. \({\mathcal Z}\ne \emptyset \) does not imply that \({\mathcal P}\) is not dominated. To see this, let \(A_n\downarrow \emptyset \) and pick a sequence of probability measures \(P_n\) such that \(P_n(A_n)=1\) for all \(n\in {\mathbb N}\), and let \({\mathcal P}=\{P_n\mid n\in {\mathbb N}\}\). Then, clearly \(\Vert 1_{A_n}\Vert _c=1\) for each n. Hence, \(\Vert \cdot \Vert _c\) is not order continuous and \({\mathcal Z}\ne \emptyset \) and thus \(M_c^*\not \subset ca\). However, we have that \(\{Q\}\thickapprox {\mathcal P}\) for \(Q=\sum _{n=1}^\infty \frac{1}{2^n}P_n\).
 (WC)

\({\mathcal A}\subset L^\infty _c\) is \(\sigma (L^\infty _c,ca_c)\)closed.
 (FC)

\({\mathcal A}\subset L^\infty _c\) is Fatou closed: for any bounded sequence \(X_n\subset {\mathcal A}\) and \(X\in L^\infty _c\) such that \(X_n\rightarrow X\) \({\mathcal P}\)q.s. we have that \(X\in {\mathcal A}\).
It is easily verified that always (WC) \(\Longrightarrow \) (FC) since any bounded \({\mathcal P}\)q.s. converging sequence also converges in \(\sigma (L^\infty _c,ca_c)\) to the same limit. However, there is in general no proof of (FC) \(\Longrightarrow \) (WC) even if \({\mathcal A}\) is convex, and also requiring monotonicity of \({\mathcal A}\), i.e. \({\mathcal A}+(L^\infty _c)_+={\mathcal A}\), in addition is not sufficient:
Theorem 3.3
Let \({\mathcal A}\subset L^\infty _c\) be convex and monotone. Without further assumptions on \({\mathcal P}\) or \({\mathcal A}\), there exists no proof of (FC) \(\Rightarrow \) (WC).
The proof of Theorem 3.3 is given by the following Example 3.4 where we give a counterexample of (FC) \(\Longrightarrow \) (WC) assuming the continuum hypothesis. So under the continuum hypothesis (FC) \(\Longrightarrow \) (WC) is indeed wrong. Note that as the continuum hypothesis does not conflict with what one perceives as standard mathematical axioms, there is of course no way to prove (FC) \(\Longrightarrow \) (WC) even if we do not believe in the continuum hypothesis.
Example 3.4
In order to make the presentation simpler, we did not require monotonicity of \({\mathcal A}\) so far, but the same arguments as above show that if \({\mathcal A}\) is the convex closure of \(C+(L^\infty _c)_+\) under bounded \({\mathcal P}\)q.s. convergence of sequences, which is convex and monotone, then \(1\not \in {\mathcal A}\) but \(1\) is an element of the \(\sigma (L^\infty _c,ca_c)\)closure of \({\mathcal A}\).
A consequence of Theorem 3.3 is that we need to ask for additional properties on \({\mathcal A}\) in order to have (FC) \(\Longleftrightarrow \) (WC).
3.1 \({\mathcal P}\)sensitivity, \(ca_c^*=L^\infty _c\), and (FC) \(\Longleftrightarrow \) (WC)
A simple property on \({\mathcal A}\) which allows to prove (FC) \(\Longleftrightarrow \) (WC) is to require that the convex set \({\mathcal A}\subset L^\infty _c\) behaves as in the dominated case, i.e. there is a reference probability \(P\in {\mathcal P}\) such that \({\mathcal A}\) is closed under bounded Pa.s. convergence. Under this assumption the whole issue can be reduced to Theorem 1.1. Clearly, this assumption is too strong. However, it gives the idea of the \({\mathcal P}\)sensitivity property we will introduce in the following.
Given a probability \(Q\in {\mathcal M}_1\) such that \(\{Q\}\ll {\mathcal P}\) we define the linear map \(j_Q:L^{\infty }_c \rightarrow L^{\infty }_Q\) by \( Q(j_Q(X)=X)=1, \) i.e. \(j_Q(X)\) is the equivalence class in \(L^\infty _Q\) such that any representative of \(j_Q(X)\) and any representative of X are Qa.s. identical. As \(ca_Q\) (which can be identified with \(L^1_Q\)) is a subset of \(ca_c\), we deduce that \(j_Q: (L^\infty _c,\sigma (L^\infty _c, ca_c))\rightarrow (L^ \infty _Q,\sigma (L^ \infty _Q,L^1_Q))\) is continuous.
Definition 3.5
Remark 3.6
Suppose that \({\mathcal P}\) is dominated. Then the Halmos Savage lemma (see [23], Lemma 7) guarantees the existence of a countable subclass \(\{P_i\}_{i=1}^{\infty }\) such that \(\{P_i\}_{i=1}^{\infty }\thickapprox {\mathcal P}\). Let \(P=\sum \frac{1}{2^i}P_i\). Then \({\mathcal P}\thickapprox \{P\}\), so the space \(L^{\infty }_c\) can be identified with \(L^{\infty }_{ P}\). Hence, in that case any set \({\mathcal A}\subset L^\infty _c\) is automatically \({\mathcal P}\)sensitive with reduction set \({\mathcal Q}=\{P\}\).
Example 3.7
The set \({\mathcal A}\) of Example 3.4 is not \({\mathcal P}\)sensitive. Since \(c(A)=0\) implies that \(A=\emptyset \), any set of probabilities \({\mathcal Q}\subset {\mathcal P}\) satisfies \({\mathcal Q}\ll {\mathcal P}\). Let \(Q\in {\mathcal M}_1\) be arbitrary and \(S:=\{\omega \in [0,1]\mid Q(\{\omega \})>0\}\) such that \(Q=\sum _{\omega \in S}a_\omega \delta _{\omega }\) with \(a_\omega >0\) and \(\sum _{\omega \in S}a_\omega =1\). Then \(1_S\in {\mathcal A}\) by definition of \({\mathcal A}\) and thus \(1\in j_Q({\mathcal A})\), or to be more precise, 1 and \(1_S\) form the same equivalence class in \(L^\infty _Q\). Since \(Q\in {\mathcal M}_1\) was arbitrary, we have \(1\in \bigcap _{Q\in {\mathcal Q}}j_Q^{1}\circ j_Q({\mathcal A})\). As we know that \(1\not \in {\mathcal A}\), the set \({\mathcal A}\) is not \({\mathcal P}\)sensitive.
Indeed \({\mathcal P}\)sensitivity is a necessary condition for (FC) \(\Longleftrightarrow \) (WC).
Proposition 3.8
Any convex set \({\mathcal A}\subset L^\infty _c\) which is \(\sigma (L^\infty _c,ca_c)\)closed (i.e. satisfies (WC)) is \({\mathcal P}\)sensitive.
Proof
Theorem 3.9
 (i)
\({\mathcal A}\) satisfies (WC).
 (ii)
\({\mathcal A}\) is \({\mathcal P}\)sensitive and satisfies (FC).
Proof
We already know that (WC) implies (FC) and \({\mathcal P}\)sensitivity. Now assume that \({\mathcal A}\) is \({\mathcal P}\) sensitive and satisfies (FC). Since \(ca_c^*=L^\infty _c\), by the Krein–Smulian theorem it is sufficient to show that \(C_K:={\mathcal A}\cap \{Z\in L^\infty \mid \Vert Z\Vert _{c,\infty }\le K\}\) is \(\sigma (L^\infty _c,ca_c)\)closed for every \(K>0\). Let \({\mathcal Q}\) be a reduction set for \(({\mathcal A},{\mathcal P})\) and fix any \(K>0\) and \(Q\in {\mathcal Q}\).
Note that Theorem 3.9 proves the socalled Cproperty introduced and discussed in [5] for convex and monotone sets.
Let \({\mathcal D}\subset L^\infty _c\). Recall that a supremum of \({\mathcal D}\) is a least upper bound of \({\mathcal D}\), that is an \(X\in L^\infty _c\) such that \(Y\le X\) for all \(Y\in {\mathcal D}\), and any \(Z\in L^\infty _c\) such \(Y\le Z\) for all \(Y\in {\mathcal D}\) satisfies \(X\le Z\). The supremum of \({\mathcal D}\) is denoted by \(\hbox {ess sup}_{Y\in {\mathcal D}}Y\). This notation is commonly used in probability theory and it is inspired by the tradition of identifying random variables with the equivalence classes they induce. Indeed for a set of random variables in \({\mathcal L}^\infty \), a supremum in the \({\mathcal P}\)q.s. order is only essentially unique—thus called essential supremum (\(\hbox {ess sup}\))—in the sense that the equivalence class generated by it in \(L^\infty _c\) is unique.
Proposition 3.10
\(ca_c^*=L^\infty _c\) if and only if \(L^\infty _c\) is order complete, i.e. there exists a supremum for any norm bounded set \({\mathcal D}\subset L^\infty _c\).
Proof
Suppose that \(L^\infty _c\) is order complete. Then \(L^\infty _c\) is in particular also monotonically complete in the sense of [25, Definition 2.4.18]. Thus [25, Theorem 2.4.22] applies which yields \(ca_c^*=L^\infty _c\).
Example 3.11
Example 3.12
Recall Example 3.4. Clearly any norm bounded set \({\mathcal D}\subset L^\infty _c={\mathcal L}^\infty \) admits an essential supremum which is simply given by \(\omega \mapsto \sup _{Y\in {\mathcal D}}Y(\omega )\). Hence \(ca^*=ca_c^*={\mathcal L}^\infty \) by Proposition 3.10. This holds without the continuum hypothesis, but is also easily directly verified using the continuum hypothesis: Let \(l\in ca_c^*\) and define \(X(\omega )=l(\delta _\omega )\), \(\omega \in [0,1]\). Then by linearity, for all \(\mu \in ca\) it follows that \(l(\mu )=\sum _{\omega \in S}a_\omega l(\delta _\omega )=\int X\, d\mu \) where \(S:=\{\omega \in [0,1]\mid \mu (\{\omega \})>0\}\) and \(a_\omega =\mu (\{\omega \})\), \(\omega \in S\). Moreover, it is also readily verified that in this case \(L^\infty _c\) does not have the countable sup property.
4 Applications of Theorem 3.9
4.1 Dual representation of (quasi) convex increasing functionals
In this section we provide a dual representation of (quasi) convex increasing functionals. Such results are key in the study of robustness of financial risk measures. An exhaustive introduction to the dual representation of convex risk measures can be found in [19] (see also [15] for the quasiconvex case and [10] for recent developments). To the best of our knowledge, in presence of model uncertainty, the only result available in the literature is [6, Theorem 3.1] which is obtained for the closure of the space of continuous functions under the norm \(\Vert \cdot \Vert _c\).
Definition 4.1

quasiconvex (resp. convex) if for every \(\lambda \in [0,1]\) and \(X,Y\in L^{\infty }\) we have \(f(\lambda X+(1\lambda )Y)\le \max \{X,Y\}\) (resp. \(f(\lambda X+(1\lambda )Y)\le \lambda f(X)+ (1\lambda )f(Y)\)).

\(\tau \)lower semicontinuous (l.s.c.) for some topology \(\tau \) on \(L^{\infty }_c\) if for every \(a\in {\mathbb R}\) the lower level set \(\{X\in L^{\infty }_c\mid f(X)\le a\}\) is \(\tau \)closed.

\({\mathcal P}\)sensitive if the lower level sets \(\{X\in L^{\infty }_c\mid f(X)\le a\}\) are \({\mathcal P}\)sensitive for every \(a\in {\mathbb R}\).
The following Lemma provides a huge class of \({\mathcal P}\)sensitive functions.
Lemma 4.2
Proof
Theorem 4.3
 (i)
f is \(\sigma (L^\infty _c,ca_c)\)lower semi continuous.
 (ii)
f has the Fatou property: for any bounded sequence \((X_n)_{n\in {\mathbb N}}\subset L^\infty _c\) converging \({\mathcal P}\)q.s. to \(X\in L^\infty _c\) we have \(f(X)\le \liminf _{n\rightarrow \infty } f(X_n)\).
 (iii)
For any sequence \((X_n)_{n\in {\mathbb N}}\subset {\mathcal A}\) and \(X\in L^\infty _c\) such that \(X_n\uparrow X\) \({\mathcal P}\)q.s. we have that \(f(X_n)\uparrow f(X)\).
 (iv)f admits a bidual representation which in the quasiconvex case iswith dual function \(R:{\mathbb R}\times ca_c\rightarrow (\infty ,\infty ]\) given by and in the convex case the dual representation is$$\begin{aligned} f(X)= & {} \sup _{P\in ca_c\cap {\mathcal M}_1}R\left( E_P[X],P\right) , \quad X\in L^\infty _c, \end{aligned}$$where the dual function \(f^{*}:ca_c\rightarrow (\infty ,\infty ]\)) is given by$$\begin{aligned} f(X)= & {} \sup _{\mu \in (ca_c)_+}\left\{ \int X\, d\mu f^*(\mu )\right\} , \quad X\in L^\infty _c, \end{aligned}$$$$\begin{aligned} f^*(\mu ):= & {} \sup _{Y\in L^\infty _c}\left\{ \int Y\, d\mu  f(Y)\right\} . \end{aligned}$$
Proof
According to Theorem 3.9 (i) holds if and only if (ii) is satisfied.
In the convex case \((i) \Leftrightarrow (iv)\) is Fenchel’s Theorem (see [18, Proposition 4.1]) together with monotonicity (see [21, Corollary 7]).
In the quasiconvex case showing \((i) \Rightarrow (iv)\) is a consequence of the PenotVolle duality Theorem (see Appendix B) and together with monotonicity (see [9, Lemma 8]), and \((iv) \Rightarrow (iii)\) follows from the monotone convergence theorem and the definition of R. \(\square \)
4.2 Fundamental Theorem of Asset Pricing
Pricing theory in mathematical finance is based on the Fundamental Theorem of Asset Pricing, which roughly asserts that in a market without arbitrage opportunities (the socalled noarbitrage condition) discounted prices are expectations under some riskneutral probability measure. This characterisation is essential to develop a pricing theory for financial instruments which are not traded in the market. In the classical dominated framework on some probability space \((\Omega ,{\mathcal F},P)\) the riskneutral probability measures are martingale measures for the discounted price process which are equivalent to the reference probability P, see [13] for a detailed review and related literature. Also note that the noarbitrage condition is necessary and sufficient the existence of an economic equilibrium, see e.g. [24].
It is well understood that the Fundamental Theorem of Asset Pricing in a classical dominated framework is highly related to duality arguments. There are also robust approaches applying duality, see e.g. [4] based on an extended order dual space, the socalled super order dual introduced in [3]. However, most recent studies of robust Fundamental Theorems of Asset Pricing do not use duality arguments given the difficulties we outlined in this paper, see e.g. [7]. However, under the conditions that we have derived in Sect. 3 we will see that it is possible to reconcile the Fundamental Theorem of Asset Pricing, the Superhedging Duality, and duality theory on the pair \((L^\infty _c, ca_c)\) using the wellknown arguments.
Definition 4.4
 NA(\({\mathcal P}\))

\((H\bullet S)_T\ge 0\) \({\mathcal P}\)q.s. implies \((H\bullet S)_T= 0\) \({\mathcal P}\)q.s..
Note that NA(\({\mathcal P}\)) is equivalent to \({\mathcal C}\cap (L^\infty _c)_+=\{0\}\).
Lemma 4.5
Under \(NA({\mathcal P})\) if \(\mathcal {C}\) is \({\mathcal P}\)sensitive then \(\mathcal {C}\) is \(\sigma (L^{\infty }_c,ca_c)\)closed.
Proof
[7, Theorem 2.2 ] shows that under \(NA({\mathcal P})\) the cone \({\mathcal C}\) is closed under \({\mathcal P}\)q.s. convergence of sequences and therefore \(\mathcal {C}\) satisfies (FC). We remark that [7, Theorem 2.2] holds in full generality without the product structure on the underlying probability space assumed in [7]. Therefore applying Theorem 3.9 we deduce that \(\mathcal {C}\) is \(\sigma (L^{\infty }_c,ca_c)\)closed. \(\square \)
Lemma 4.6
Proof
The proof is wellknown and straightforward, so we just give the basic arguments: indeed choose any \(Q\in \{ Q\ll {\mathcal P}\mid S \text { is a } Q\text {martingale} \}\), and let \(X\in \mathcal {C}\) and \(H\in \mathcal {H}\) such that \(X\le (H\bullet S)_T\) \({\mathcal P}\)q.s. Then \(E_Q[X]\le E_Q[(H\bullet S)_T]= (H\bullet S)_0=0\) since \(((H\bullet S)_t)_{t\in I}\) is a Qmartingale (using generalized conditional expectations, see [7, Appendix]). Thus \(Q\in \mathcal {C}_1^0\).
If \(Q\in \mathcal {C}_1^0\) then \(E_Q[(H\bullet S)_T]=0\) for any \(H\in \mathcal {H}\) and by choosing appropriate strategies in \(\mathcal {H}\) such as \(H_t^{j}=1_A\) for \(A\in {\mathcal F}_{t1}\), \(H_t^i=0\) for \(i\ne j\) and \(H_s=0\) for \(s\ne t\) one verifies that Q is a martingale measure for S.
Theorem 4.7
 (i)
\(NA({\mathcal P})\)
 (ii)
\(\mathcal {C}^0_1\approx {\mathcal P}\)
Proof
(i) \(\Rightarrow \) (ii): Clearly, \(c(A)=0\) implies \(\sup _{Q\in \mathcal {C}^0_1}Q(A)=0\) as \(\mathcal {C}^0_1\subset ca_c\). Let \(B\in {\mathcal F}\) such that \(Q(B)=0\) for all \(Q\in \mathcal {C}^0_1\). Thus \(1_B\in {\mathcal C}\) by (4.2), so \(1_B=0\) in \(L^\infty _c\) by \(NA({\mathcal P})\), i.e. \(c(B)=0\).
(ii) \(\Rightarrow \) (i): let \(H\in {\mathcal H}\) such that \((H\bullet S)_T\ge 0\) \({\mathcal P}\)q.s. Then \(Q\{(H\bullet S)_T\ge 0\}=0\) for every \(Q\in \mathcal {C}^0_1\), because \((H\bullet S)_t\) is a Qmartingale with expectation 0, and therefore \((H\bullet S)_T= 0\) \({\mathcal P}\)q.s.
As for the Superhedging Duality note that clearly \(\pi (X)\le \Vert X\Vert _{c,\infty }\) since \(0\in {\mathcal H}\), and as \({\mathcal C}_1^0\ne \emptyset \) (\({\mathcal C}\ne L^\infty _c\)) it follows that \(\pi (X)>\infty \). Moreover, by (4.2) we have for any \(y\in {\mathbb R}\) that \(Xy\in {\mathcal C}\) if and only if \(0\ge \sup _{Q\in \mathcal {C}^0_1}E_{Q}[Xy]= y + \sup _{Q\in \mathcal {C}^0_1}E_{Q}[X]\) which proves (4.3). \(\square \)
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