We consider
superreplication under proportional transaction costs and adopt the discrete version of the model by Kabanov [
33]. For every
\(t\in \{0,\dots ,T\}\), we say that a set-valued map
\(S:\Omega \rightrightarrows {\mathbb{R}}^{d}\) is
\({\mathcal {F}}_{t}\)-measurable provided that
$$ \{\omega \in \Omega : S(\omega )\cap {\mathcal {U}}\neq \emptyset \} \in {\mathcal {F}}_{t} $$
for every open set
\({\mathcal {U}}\subseteq {\mathbb{R}}^{d}\). In this case, we denote by
\(L^{0}(S)\) the set of all random vectors
\(X\in L^{0}({\mathbb{R}}^{d})\) such that
\(\mathbb{P}[X\in S]=1\). This set is always nonempty if
\(S\) has closed values; see Rockafellar and Wets [
41, Corollary 14.6]. Now let
\(K_{t}:\Omega \rightrightarrows {\mathbb{R}}^{d}\) be an
\({\mathcal {F}}_{t}\)-measurable set-valued map such that
\(K_{t}(\omega )\) is a polyhedral convex cone (hence
\(K_{t}(\omega )\) is closed) containing
\({\mathbb{R}}^{d}_{+}\) for every
\(\omega \in \Omega \) and set
$$ {\mathcal {C}}_{t} = L^{0}(K_{t}). $$
Moreover, we consider the worst-case acceptance set
$$ {\mathcal {A}}= L^{1}({\mathbb{R}}^{d}_{+}). $$
Assumptions
5.5, (1) and (2) are easily seen to be satisfied. Moreover,
\({\mathcal {A}}\) as well as each of the sets
\({\mathcal {C}}_{t}\) is a cone. As proved in Schachermayer [
42, Theorem 2.1], Assumption
5.5, (3) always holds under the so-called “robust no-arbitrage” condition. Finally, as
\(0\in R(0)\), Assumption
5.5, (4) holds if and only if
\({\mathbb{R}}^{d}\) is not entirely contained in
\(\sum _{t=0}^{T}{\mathcal {C}}_{t}\). Note also that
\({\mathcal {A}}^{+}=L^{\infty }({\mathbb{R}}^{d}_{+})\). As a result, Proposition
5.6 yields
$$ R(X) = \bigcap _{w\in K_{0}^{+}\setminus \{0\}} \bigcap _{ \substack{Z\in L^{\infty }({\mathbb{R}}^{d}_{+}),\\ Z\in ({\mathcal {C}}_{1:T}^{1})^{+}, {\mathbb{E}}[Z]=w}} \{m\in {\mathbb{R}}^{d} : \langle m,w\rangle \geq -{\mathbb{E}}[ \langle X,Z\rangle ]\} $$
for every
\(X\in L^{1}({\mathbb{R}}^{d})\). The dual elements
\(Z\) in the above representation can be linked to consistent pricing systems; see e.g. Schachermayer [
42]. To see this, note that for every
\(t\in \{0,\dots ,T\}\), the set-valued map
\(K^{+}_{t}:\Omega \rightrightarrows {\mathbb{R}}^{d}\) defined by
$$ K^{+}_{t}(\omega )=\big(K_{t}(\omega )\big)^{+} $$
is
\({\mathcal {F}}_{t}\)-measurable, see e.g. Rockafellar and Wets [
41, Exercise 14.12], and such that
$$ \big({\mathcal {C}}_{t}\cap L^{1}({\mathbb{R}}^{d})\big)^{+}=L^{0}(K^{+}_{t}) \cap L^{\infty }({\mathbb{R}}^{d}) $$
by measurable selection; see the argument in the proof of Schachermayer [
42, Theorem 1.7]. As a result, every dual element
\(Z\) in the above dual representation satisfies
$$ {\mathbb{E}}[Z]\in K_{0}^{+}, \qquad Z \in ({\mathcal {C}}_{1:T}^{1})^{+} \subseteq \bigcap _{t=1}^{T}\big({\mathcal {C}}_{t}\cap L^{1}({ \mathbb{R}}^{d})\big)^{+} \subseteq \bigcap _{t=1}^{T}L^{0}(K^{+}_{t}). $$
This shows that the
\(d\)-dimensional adapted process
\(({\mathbb{E}}[Z\vert {\mathcal {F}}_{t}])\), with the conditional expectations taken componentwise, satisfies
\({\mathbb{E}}[Z\vert {\mathcal {F}}_{T}]=Z\) and
\({\mathbb{E}}[Z \vert {\mathcal {F}}_{t}] \in L^{0}(K^{+}_{t})\) for every
\(t\in \{0,\dots ,T\}\) and thus qualifies as a consistent pricing system. In other words, the above dual elements
\(Z\) can be viewed as the terminal values of consistent pricing systems.