Consistent price systems under model uncertainty

Abstract

We develop a version of the fundamental theorem of asset pricing for discrete-time markets with proportional transaction costs and model uncertainty. A robust notion of no-arbitrage of the second kind is defined and shown to be equivalent to the existence of a collection of strictly consistent price systems.

Appendix

1.1 A.1 Measure theory

Given a measurable space $(\varOmega,\mathcal{A})$, let $\mathfrak {P}(\varOmega)$ be the set of all probability measures on $\mathcal{A}$. The universal completion of $\mathcal{A}$ is the $\sigma$-field $\bigcap_{P\in \mathfrak{P} (\varOmega)} \mathcal{A}^{P}$, where $\mathcal{A}^{P}$ is the $P$-completion of $\mathcal{A}$. When $\varOmega$ is a topological space with Borel $\sigma$-field $\mathcal{B}(\varOmega)$, we always endow $\mathfrak{P}(\varOmega )$ with the topology of weak convergence. If $\varOmega$ is Polish, then $\mathfrak {P}(\varOmega )$ is also Polish. A subset $A\subset\varOmega$ is analytic if it is the image of a Borel subset of another Polish space under a Borel-measurable mapping. Analytic sets are stable under countable union and intersection, under forward and inverse images of Borel functions, but not under complementation; the complement of an analytic set is called co-analytic and is not analytic unless it is Borel. Any Borel set is analytic, and any analytic set is universally measurable, i.e., measurable for the universal completion of $\mathcal{B} (\varOmega)$. We refer to [4, Chap. 7] for these results and further background.

1.2 A.2 Random sets

Let $(\varOmega,\mathcal{A})$ be a measurable space. A mapping $\varPsi$ from $\varOmega$ into the power set $2^{\mathbb{R}^{d}}$ is called a random set in $\mathbb{R}^{d}$ and its graph is defined as

$$\operatorname{graph}(\varPsi)=\{(\omega,x):\, \omega\in\varOmega,\, x\in\varPsi(\omega )\}\subset\varOmega\times\mathbb{R}^{d}. $$

We say that $\varPsi$ is $\mathcal{A}$ -measurable (weakly $\mathcal{A}$ -measurable) if

$$\{\omega\in\varOmega:\, \varPsi(\omega)\cap O\neq\emptyset\}\in \mathcal{A} \quad\mbox{for all closed (open)}\ O\subset\mathbb{R}^{d}. $$

Moreover, $\varPsi$ is called closed (convex, etc.) if $\varPsi(\omega)$ is closed (convex, etc.) for all $\omega\in\varOmega$. We emphasize that measurability is not defined via the measurability of the graph, as it is sometimes done in the literature.

Lemma A.1

Let $(\varOmega,\mathcal{A})$ be a measurable space and $\varPsi$ a closed, nonempty random set in $\mathbb{R}^{d}$. The following are equivalent:

• (i) $\varPsi$ is $\mathcal{A}$-measurable.

• (ii) $\varPsi$ is weakly $\mathcal{A}$-measurable.

• (iii) The distance function $d(\varPsi,y)=\inf\{x\in\varPsi:\, |x-y|\}$ is $\mathcal{A}$-measurable for all $y\in\mathbb{R}^{d}$.

• (iv) There exist $\mathcal{A}$-measurable functions $(\psi _{n})_{n\geq1}$ with $\varPsi=\overline{\{\psi_{n}, n \geq1\}}$ (“Castaing representation”).

Moreover, (i)–(iv) imply that

• (v) $\mathrm{graph}(\varPsi)$ is $\mathcal{A}\times\mathcal{B}(\mathbb {R}^{d})$-measurable;

• (vi) The dual cone $\varPsi^{*}$ is $\mathcal{A}$-measurable;

• (vii) $\operatorname{graph}(\operatorname{int}\varPsi^{*})$ is $\mathcal{A}\times\mathcal{B}(\mathbb{R} ^{d})$-measurable;

• (viii) There exists an $\mathcal{A}$-measurable selector $\psi$ of $\varPsi^{*}$ which satisfies $\psi\in\operatorname{int}\varPsi^{*}$ on $\{\operatorname{int} \varPsi^{*}\neq\emptyset\}$.

If $\mathcal{A}$ is universally complete, then (v) is equivalent to (i)–(iv).

Proof

We refer to [23, Sect. 1] for the results concerning (i)–(vi). If (iv) holds, then we have the representation

$$\begin{aligned} \operatorname{graph}(\operatorname{int}\varPsi^{*}) &=\big\{ (\omega,y)\in\varOmega\times\mathbb{R}^{d}:\, \langle x,y \rangle>0\mbox{ for all } x\in\varPsi(\omega)\setminus\{0\}\big\} \\ &=\bigcap_{n\geq1}\{(\omega,y)\in\varOmega\times\mathbb{R}^{d}:\, \langle\psi _{n}(\omega),y \rangle>0 \mbox{ or }\psi_{n}(\omega )=0\} \end{aligned}$$

which readily implies (vii).

Finally, let (vi) hold; then $\varPsi^{*}$ has a Castaing representation $(\phi_{n})$. If we define $\phi:=\sum_{n} 2^{-n} \phi_{n}$, then $\phi$ is $\mathcal{A}$-measurable and $\varPsi^{*}$-valued since $\varPsi^{*}$ is closed and convex. Let $\omega\in\varOmega$ be such that $\operatorname{int}\varPsi ^{*}(\omega)\neq\emptyset$. By their denseness, at least one of the points $\phi_{n}(\omega)\in\varPsi^{*}(\omega)$ must lie in the interior of $\varPsi^{*}(\omega)$. Moreover, since $\varPsi^{*}(\omega)$ is convex, we observe that a nondegenerate convex combination of a point in $\varPsi^{*}(\omega)$ with an interior point of $\varPsi^{*}(\omega)$ is again an interior point. These two facts yield that $\phi(\omega)\in \operatorname{int}\varPsi^{*}(\omega)$ as desired. (This applies to any closed and nonempty convex random set, not necessarily of the form $\varPsi^{*}$.) □

In some cases, we need to select from random sets in infinite-dimensional spaces, or random sets that are not closed. The following is sufficient for our purposes.

Lemma A.2

(Jankov–von Neumann)

Let $\varOmega, \varOmega'$ be Polish spaces and $\varGamma\! \subset \varOmega\times\varOmega'$ an analytic set. Then the projection $\pi_{\varOmega}(\varGamma)\subset \varOmega$ is universally measurable, and there exists a universally measurable function $\psi:\pi_{\varOmega}(\varGamma)\to\varOmega'$ whose graph is contained in $\varOmega'$.

We refer to [4, Proposition 7.49] for a proof. In many applications, we start with a random set $\varPsi:\varOmega\to 2^{\varOmega'}$ such that $\varGamma:=\operatorname{graph}(\varPsi)$ is analytic. Noting that $\pi_{\varOmega}(\varGamma)=\{\varPsi\neq\emptyset\}$, Lemma A.2 then yields a universally measurable selector for $\varPsi$ on the set $\{\varPsi\neq\emptyset\}$.