Consistent price systems under model uncertainty


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.

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    Before the final revision of the present paper was completed, we received the revised version of [3] in which a multivariate extension is provided using the strict no-arbitrage condition of [18].

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    As can be seen here, we can dispense with a condition that is slightly weaker than (2.2), but aesthetically less pleasant: the existence of a constant \(c\) and selectors \(S_{t}\) such that \(y^{i}/S^{i}_{t+1}(\omega,\cdot) \leq cy^{1}/S^{1}_{t+1}(\omega ,\cdot)\) for all \(y\in K^{*}_{t+1}(\omega,\cdot)\).


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Corresponding author

Correspondence to Bruno Bouchard.

Additional information

Research of first author supported by ANR Liquirisk and Investissements d’Avenir (ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047).

Research of second author supported by NSF Grant DMS-1208985.



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.

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:

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

  4. (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

  1. (v)

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

  2. (vi)

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

  3. (vii)

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

  4. (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).


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 \(\{\varPsi\neq\emptyset\}\).

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Bouchard, B., Nutz, M. Consistent price systems under model uncertainty. Finance Stoch 20, 83–98 (2016).

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  • Transaction costs
  • Arbitrage of the second kind
  • Consistent price system
  • Model uncertainty

Mathematics Subject Classification (2010)

  • 60G42
  • 91B25
  • 93E20
  • 49L20

JEL Classification

  • G13