Fatou closedness under model uncertainty

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 non-dominated) 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.

1 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

Let ${\mathcal A}\subset L_P^\infty$ be convex. Equivalent are:

• (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 P-almost 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 first-countable, 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 so-called 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 non-dominated class of probability measures, see [6,7,8, 26] and the reference therein.

We will show that there is a version of Theorem 1.1 in a robust probabilistic framework $(\Omega ,{\mathcal F},{\mathcal P})$, see Theorem 3.9. Let

$$\begin{aligned} c(A):=\sup _{P\in {\mathcal P}} P(A), \quad A\in {\mathcal F}, \end{aligned}$$

denote the capacity generated by ${\mathcal P}$. Under some conditions on the convex set ${\mathcal A}$ and on $L^\infty _c$ we obtain equivalence between

(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 $X\in {\mathcal A}$,

where $L^\infty _c$ and $ca_c$ are the robust analogues of $L^\infty _P$ and $L^1_P$ given by the capacity c, respectively, and ${\mathcal P}$ quasi sure convergence means Q-almost sure convergence under each $Q\in {\mathcal P}$. The conditions we have to require on ${\mathcal A}$ are monotonicity (${\mathcal A}={\mathcal A}+(L^\infty _c)_+$) and a property called ${\mathcal P}$-sensitivity. Monotonicity is typically satisfied in economic applications, and we show that ${\mathcal P}$-sensitivity is indeed a necessary condition to have (WC) $\Leftrightarrow$ (FC), see Proposition 3.8. If ${\mathcal P}$ is dominated, ${\mathcal P}$-sensitivity is always fulfilled.

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 non-dominated ${\mathcal P}$ 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.

Let $(\Omega ,{\mathcal F})$ be any measurable space.

• (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 variables

  $$\begin{aligned} {\mathcal L}:= & {} \{f:\Omega \rightarrow {\mathbb R}\mid f\, \text{ is }\, {\mathcal F}\text{-measurable }\}, \\ {\mathcal L}_+:= & {} \{f\in {\mathcal L}\mid f(\omega )\ge 0, \forall \, \omega \in \Omega \}, \\ {\mathcal L}^\infty:= & {} \{f\in {\mathcal L}\mid \text{ f } \text{ is } \text{ bounded }\}. \end{aligned}$$

  In particular ${\mathcal L}^\infty$ is a Banach space under the (pointwise) supremum norm $\Vert \cdot \Vert _{\infty }$ with dual space ba.

• (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 expectation

  $$\begin{aligned} c(f):=\sup _{Q\in {\mathcal P}}E_Q[f], \quad f\in {\mathcal L}_+ \end{aligned}$$

  and by some abuse of notation we define the capacity $c(A):=c(1_A)$ for $A\in {\mathcal F}$.

• (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 $A\in {\mathcal F}$:

  $$\begin{aligned} \sup _{P\in {\widehat{{\mathcal P}}}} P(A)=0 \quad \Rightarrow \quad \sup _{P\in {\widetilde{{\mathcal P}}}} P(A)=0 . \end{aligned}$$

  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 ${\widehat{{\mathcal P}}}\ll {\widetilde{{\mathcal P}}}$.

• (viii) A statement holds ${\mathcal P}$-quasi surely (q.s.) if the statement holds Q-almost 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 by

  $$\begin{aligned} f\sim g\quad \Leftrightarrow \quad \forall P\in {\mathcal P}: P(f=g)=1 . \end{aligned}$$

  We 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 well-known consequence of the Radon-Nikodym theorem ([1, Theorem 13.18]) that $ca_Q$ may be identified with $L_Q^1$.

• (xi) For any $f,g\in {\mathcal L}$ and $P\in {\mathcal M}_1$, we write $f\le g$ P-a.s. if and only if $P(f\le g)=1$. Similarly $f\le g$ ${\mathcal P}$-q.s. if and only if $f\le g$ P-a.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 ${\mathcal P}$.

For simplicity of presentation, if there is no risk of confusion, we will follow the usual convention of identifying random variables in ${\mathcal L}$ with the equivalence classes they induce (in $L_c$, $L^\infty _c$, $L^1_Q$ or $L_Q^\infty$) and vice versa.

3 Towards a robust version of Theorem 1.1

We start by recalling the proof of the non-trivial implication (ii) $\Rightarrow$ (i) of Theorem 1.1: the idea is to apply the Krein–Smulian theorem which implies that we only need to show that the sets

$$\begin{aligned} C_K:={\mathcal A}\cap \{X\in L^\infty _P\mid \Vert X\Vert _{P,\infty } \le K\} \end{aligned}$$

are weak*-closed for any constant $K>0$. Now we could invoke the countable sup property of $L^\infty _P$ to find that (ii) implies (i), see e.g. [2, Definition 1.43 and following discussion]. But as, in the robust setting we envisage, $L^\infty _c$ typically does not possess this property (see for instance Examples 3.11 and 3.12), we present an alternative argument by means of the following inclusion:

$$\begin{aligned} i:\left( L_P^\infty ,\sigma \left( L_P^\infty ,L_P^1\right) \right) \rightarrow \left( L^1_P,\sigma \left( L^1_P,L_P^\infty \right) \right) \end{aligned}$$

(3.1)

Note that i is continuous. Now, as ${\mathcal A}$ is Fatou closed, i.e. closed under bounded P-a.s. convergence, it follows that $i(C_K)$ is a closed subset of the Banach space $(L^1_P, E_P[|\cdot |])$, and thus $i(C_K)$ is also weakly (i.e. $\sigma (L^1_P,L_P^\infty )$) closed by convexity, so eventually $C_K$ must be weak*-closed by continuity of i.

A natural approach to prove a robust version of Theorem 1.1 is to ’robustify’ the spaces $L^1_P$ and try to repeat the argument above. There are two natural candidates for this: Let $H_c:=\{X\in L\mid c(|X|)<\infty \}$, with norm $\Vert X\Vert _c:=c(|X|)$. Then it is readily verified that $(H_c,\Vert \cdot \Vert _c)$ is a Banach lattice. But in the robust case there is also another candidate, namely $M_c:=\overline{L^\infty _c}^{\Vert \cdot \Vert _c}$ which is also a Banach lattice with the norm $\Vert \cdot \Vert _c$. These spaces have recently been studied in the literature, see e.g. [14, 26], since they appear as natural environments to embed financial modelling under uncertainty. Clearly, $L_c^\infty \subset M_c\subset H_c\subset L_c$. Note that the trick with the inclusion (3.1) requires that the norm dual of $L^1_P$ can be identified with $L^\infty _P$, so in particular with a subset of $L^1_P$ where in this latter case $L^1_P$ is viewed as a representation of $ca_P$. Thus the reader may readily check that we could save the above argument if the norm duals $M_c^*$ and $H_c^*$ of $M_c$ and $H_c$, respectively, would satisfy $M_c^*\subset ca$ or $H_c^*\subset ca$. The following Theorem 3.1 shows that this is the case only if ${\mathcal P}$ is dominated. To this end, denote by

$$\begin{aligned} {\mathcal Z}:=\{ (A_n)_{n\in {\mathbb N}}\subset {\mathcal F}\mid A_n\downarrow \emptyset \; \text{ and }\; c(A_n)\not \rightarrow 0\}, \end{aligned}$$

(3.2)

where $A_n\downarrow \emptyset$ means that $A_n\supset A_{n+1}$, $A_n\ne \emptyset$, $n\in {\mathbb N}$, and $\bigcap _{n\in {\mathbb N}}A_n=\emptyset$, the decreasing sequences of sets on which c is not continuous.

Theorem 3.1

Consider the following conditions:

• (i) ${\mathcal Z}=\emptyset$.

• (ii) $M_c^*\subset ca$.

• (iii) $H_c^*\subset ca$.

Then (i) $\Longleftrightarrow$ (ii) $\Longleftarrow$ (iii).

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

(i) $\Rightarrow$ (ii): By Proposition A.2 for any $l\in M_c^*$ there is $\mu \in ca$ such that $l(X)=\int X\, d\mu$ for all simple random variables X. Moreover, $\mu \in ca_c$, because $c(A)=0$ implies $l(1_A)=0$, $A\in {\mathcal F}$. Since for any $X\in L^\infty _c$ and any $n\in {\mathbb N}$ by the usual approximation method from integration theory there is a simple random variable $X_n$ such that $|X-X_n|<1/n$ ${\mathcal P}$-q.s., so $\Vert X-X_n\Vert _c<1/n$, continuity of l and the dominated convergence theorem yield

$$\begin{aligned} l(X)=\lim _{n\rightarrow \infty }l(X_n)=\lim _{n\rightarrow \infty }\int X_n\, d\mu =\int X\, d\mu \end{aligned}$$

for all $X\in L^\infty _c$. We recall that in [14] Proposition 18 the following relation was shown

$$\begin{aligned} M_c= \{X\in H_c\mid \lim _{n\rightarrow \infty }\Vert X1_{\{|X|\ge n\}}\Vert _c=0 \}. \end{aligned}$$

Hence, for $X\in (M_c)_+$ we have by monotone convergence that

$$\begin{aligned} l(X)=\lim _{n\rightarrow \infty }l(X1_{\{|X|\le n\}})= \lim _{n\rightarrow \infty }\int X1_{\{|X|\le n\}}\, d\mu = \int X\, d\mu . \end{aligned}$$

Finally, decomposing $X\in M_c$ into $X^+-X^-$ with $X^+,X^-\in (M_c)_+$ and linearity of l and the integral shows (ii).

(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 $Q=\sum _{n=1}^\infty \frac{1}{2^n}P_n$.

Recall the conditions

(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 counter-example 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

Consider the measure space $(\Omega ,{\mathcal F})=([0,1], \mathfrak {P}([0,1])$, where $\mathfrak {P}([0,1])$ denotes the power set of [0, 1]. Assume the continuum hypothesis. Banach and Kuratowski have shown that for any set I with the same cardinality as ${\mathbb R}$ there is no measure $\mu$ on $(I,\mathfrak {P}(I))$ such that $\mu (I)=1$ and $\mu (\{\omega \})=0$ for all $\omega \in I$; see for instance [16, Theorem  C.1]. It follows that any probability measure $\mu$ over $(\Omega ,{\mathcal F})$ must be a countable sum of weighted Dirac-measures, i.e. $\mu =\sum _{i=1}^\infty a_i\delta _{\omega _i}$ where $a_i\ge 0$, $\sum _{i=1}^na_i=1$, $\omega _i\in \Omega$, $i\in {\mathbb N}$. (Recall that for $\omega \in \Omega$ and $A\in {\mathcal F}$: $\delta _\omega (A)=1$ if and only if $\omega \in A$ and $\delta _\omega (A)=0$ otherwise.) Indeed, let $\mu \in {\mathcal M}_1$, and let

$$\begin{aligned} S:=\{\omega \in \Omega \mid \mu (\{\omega \})>0\}. \end{aligned}$$

Then S can at most be countable (consider the sets $S_n:=\{\omega \in \Omega \mid \mu (\{\omega \})>1/n\}$, $n\in {\mathbb N}$, and note that $S=\bigcup _{n\in {\mathbb N}}S_n$). Now suppose that $\mu ([0,1]\setminus S)>0$, then as $[0,1]\setminus S$ has the same cardinality as [0, 1], this implies the existence of an atom for the measure $\mu$ restricted to $[0,1]\setminus S$, i.e. there exists ${\hat{\omega }}\in [0,1]\setminus S$ such that

$$\begin{aligned} \frac{1}{\mu ([0,1]\setminus S)}\mu (\{{\hat{\omega }}\})>0. \end{aligned}$$

This clearly contradicts the definition of S.

Let ${\mathcal P}:=\{\delta _\omega \mid \omega \in [0,1]\}$ be the set of all Dirac measures. Then

$$\begin{aligned} c(|X|)=\sup _{\omega \in [0,1]}|X(\omega )|, \end{aligned}$$

so it turns out that $L^\infty _c=M_c=H_c={\mathcal L}^\infty$. Hence, $(L^\infty _c)^*=M_c^*=H_c^*=ba$, and, as $c(A)=0$ is equivalent to $A=\emptyset$, we also have that $ca_c=ca$. Consider the set

$$\begin{aligned} C:=\{1_A \mid \emptyset \ne A \subset [0,1]\;\text{ is } \text{ countable }\}, \end{aligned}$$

and let ${\mathcal A}$ be the convex closure of C under bounded ${\mathcal P}$-q.s. convergence of sequences. Then $1\not \in {\mathcal A}$: Indeed, any $X=\sum _{i=1}^na_i1_{A_i}$, $a_i\ge 0$, $\sum _{i=1}^na_i=1$, $1_{A_i}\in C$, in the convex hull of C satisfies $0\le X\le 1_{A_X}$ where $A_X:=\bigcup _{i=1}^nA_i$ is countable. Let $X_k$ be any sequence in the convex hull of C, then $0\le X_k\le 1_B$, $k\in {\mathbb N}$, where $B:=\bigcup _{k\in {\mathbb N}}A_{X_k}$ is countable. Hence, $X_k(\omega )=0$ for all $\omega \in [0,1]\setminus B$, so $1\not \in {\mathcal A}$. Now consider the family ${\mathcal G}$ of all countable subsets of [0, 1] directed by $A\le B$ if and only if $A\subset B$. Consider the net $\{1_{A}\mid A\in {\mathcal G}\}\subset C$. Then for any probability measure $\mu$ there is $A\in {\mathcal G}$ (namely $A=S$) such that for all $B\in {\mathcal G}$ with $B\ge A$ we have $\int 1_B\, d\mu =1=\int 1\,d\mu$. Thus 1 lies in the $\sigma (L^\infty _c,ca_c)$-closure of ${\mathcal A}$.

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 P-a.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 Q-a.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

A set ${\mathcal A}\subset L^\infty _c$ is called ${\mathcal P}$-sensitive if there exists a set ${\mathcal Q}\subset {\mathcal M}_1$ with ${\mathcal Q}\ll {\mathcal P}$ such that

$$\begin{aligned} j_Q(X)\in j_Q({\mathcal A}) \text{ for } \text{ all } Q\in {\mathcal Q} \text{ implies } X\in {\mathcal A} \end{aligned}$$

or equivalently

$$\begin{aligned} {\mathcal A}= \bigcap _{Q\in {\mathcal Q}} j^{-1}_Q\circ j_Q({\mathcal A}). \end{aligned}$$

The set ${\mathcal Q}$ will be called reduction set for $({\mathcal A},{\mathcal P})$.

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

If ${\mathcal A}=\emptyset$ or ${\mathcal A}= L^{\infty }_c$, the assertion is trivial. Now assume that ${\mathcal A}\ne \emptyset$ and ${\mathcal A}\ne L^{\infty }_c$. As ${\mathcal A}$ is $\sigma (L^\infty _c,ca_c)$-closed and convex, the function

$$\begin{aligned} \rho (X):=\delta (X\mid {\mathcal A}):= {\left\{ \begin{array}{ll} 0 &{} \text{ if } \, X\in {\mathcal A}\\ \infty &{}\text{ else } \end{array}\right. } ,\quad X\in L^\infty _c, \end{aligned}$$

is convex and $\sigma (L^\infty _c,ca_c)$ lower-semicontinuous. Hence, by the Fenchel–Moreau theorem (see [18, Proposition 4.1]) there exists a dual representation of $\rho$, i.e.

$$\begin{aligned} \rho (X)=\sup _{\mu \in {\mathcal Q}}\left\{ \int X\, d\mu -\rho ^*(\mu )\right\} \end{aligned}$$

where ${\mathcal Q}:=\{\mu \in ca_c\mid \rho ^*(\mu )<\infty \}$ is a convex set and

$$\begin{aligned} \rho ^*(\mu ):=\sup _{X\in {\mathcal A}}\int X\, d\mu , \quad \mu \in ca_c . \end{aligned}$$

${\mathcal A}\ne L^{\infty }_c$ implies ${\mathcal Q}\supsetneqq \{0\}$ and therefore,

$$\begin{aligned} {\mathcal A}=\bigcap _{\mu \in {\mathcal Q}}\left\{ X\in L^\infty _c\mid \int X\, d\mu \le \rho ^*(\mu )\right\} =\bigcap _{\mu \in {\mathcal Q}\setminus \{0\}}\left\{ X\in L^\infty _c\mid \int X\, d\mu \le \rho ^*(\mu )\right\} . \end{aligned}$$

Let ${\tilde{{\mathcal Q}}}:=\{\frac{|\mu |}{|\mu |(\Omega )} \mid \mu \in {\mathcal Q}\setminus \{0\}\}\subset {\mathcal M}_1$ and note that ${\tilde{{\mathcal Q}}}\ll {\mathcal P}$ since ${\mathcal Q}\subset ca_c$. Consider

$$\begin{aligned} X\in \bigcap _{Q\in {\tilde{{\mathcal Q}}}}j_Q^{-1}\circ j_Q({\mathcal A}). \end{aligned}$$

Fix $Q\in {\tilde{{\mathcal Q}}}$ and $\nu \in {\mathcal Q}$ such that $Q=\frac{|\nu |}{|\nu |(\Omega )}$. Then, $j_Q(X)\in j_Q({\mathcal A})$, i.e. there is $Y\in {\mathcal A}$ such that $j_Q(X)=j_Q(Y)$. Noting that $X=j_Q(X)$ and $Y=j_Q(Y)$ under $\nu$, it follows that

$$\begin{aligned} \int X\, d\nu =\int j_Q(X)\, d\nu =\int j_Q(Y)\, d\nu =\int Y\, d\nu \le \rho ^*(\nu ), \end{aligned}$$

where the inequality follows from $Y\in {\mathcal A}$. Since $Q\in {\tilde{{\mathcal Q}}}$ was arbitrary, we conclude that indeed $\int X\, d\mu \le \rho ^*(\mu )$ for all $\mu \in {\mathcal Q}$, and hence that $X\in {\mathcal A}$. This shows that $\bigcap _{Q\in {\mathcal Q}}j_Q^{-1}\circ j_Q({\mathcal A})\subset {\mathcal A}$. The other inclusion $\bigcap _{Q\in {\mathcal Q}}j_Q^{-1}\circ j_Q({\mathcal A})\supset {\mathcal A}$ is trivially satisfied, so we have that ${\mathcal A}$ is ${\mathcal P}$-sensitive with reduction set ${\tilde{{\mathcal Q}}}$. $\square$

The following Theorem 3.9 gives conditions under which (FC) $\Longleftrightarrow$ (WC) for a convex set ${\mathcal A}\subset L^\infty _c$. Besides ${\mathcal P}$-sensitivity we have to require that the norm dual $ca_c^*$ of $(ca_c, TV)$, where TV denotes the total variation norm on $ca_c$, may be identified with $L^\infty _c$. Clearly any $X\in L^\infty _c$ may be identified with a continuous linear functional on $ca_c$ by

$$\begin{aligned} ca_c\ni \mu \mapsto \int X \,d\mu , \end{aligned}$$

(3.3)

so we always have $L^\infty _c\subset ca_c^*$. However, $ca_c^*=L^\infty _c$ is obviously a very strong condition which we will characterize in Proposition 3.10 in terms of order closedness of $L^\infty _c$.

Theorem 3.9

Suppose that $ca_c^*=L^\infty _c$ and let ${\mathcal A}\subset L^\infty _c$ be convex and monotone (${\mathcal A}+(L^\infty _c)_+={\mathcal A}$). Equivalent are

• (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 $Q\in {\mathcal Q}$.

Consider the continuous inclusion

$$\begin{aligned} i: (L^\infty _Q,\sigma (L^ \infty _Q,L^1_Q))\rightarrow (L^1_Q,\sigma (L^1_Q,L^{\infty }_Q)). \end{aligned}$$

In a first step we show that $C_{Q,K}:=i\circ j_Q(C_K)$ is $\Vert \cdot \Vert _Q:=E_Q[|\cdot |]$-closed in $L^1_Q$, because being convex it then follows that $C_{Q,K}$ is $\sigma (L^1_Q,L^{\infty }_Q)$-closed and therefore $j_Q(C_K)$ is $\sigma (L^{\infty }_Q,L^1_Q)$-closed by continuity of i. To this end let $(Y_n)_{n\in {\mathbb N}}\subset C_{Q,K}$ and $Y\in L^1_Q$ such that $\Vert Y_n-Y\Vert _Q\rightarrow 0$, and without loss of generality we may also assume that $Y_n\rightarrow Y$ Q-a.s. Note that Y is necessarily bounded by K. Choose $X_n\in C_K$ such that $Y_n=j_Q(X_n)$ for all $n\in {\mathbb N}$ and $X\in L^\infty _c$ such that $Y=j_Q(X)$. Consider now the set

$$\begin{aligned} F:=\{\omega \in \Omega \mid X_n(\omega )\rightarrow X(\omega )\} \end{aligned}$$

(by the usual abuse of notation, in the definition of F we still write $X_n$ and X for arbitrary representatives of the equivalence classes $X_n$ and X). By monotonicity of ${\mathcal A}$ we have that $\widetilde{X}_n:= X_n 1_F + K1_{F^c}\in C_K$ for all $n\in {\mathbb N}$, and $\widetilde{X}_n\rightarrow X 1_F +K1_{F^c}=:{\widetilde{X}}$ ${\mathcal P}$-q.s. Consequently ${\widetilde{X}} \in C_K$ and since $Q(F)=1$ we have $Y=j_Q(X)=j_Q({\widetilde{X}})\in C_{Q,K}$. Hence, $j_Q(C_{K})$ is $\sigma (L^\infty _Q,L^{1}_Q)$ closed.

By continuity of $j_Q$, the preimage $j_Q^{-1}\circ j_Q(C_{K})$ is $\sigma (L^\infty _c,ca_c)$-closed, and as also $\{X\mid \Vert X\Vert _{c,\infty }\le K\}$ is $\sigma (L^\infty _c,ca_c)$-closed, we conclude that

$$\begin{aligned} A_{Q,K}:= j_Q^{-1}\circ j_Q(C_{K}) \cap \{X\mid \Vert X\Vert _{c,\infty }\le K\} \supset C_K \end{aligned}$$

and finally also $\bigcap _{Q\in {\mathcal Q}} A_{Q,K}$ are $\sigma (L^\infty _c,ca_c)$-closed. Clearly, $\bigcap _{Q\in {\mathcal Q}} A_{K,Q}\supset C_K$. If we can show $\bigcap _{Q\in {\mathcal Q}} A_{Q,K}\subset C_K$, then we are done, because then $\bigcap _{Q\in {\mathcal Q}} A_{Q,K}= C_K$, and thus $C_K$ is $\sigma (L^\infty _c,ca_c)$-closed. To this end, let $X\in \bigcap _{Q\in {\mathcal Q}} A_{Q,K}$. Then $j_Q(X)\in j_{Q}({\mathcal A})$ for any $Q\in {\mathcal Q}$ and therefore $X\in {\mathcal A}$ by ${\mathcal P}$-sensitivity. Moreover by definition of $A_{K, Q}$ we also have $\Vert X\Vert _{c,\infty }\le K$. $\square$

Note that Theorem 3.9 proves the so-called C-property 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 $ca_c^*=L^\infty _c$.

In order to prove that $ca_c^*=L^\infty _c$ implies the existence of a supremum for any norm bounded set ${\mathcal D}\subset L^\infty _c$, we recall that ca and thus also $ca_c$ is an AL-space ([1, Theorem 10.56]), so $ca_c^*$ is an AM-space ([1, Theorem 9.27]). In particular $ca_c^*$ is order complete. Here, the order $\ge _*$ on $ca^*_c$ is given by $l\ge _*0$ if and only if $l(\mu )\ge 0$ for all $\mu \in (ca_c)_+$, and a set ${\mathcal S}\subset ca_c^*$ is order bounded from above if there is $h\in ca_c^*$ such that $h-l\ge _*0$ for all $l\in {\mathcal S}$. Any norm bounded ${\mathcal D}\subset L^\infty _c$ is order bounded from above in $ca_c^*$, because $K\mu (\Omega )-\int X\, d\mu \ge 0$, $\mu \in (ca_c)_+$, for a constant $K>0$ which is an upper bound of the norm on ${\mathcal D}$, so $(\mu \mapsto K\mu (\Omega ))\in ca_c^*$ is an upper bound with respect to $\ge _*$. Thus there is a least upper bound of ${\mathcal D}$ viewed as a subset of $ca_c^*$. Now suppose that $ca_c^*$ can be identified with $L^\infty _c$. Then this least upper bound of ${\mathcal D}$ may be identified with an element in $X\in L^\infty _c$, that is

$$\begin{aligned} \int X\, d\mu \ge \int Y\, d\mu \quad \text{ for } \text{ all } \mu \in (ca_c)_+ \text{ and } \text{ all } Y\in {\mathcal D}. \end{aligned}$$

Considering measures $\mu$ of type $1_A dP$ for $P\in {\mathcal P}$ and $A\in {\mathcal F}$ shows that $X\ge Y$ for all $Y\in {\mathcal D}$, and $\mu \mapsto \int X\, d\mu$ being the least amongst the upper bounds of ${\mathcal D}$ in the $\ge _*$-order implies that X is a supremum of ${\mathcal D}$. $\square$

Example 3.11

In this example we fix a measure space $(\Omega ,{\mathcal F})$ and an uncountable family ${\mathcal P}=\{P_{\sigma }\}_{\sigma \in \Sigma }$ of probability measures. Consider the enlarged sigma algebra ${\mathcal F}^{\Sigma }=\bigcap _{\sigma \in \Sigma } {\mathcal F}^{\sigma }$ where ${\mathcal F}^{\sigma }$ is the $P^{\sigma }$ completion of ${\mathcal F}$, and notice that any $P^{\sigma }$ uniquely extends to ${\mathcal F}^{\Sigma }$. Assume that there exists a family of sets $\{\Omega ^{\sigma }\}_{\sigma \in \Sigma }\subset {\mathcal F}^{\Sigma }$ such that for any $\sigma \in \Sigma$, $P^{\sigma }(\Omega ^{\sigma })=1$ and $P^{\tilde{\sigma }}(\Omega ^{\sigma })=0$ for $\tilde{\sigma }\ne \sigma$. In this case it is easily seen that any norm bounded set ${\mathcal D}\subset L^{\infty }_c(\Omega ,{\mathcal F}^{\Sigma })$ admits a supremum given by

$$\begin{aligned} \hbox {ess sup}_{Y\in {\mathcal D}}Y = \sum _{\sigma \in \Sigma } j_{P^{\sigma }}^{-1}(\hbox {ess sup}_{Y\in {\mathcal D}}j_{P^{\sigma }}(Y))1_{\Omega ^{\sigma }}. \end{aligned}$$

Note that $\hbox {ess sup}_{Y\in {\mathcal D}}j_{P^{\sigma }}(Y)$ in $L^{\infty }_{P^{\sigma }}$ is well-defined for every $\sigma \in \Sigma$. Also notice that $\hbox {ess sup}_{Y\in {\mathcal D}}Y$ is ${\mathcal F}^{\sigma }$-measurable for any $\sigma \in \Sigma$ and therefore is also ${\mathcal F}^{\Sigma }$-measurable. Therefore $L^{\infty }_c(\Omega ,{\mathcal F}^{\Sigma })=ca_c^{*}(\Omega ,{\mathcal F}^{\Sigma })$. Notice that, as $\Sigma$ is not countable, $L^{\infty }_c(\Omega ,{\mathcal F}^{\Sigma })$ does not possess the countable sup property, think for instance of the essential supremum of the set $\{1_{\Omega ^ \sigma }\mid \sigma \in \Sigma \}$. We refer to [11] for a deeper study of this example and applications to mathematical finance.

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

A function $f:L^\infty _c\rightarrow (-\infty ,\infty ]$ is

• 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

Consider a function $f:L^\infty _c\rightarrow [-\infty ,\infty ]$ such that

$$\begin{aligned} f(X)=\sup _{P\in {\mathcal Q}} f_P(j_P(X)), \end{aligned}$$

(4.1)

for some ${\mathcal Q}\subset {\mathcal M}_1$ and $f_P:L^{\infty }_P\rightarrow [-\infty ,\infty ]$. If $\mathcal {Q}\ll \mathcal {P}$ then f is ${\mathcal P}$-sensitive with reduction set ${\mathcal Q}$.

Proof

From representation (4.1) we automatically have

$$\begin{aligned} \{X\in L^{\infty }_c\mid f(X)\le a\}=\bigcap _{P\in {\mathcal Q}}\{X\in L^{\infty }_c\mid f_P(j_P(X))\le a\}. \end{aligned}$$

As $\{X\in L^{\infty }_c\mid f_P(j_P(X))\le a\}= j_P^{-1}\circ j_P\{X\in L^{\infty }_c\mid f_P(j_P(X))\le a\}$, we conclude that f is ${\mathcal P}$-sensitive with reduction set ${\mathcal Q}$. $\square$

Theorem 4.3

Assume that $ca_c^*=L^\infty _c$. Let $f:L^\infty _c\rightarrow (-\infty ,\infty ]$ be a quasiconvex (resp. convex), monotone non-decreasing ( $X\le Y$ ${\mathcal P}$-q.s. implies $f(X)\le f(Y)$) and ${\mathcal P}$-sensitive function. The following are equivalent:

• (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 is

  $$\begin{aligned} f(X)= & {} \sup _{P\in ca_c\cap {\mathcal M}_1}R\left( E_P[X],P\right) , \quad X\in L^\infty _c, \end{aligned}$$

  with dual function $R:{\mathbb R}\times ca_c\rightarrow (-\infty ,\infty ]$ given by

  

  and in the convex case the dual representation is

  $$\begin{aligned} f(X)= & {} \sup _{\mu \in (ca_c)_+}\left\{ \int X\, d\mu -f^*(\mu )\right\} , \quad X\in L^\infty _c, \end{aligned}$$

  where the dual function $f^{*}:ca_c\rightarrow (-\infty ,\infty ]$) is given by

  $$\begin{aligned} f^*(\mu ):= & {} \sup _{Y\in L^\infty _c}\left\{ \int Y\, d\mu - f(Y)\right\} . \end{aligned}$$

In addition, if $f(X+c)=f(X)+c$ for every $X\in L^{\infty }_c$ and $c\in {\mathbb R}$ then f is necessarily convex and

$$\begin{aligned} f(X)= \sup _{P\in ca_c\cap {\mathcal M}_1} \left\{ E_P[X]-f^*(P)\right\} , \quad X\in L^\infty _c. \end{aligned}$$

Proof

According to Theorem 3.9 (i) holds if and only if (ii) is satisfied.

(ii) $\Rightarrow$ (iii) is due to

$$\begin{aligned} f(X)\le \liminf _{n\rightarrow \infty }f(X_n)\le f(X) \end{aligned}$$

where the last ineqaulity follows from monotonicity. Conversely (iii) $\Rightarrow$ (ii) follows by considering $Y_n:=\hbox {ess inf}_{k\ge n}X_k$ and noting that $Y_n\uparrow X$ ${\mathcal P}$-q.s. and $f(Y_n)\le f(X_n)$; see also [19, Lemma 4.16].

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 Penot-Volle 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 so-called no-arbitrage condition) discounted prices are expectations under some risk-neutral 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 risk-neutral 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 no-arbitrage 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 so-called 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 well-known arguments.

Throughout this section we assume that $ca_c^*=L^\infty _c$ holds true. We consider a discrete time market model with terminal time horizont $T\in \mathbb {N}$, and trading times $I:=\left\{ 0,\ldots ,T\right\}$. The price process is given by a ${\mathcal P}$-q.s. bounded $\mathbb {R}^{d}$-valued stochastic process $S=(S_{t})_{t\in I}=(S^j_{t})_{t\in I}^{j=1,\ldots ,d}$ on $(\Omega ,\mathcal {F})$, and we also assume the existence of a numeraire asset $S_{t}^{0}=1$ for all $t\in I$. Moreover, we fix a filtration $\mathbb {F}:=\{\mathcal {F}_{t}\}_{t\in I}$ such that the process S is $\mathbb {F}$-adapted. Denote by $\mathcal {H}$ the class of ${\mathbb R}^d$-valued, $\mathbb {F}$-predictable stochastic processes, which is the class of all admissible trading strategies. Let

$$\begin{aligned} {\mathcal C}:=\left\{ X\in L^\infty _c\mid X\le (H\bullet S)_T \;{\mathcal P}\text {-q.s. for some } H\in {\mathcal H}\right\} \end{aligned}$$

where

$$\begin{aligned} (H\bullet S)_t:=\sum _{k=1}^{t}\sum _{j=1}^{d}H_{k}^{j}(S_{k}^{j}-S_{k-1}^{j}) \end{aligned}$$

is the payoff of the self-financing trading strategy at time $t\in I\setminus \{0\}$ with initial investment $(H\bullet S)_0=0$ given by the predictable process $H=(H_t)_{t\in I\setminus \{0\}}$. In this framework the no-arbitrage condition (NA(${\mathcal P}$)) was introduced by [7] as given by the following definition.

Definition 4.4

The described market model is called arbitrage-free, if it satisfies the no-arbitrage condition

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. $\square$

Suppose that ${\mathcal C}$ is ${\mathcal P}$-sensitive. As ${\mathcal C}$ is a $\sigma (L^{\infty }_c,ca_c)$-closed convex cone, the bipolar Theorem yields

$$\begin{aligned} \mathcal {C}= & {} \mathcal {C}^{00}=\left\{ Y\in L^{\infty }_c\mid \forall Q\in \mathcal {C}^0_1: \,E_Q[ Y ] \le 0\right\} \nonumber \\ \text { where } \mathcal {C}^0_1:= & {} \mathcal {C}^{0}\cap {\mathcal M}_1=\left\{ \mu \in \mathcal {C}^{0}\mid \mu (1_{\Omega })=1\right\} \nonumber \\ \text { and } \mathcal {C}^0:= & {} \left\{ \mu \in ca_c\mid \forall X\in \mathcal {C}: \, \int X\, d\mu \le 0\right\} . \end{aligned}$$

(4.2)

Notice that since $\mathcal {C}\supset -(L^{\infty }_c)_+$ then $\mu \in (ca_c)_+$ for every $\mu \in \mathcal {C}^0$ which explains $\mathcal {C}^0_1$.

Lemma 4.6

$\mathcal {C}^0_1$ is the set of all martingale measures dominated by the capacity c, that is

$$\begin{aligned} \mathcal {C}^0_1 = \{ Q\ll {\mathcal P}\mid S \text { is a } Q\text {-martingale} \} \end{aligned}$$

Proof

The proof is well-known 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 Q-martingale (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 $A\in {\mathcal F}_{t-1}$, $H_t^i=0$ 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

(First Fundamental Theorem of Asset Pricing) Suppose ${\mathcal C}$ is ${\mathcal P}$-sensitive. The following are equivalent:

• (i) $NA({\mathcal P})$

• (ii) $\mathcal {C}^0_1\approx {\mathcal P}$

Moreover, the Superhedging Duality holds, that is for any $X\in L^{\infty }_c$ the minimal superhedging price

satisfies

$$\begin{aligned} \pi (X)=\sup _{Q\in \mathcal {C}^0_1}E_{Q}[X]. \end{aligned}$$

(4.3)

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 Q-martingale 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 $X-y\in {\mathcal C}$ if and only if $0\ge \sup _{Q\in \mathcal {C}^0_1}E_{Q}[X-y]= -y + \sup _{Q\in \mathcal {C}^0_1}E_{Q}[X]$ which proves (4.3). $\square$

Appendices

Auxiliary results for Theorem 3.1

Recall the set ${\mathcal Z}$ defined in (3.2).

Proposition A.1

If ${\mathcal Z}=\emptyset$, then there exists a countable subset ${\widetilde{{\mathcal P}}}\subset {\mathcal P}$ such that ${\widetilde{{\mathcal P}}}\thickapprox {\mathcal P}$. The latter implies that there is a probability measure $Q\in {\mathcal M}_1$ such that $\{Q\}\thickapprox {\mathcal P}$.

Proof

We claim that for each $\varepsilon >0$, there exists $P_1,\ldots , P_n\in {\mathcal P}$ and $\delta >0$ such that $P_i(A)<\delta$ for all $i=1,\ldots , n$ implies that for all $P\in {\mathcal P}$ we have $P(A)<\varepsilon$. Suppose this is not the case. Then there exists $\varepsilon >0$ such that for any $P_1\in {\mathcal P}$ there is $A_1\in {\mathcal F}$ and $P_2\in {\mathcal P}$ satisfying

$$\begin{aligned} P_1(A_1)<1/2 \quad \text{ and }\quad P_2(A_1)\ge \varepsilon . \end{aligned}$$

Then there also exists $A_2\in {\mathcal F}$ and $P_3\in {\mathcal P}$ such that

$$\begin{aligned} P_1(A_2)<1/4,\, P_2(A_2)<1/4 \quad \text{ while }\quad P_3(A_2)\ge \varepsilon . \end{aligned}$$

Continuing this procedure we find sequences $(A_n)_{n\in {\mathbb N}}\subset {\mathcal F}$ and $(P_{n})_{n\in {\mathbb N}}\in {\mathcal P}$ such that

$$\begin{aligned} P_i(A_n)<\frac{1}{2^n}, \, i=1,\ldots , n, \quad \text{ and }\quad P_{n+1}(A_n)\ge \varepsilon . \end{aligned}$$

Consider $N:=\bigcap _{n\in {\mathbb N}}\bigcup _{k\ge n} A_k$. Then $P_i(N)=0$ for each $i\in {\mathbb N}$, because for all $n>(i-1)$

$$\begin{aligned} P_i(N)\le \sum _{k=n}^\infty P_i(A_k)\le \frac{1}{2^{n-1}}. \end{aligned}$$

Hence, replacing the above sequence $A_n$ by $B_n:=A_n\setminus N$, $n\in {\mathbb N}$, we still have

$$\begin{aligned} P_i(B_n)<\frac{1}{2^n}, \, i=1,\ldots , n, \quad \text{ and }\quad P_{n+1}(B_n)\ge \varepsilon . \end{aligned}$$

Now let $E_n:=\bigcup _{k\ge n}B_k$, $n\in {\mathbb N}$. It follows that $E_n\downarrow \emptyset$. However, for each $n\in {\mathbb N}$

$$\begin{aligned} c(E_n)\ge P_{n+1}(E_n)\ge P_{n+1}(B_n)\ge \varepsilon \end{aligned}$$

which contradicts ${\mathcal Z}=\emptyset$.

Now let $\delta _n>0$ and let $P^{(n)}_1,\ldots , P^{(n)}_{m(n)}\in {\mathcal P}$ be such that for all $P \in {\mathcal P}$ it holds $P(A)<1/n$ whenever $P^{(n)}_i(A)<\delta _n$ for all $i=1,\ldots , m(n)$. Define

$$\begin{aligned} \mu :=\sum _{n=1}^\infty \sum _{i=1}^{m(n)}\frac{1}{2^n}\frac{1}{2^i}P_i^{(n)}. \end{aligned}$$

Then $\mu \in ca_+$, and $\mu (A)=0$ implies that $P_i^{(n)}(A)=0$ for all $i=1,\ldots , m(n)$ and $n\in {\mathbb N}$. Eventually this implies that for all $P\in {\mathcal P}$ we have $P(A)<1/n$ for all $n\in {\mathbb N}$, hence $P(A)=0$. Thus

$$\begin{aligned} {\widetilde{{\mathcal P}}}:=\{P_i^{(n)}\mid i\in \{1,\ldots , m(n)\}, n\in {\mathbb N}\}\quad \text{ and }\quad Q:=\frac{1}{\mu (\Omega )}\mu \end{aligned}$$

satisfy the assertion. $\square$

Proposition A.2

Let $(B,\Vert \cdot \Vert )$ be a Banach lattice of (equivalence classes of) random variables on $(\Omega ,{\mathcal F})$ containing all simple random variables such that the order $\le$ on B satisfies $0\le 1_A\le 1_{A'}$ whenever $A\subset A'$ for $A,A'\in {\mathcal F}$. If $B^*\subset ca$, in the sense that every $l\in B^*$ is of type

$$\begin{aligned} l(X)=\int X\,d\mu , \quad X\in B, \end{aligned}$$

for some $\mu \in ca$, then $\Vert 1_{A_n}\Vert \rightarrow 0$ $(n\rightarrow \infty )$ for all $(A_n)_{n\in {\mathbb N}}\subset {\mathcal F}$ such that $A_n\downarrow \emptyset$.

Conversely, if $\Vert 1_{A_n}\Vert \rightarrow 0$ $(n\rightarrow \infty )$ for all $(A_n)_{n\in {\mathbb N}}\subset {\mathcal F}$ such that $A_n\downarrow \emptyset$, then for every $l\in B^*$ there is a $\mu \in ca$ such that $l(Y)=\int Y\, d\mu$ for all simple random variables Y.

Proof

Suppose that $B^*\subset ca$ and let $(A_n)_{n\in {\mathbb N}}\subset {\mathcal F}$ such that $A_n\downarrow \emptyset$. Then $1_{A_n}\rightarrow 0$ with respect to $\sigma (B,B^*)$ since every element in $B^*$ corresponds to a $\sigma$-additive measure. Hence,

$$\begin{aligned} 0\in \overline{co\{1_{A_n}\mid n\in {\mathbb N}\}} \end{aligned}$$

where the closure is taken in the $\sigma (B,B^*)$-topology. As the closed convex set in the $\sigma (B,B^*)$-topology and in the norm topology coincide, we have that there is a sequence of convex combinations

$$\begin{aligned} c_k:=\sum _{i=1}^{m(k)}a_i(k)1_{A_{n_i(k)}}, \quad k\in {\mathbb N}, \end{aligned}$$

where $a_i(k)\in {\mathbb R}$ and $n_1(k)\le n_2(k)\le \ldots \le n_{m(k)}(k)$ for all $k\in {\mathbb N}$ such that $\Vert c_k\Vert \rightarrow 0$ for $k\rightarrow \infty$. Moreover, since $0\in \overline{co\{1_{A_n}\mid n\ge N\}}$ for any $N\in {\mathbb N}$, we may assume that $n_1(k)\le n_1(k+1)$ for all $k\in {\mathbb N}$. However, $c_k\ge 1_{A_k}$ where $A_k=A_{n_{m(k)}(k)}$, because $A_n\supset A_{n+1}$ for all $n\in {\mathbb N}$. Thus, as $\Vert \cdot \Vert$ is a lattice norm, the subsequence $1_{A_ k}$ converges to 0 in norm and hence also $1_{A_{n}}$ converges to 0 in the norm topology (again due to $A_n\supset A_{n+1}$ for all $n\in {\mathbb N}$).

Finally suppose that $\Vert 1_{A_n}\Vert \rightarrow 0$ $(n\rightarrow \infty )$ for all $(A_n)_{n\in {\mathbb N}}\subset {\mathcal F}$ such that $A_n\downarrow \emptyset$. Then for any $l\in B^*$, the set function

$$\begin{aligned} \mu (A):=l(1_A), \quad A\in {\mathcal F}, \end{aligned}$$

is $\sigma$-additive. By linearity of l we deduce that $l(X)=\int X\, d\mu$ for all simple random variables X. $\square$

Penot–Volle duality theorem

Theorem B.1

(see e.g. [20, Theorem 1.1]) Let L be a locally convex topological vector space, $L^{\prime }$ be its dual space and $f:L\rightarrow \overline{\mathbb {R}}:= \mathbb {R}\cup \left\{ -\infty \right\} \cup \left\{ \infty \right\}$ be quasiconvex and lower semicontinuous. Then

$$\begin{aligned} f(X)=\sup _{X^{\prime }\in L^{\prime }}R(X^{\prime }(X),X^{\prime }) \end{aligned}$$

(B.1)

where $R:\mathbb {R\times }L^{\prime }\rightarrow \overline{\mathbb {R}}$ is defined by

(B.2)