# On the quasi-sure superhedging duality with frictions

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## Abstract

We prove the superhedging duality for a discrete-time financial market with proportional transaction costs under model uncertainty. Frictions are modelled through solvency cones as in the original model of Kabanov (Finance Stoch. 3:237–248, 1999) adapted to the quasi-sure setup of Bouchard and Nutz (Ann. Appl. Probab. 25:823–859, 2015). Our approach allows removing the restrictive assumption of no arbitrage of the second kind considered in Bouchard et al. (Math. Finance 29:837–860, 2019) and showing the duality under the more natural condition of strict no arbitrage. In addition, we extend the results to models with portfolio constraints.

## Keywords

Model uncertainty Superhedging Proportional transaction costs Portfolio constraints Robust finance## Mathematics Subject Classification (2010)

90C15 90C39 91G99 28A05 46A20## JEL Classification

C61 G13## 1 Introduction

It is more often the rule, rather than the exception, that socio-economic phenomena are influenced by a strong component of randomness. Starting from the pioneering work of Knight (see e.g. Knight [22, Chap. VII]), a distinction between *risk* and *uncertainty* has been widely accepted with respect to the nature of such a randomness. We often call a situation *risky* if a probabilistic description is available (e.g. the toss of a fair coin). In contrast, we call a situation *uncertain* if it cannot be fully described in probabilistic terms. Simple reasons could be the absence of an objective model (e.g. the result of a horse race; see Bayraktar and Munk [6] and the references therein) or the lack of information (e.g. the draw from an urn whose composition is unknown). The classical literature in mathematical finance has been mainly focusing on risk, and the attention to problems of Knightian uncertainty has been drawn only relatively recently starting from Avellaneda et al. [4]. In particular, fundamental topics such as the theory of arbitrage and the related superhedging duality have been systematically studied in frictionless discrete-time markets in Bayraktar and Zhou [8], Bouchard and Nutz [12] in a quasi-sure framework, and in Acciaio et al. [1], Burzoni et al. [15], Cheridito et al. [17] in a pointwise framework.

Under risk, the classical model of a discrete-time market with proportional transaction costs has been introduced in Kabanov [20]. The model is described by a collection of cones \(\mathbb {K}:=(K_{t})_{t=0,\ldots ,T}\) which determines (i) admissible strategies, (ii) solvency requirements, (iii) pricing mechanisms. More precisely, the latter are called *consistent price systems*, and they are essentially martingale processes taking values in the dual cones \(K^{*}_{t}\). Instances of such models have been considered, in the uncertainty case, in Bartl et al. [5], Bayraktar and Zhang [7], Bouchard and Nutz [13], Burzoni [14], Dolinsky and Soner [19]; nevertheless, the problem of establishing a quasi-sure superhedging duality has remained open. Recently, a first duality result was obtained in Bouchard et al. [11] using a randomisation approach (see also Aksamit et al. [2], Bayraktar and Zhou [9], Deng et al. [18] for other applications). The idea is to construct a fictitious frictionless price process \(\hat{S}\) for which (i) the superhedging price of an option in the market with frictions coincides with the corresponding superhedging price in the frictionless one, and (ii) the class of martingale measures for \(\hat{S}\) produces the same prices for the option as the class of consistent price systems for the original market. When these two properties are achieved, the duality follows from the frictionless results of Bouchard and Nutz [12]. In order to perform this program, a crucial role is played by the assumption of no arbitrage of the second kind (\(\operatorname{NA}_{2}(\mathcal {P})\)), which ensures that the construction of the fictitious frictionless market is automatically arbitrage-free. \(\operatorname{NA}_{2}(\mathcal {P})\) prescribes that if a position is quasi-surely solvent at time \(t+1\), it must be quasi-sure solvent at time \(t\). Such a condition is quite restrictive as it fails in very basic examples of one-period markets, even though no sure profit can be made by market participants (see [7, Remark 11]).

In this paper, we do not require the strong assumption \(\operatorname{NA}_{2}(\mathcal {P})\) and we show the superhedging duality under the more natural condition of strict no arbitrage (\(\operatorname{NA}^{\mathrm{s}}(\mathcal {P})\)). The latter ensures that it is not possible to make profits without taking any risk; thus it generalises the classical no-arbitrage condition in frictionless markets. From a technical perspective, we also do not assume other unnecessary hypotheses taken in Bouchard et al. [11]: (i) we do not require that transaction costs are uniformly bounded, or stated differently, the bid–ask spreads relative to a chosen numéraire are not necessarily subsets of \([1/c,c]\) for some \(c>0\); (ii) we do not require the technical assumption \(K^{*}_{t}\cap \partial \mathbb {R}^{d}_{+}=\{0\}\) for any \(t=0,\ldots , T\). From a modelling perspective, our approach allows extending the previous results to models where a process of portfolio constraints \(C:=(C_{t})_{t=0,\ldots ,T}\) defines the admissible strategies in the market. To the best of our knowledge, these results are new even in the classical case where a reference probability measure ℙ is fixed. As in Bouchard et al. [11], we assume the so-called *efficient friction hypothesis* and adopt a randomisation approach.

We first construct a backward procedure similar to the one of Bayraktar and Zhang [7] and based on a dynamic programming approach (see also Burzoni and Šikić [16] for an extensive study of the related *martingale selection problem*). This procedure yields a new collection of cones \(\tilde{\mathbb {K}}^{*}=(\tilde{K}^{*}_{t})_{t=0,\ldots , T}\) which is in general different from the original \(\mathbb {K}\) and is shown to be nonempty under the condition \(\operatorname{NA}^{\mathrm{s}}(\mathcal {P})\). Notably, it is not possible to apply directly the results of Bouchard et al. [11] (or a straightforward adaptation of them) to \(\tilde{\mathbb {K}}^{*}\). Indeed, in general, \(\tilde{K}^{*}_{t}\) will only have an analytic graph as opposed to the Borel-measurability of \(K^{*}_{t}\). The Borel-measurability assumption is crucial in order to apply the results of Bouchard and Nutz [12] in frictionless markets. To overcome this difficulty, we propose a new randomisation method. We do not design the frictionless process \(\hat{S}\) to take values in \(\tilde{\mathbb {K}}^{*}\), but instead consider a suitable class \(\hat{\mathcal {P}}\) of probabilities in order to have \(\hat{S}_{t}\in \tilde{K}^{*}_{t}\)\(\hat{\mathcal {P}}\)-q.s. at each time. Similarly to Bouchard et al. [11], we finally prove that the desired duality can be deduced from duality results in frictionless market. In particular, we use here results of Bayraktar and Zhou [8], which take into account possible portfolio constraints.

We conclude the introduction by specifying the frequently used notation and the setup. The superhedging duality is stated in Sect. 2. The construction of the fictitious frictionless market is the content of Sect. 3. Finally, we prove the main result in Sect. 4 where we also show how it extends to semi-static trading.

### Notation

*multifunction*and denoted by \(U:X\rightrightarrows Y\). For \(Y=\mathbb {R}^{d}\) and a sigma-algebra \(\mathcal {G}\) on \(X\), a multifunction \(U\) is called \(\mathcal {G}\)

*-measurable*if for any open set \(O\subseteq \mathbb {R}^{d}\), we have

*selector of*\(U\). We denote by \(\mathcal {L}^{0}(\mathcal {G};U)\) the class of \(\mathcal {G}\)-measurable selectors of \(U\). For \(U:X_{1}\times X_{2}\rightrightarrows \mathbb {R}^{d}\) and \(x\in X_{1}\) fixed, the notation \(U(x;\cdot )\) refers to the multifunction \(U\) viewed as a (multi)function on \(X_{2}\). Given a class of probabilities \(\mathcal {R}\subseteq \mathfrak{P}(X_{2})\), the (conditional) quasi-sure support of \(U(x;\cdot )\), denoted by \(\operatorname{supp}_{\mathcal {R}}U(x;\cdot )\), is the smallest closed set \(F\subseteq \mathbb {R}^{d}\) such that \(U(x;\cdot )\subseteq F\) ℛ-q.s. For a collection of multifunctions \(U:=(U_{t})_{t=0}^{T}\) adapted to a given filtration \(\mathbb {G}\), we denote by \(\mathcal {L}^{0}(\mathbb {G}_{-};U)\) the class of processes \(H\) such that \(H_{t+1}\in \mathcal {L}^{0}(\mathcal {G}_{t};U_{t})\) for every \(t=0,\ldots , T-1\). Finally, for two \(\mathbb {R}^{d}\)-valued processes \(H\) and \(S\), we define \((H\bullet S)_{t}:=\sum _{u=0}^{t-1} H_{u+1}\cdot (S_{u+1}-S_{u})\).

### Setup

Let \(T\in \mathbb {N}\) be a fixed time horizon and \(\mathcal {I}:=\{0,\ldots , T\}\). For later use, we also define \(\mathcal {I}_{-1}:=\{-1,\ldots ,T-1\}\). We consider a filtered space \((\Omega ,\mathcal {F},\mathcal {F}^{u},\mathbb {F},\mathbb {F}^{u})\) endowed with a (possibly nondominated) class of priors \(\mathcal {P}\subseteq \mathfrak{P}(\Omega )\) described as follows:

• \(\Omega _{0}\) is a given Polish space, \(\Omega :=\Omega _{T}\), where \(\Omega _{t}\) denotes the \((t+1)\)-fold product of \(\Omega _{0}\). Any \(\omega \in \Omega _{t}\) is denoted by \(\omega =(\omega _{0},\ldots ,\omega _{t})\) with \(\omega _{s}\in \Omega _{0}\) for any \(0\le s\le t\).

• We set \(\mathcal {F}:=\mathcal {B}_{\Omega }\) and call \(\mathcal {F}^{u}\) its universal completion. Similarly, the filtrations \(\mathbb {F}=(\mathcal {F}_{t})_{t\in \mathcal {I}}\) and \(\mathbb {F}^{u}=(\mathcal {F}^{u}_{t})_{t\in \mathcal {I}}\) are given by \(\mathcal {F}_{t}:=\mathcal {B}_{\Omega _{t}}\) and \(\mathcal {F}^{u}_{t}\) is its universal completion.

^{1}with \(\mathcal{P}_{0}\) a constant multifunction. We set

## 2 Main result

We consider the general model of financial markets with proportional transaction costs introduced in Kabanov [20]. The model is fully described by a collection of multifunctions \(\mathbb {K}:=(K_{t})_{t\in \mathcal {I}}\) with values in the family of convex closed cones in \(\mathbb {R}^{d}\) with \(d\ge 2\), called *solvency cones*. These represent the sets of positions, in terms of physical units of \(d\) underlying assets, which can be liquidated to the zero portfolio at zero cost. We assume that any position with nonnegative coordinates is solvent, i.e., \(\mathbb {R}^{d}_{+}\subseteq K_{t}\). The set \(-K_{t}\) represents the class of portfolios which are available at zero cost. We assume that \(K_{t}\) is \(\mathcal {F}_{t}\)-measurable for any \(t\in \mathcal {I}\) with \(K_{0}\) nonrandom. Following standard notation, for a cone \(K\subseteq \mathbb {R}^{d}\), we denote by \(K^{*}:=\{x\in \mathbb {R}^{d}: x \cdot k\ge 0,\, \forall k\in K\}\) its dual cone and by \(K^{\bullet }:=-K^{*}\) its polar cone.

*admissible*if it satisfies \(\eta _{t}\in A_{t}\) for any \(t\in \mathcal {I}\), where

### Assumption 2.1

We assume that \(\operatorname{int}(K^{*}_{t})\neq \emptyset \) for any \(t\in \mathcal {I}\). Moreover, we assume that \(C_{t}\subseteq C_{t+1}\) for any \(t=0,\ldots , T-1\).

The first assumption is known as *efficient friction hypothesis*. The second means that it is allowed to not trade between two periods, which is obviously satisfied in the unconstrained case where \(C_{t}\equiv \mathbb {R}^{d}\) for any \(t\in \mathcal {I}\).

### Definition 2.2

The *strict no arbitrage condition*\(\operatorname{NA}^{\mathrm{s}}(\mathcal {P})\) holds if for all \(t\in \mathcal {I}\), we have \(A_{t}\cap \mathcal {L}^{0}(\mathcal {F}^{u}_{t};K_{t})=\{0\}\).

This condition is the straightforward generalisation to the quasi-sure setting of the classical one (see e.g. Kabanov et al. [21] and the recent paper Kühn and Molitor [23] for a slightly weaker variant of this concept).

### Definition 2.3

A couple \((Z,\mathbb {Q})\) with \(\mathbb {Q}\ll \mathcal {P}\) is called a *(strictly) consistent price system (SCPS)* if \(Z_{t}\in \operatorname{int}(K^{*}_{t})\) ℚ-a.s., \(\forall t\in \mathcal {I}\), and \(H\bullet Z\) is a local ℚ-supermartingale for all \(H\in \mathcal {L}^{0}(\mathbb {F}^{u}_{-};C)\).

The interpretation is that \((Z,\mathbb {Q})\) defines a frictionless arbitrage-free price process which is compatible with the model of transaction costs defined by \((K_{t},C_{t})_{t\in \mathcal {I}}\). We shortly denote by \(\mathcal {S}\) the set of SCPSs and by \(\mathcal {S}^{0}\) the class of normalised SCPSs, i.e., those satisfying \(Z^{d}_{t}=1\) for any \(t\in \mathcal {I}\).

### Theorem 2.4

*Assume*\(\operatorname{NA}^{\mathrm{s}}(\mathcal {P})\).

*For any Borel*-

*measurable random vector*\(G\),

*Moreover*,

*the superhedging price is attained when*\(\pi _{\mathbb {K}}(G)<\infty\).

The proof of Theorem 2.4 is given in Sect. 4. The main difficulty is to establish the result when only dynamic trading is allowed. In Theorem 4.12 below, we extend the duality to the case where also buy-and-hold positions in a finite number of options are allowed. In the following, it will be more convenient to extend the original market with an extra unconstrained component. More precisely, we consider the market \(\bar{\mathbb {K}}\) with \(\bar{K}_{t}:=K_{t}\times \mathbb {R}_{+}\) and \(\bar{C}_{t}:=C_{t}\times \mathbb {R}\) for \(t\in \mathcal {I}\), which also satisfies Assumption 2.1. It is easy to see that \(\pi _{\mathbb {K}}(G)=\pi _{\bar{\mathbb {K}}}(\bar{G})\) with \(\bar{G}=(G,0)\). On the dual side, \(i:\mathcal {S}\to \bar{\mathcal {S}}^{0}\) with \(i(Z,\mathbb {Q})=((Z_{T},1),\mathbb {Q})\) is clearly a bijection and \(\mathbb {E}_{\mathbb {Q}}[G\cdot Z_{T}]=\mathbb {E}_{\mathbb {Q}}[\bar{G} \cdot (Z_{T},1)]\).

Without loss of generality, we assume that \((K_{t},C_{t})\) have one unconstrained component for any \(t\in \mathcal {I}\).

### Lemma 2.5

\(\tilde{K}^{*}_{t}\)*has analytic graph for every*\(t\in \mathcal {I}\).

### Proposition 2.6

*If*\(\mathbb {K}\)*satisfies Assumption *2.1*and*\(\operatorname{NA}^{\mathrm{s}}(\mathcal {P})\), *the same holds for*\(\tilde{\mathbb {K}}\). *In particular*, \(\operatorname{int}(\tilde{K}^{*}_{t})\neq \emptyset \)\(\mathcal {P}\)-*q*.*s*.* for all*\(t\in \mathcal {I}\).

## 3 The randomisation approach

In this section, we construct an enlarged measurable space \((\hat{\Omega },\hat{\mathcal {F}},\hat{\mathcal {F}}^{u},\hat{\mathbb {F}},\hat{\mathbb {F}}^{u})\) endowed with a suitable class \(\hat{\mathcal {P}}\) of probabilities. On this space, we construct a price process \(\hat{S}=(\hat{S}_{t})_{t\in \mathcal {I}}\) which represents a *frictionless* financial market with the property that \(\hat{S}_{t}\in \tilde{K}^{*}_{t}\)\(\hat{\mathcal {P}}\)-q.s. for any \(t\in \mathcal {I}\) (Corollary 3.5 below) and which is arbitrage-free (Proposition 3.9 below).

### Lemma 3.1

*For every*\(t\in \mathcal {I}\), \(\Theta _{t}\)*has an analytic graph*.

### Proof

In the proof, we repeatedly use the fact that the class of analytic sets is closed under countable unions and intersections and that the image of an analytic set under a Borel-measurable function is again analytic.

### Corollary 3.2

*For any*\(t\in \mathcal {I}\),

*the multifunction*

*has analytic graph*.

### Proof

The graph of \(\delta _{\Theta _{t}}\) is the image of the graph of \(\Theta _{t}\) via the map \((\omega ,\theta )\mapsto (\omega ,\delta _{\theta })\), which is an embedding (see [3, Theorem 15.8]). Since the image of an analytic set under a continuous function is again analytic, the claim follows. □

### Proposition 3.3

*The multifunctions*\(\hat{\mathcal {P}}_{t}\)*defined in* (3.5) *have analytic graphs*.

### Proof

We now show that \(\hat{\mathcal {P}}_{t}\) is nonempty on a sufficiently rich set of events.

### Lemma 3.4

*Assume*\(\operatorname{NA}^{\mathrm{s}}(\mathcal {P})\). *The set*\(N_{t}:=\{\omega \in \Omega _{t}: \hat{\mathcal {P}}_{t}(\omega )=\emptyset \}\)*is a universally measurable*\(\mathcal {P}\)-*polar set for any*\(t\in \mathcal {I}_{-1}\). *In particular*, *the same holds for the set*\(N:=\bigcup _{t\in \mathcal {I}_{-1}}N_{t}\).

### Proof

It remains to show that \(N'_{t}=N_{t}\). The inclusion “⊆” follows from the definition of \(B_{t}\). Take now an \(\mathcal {F}^{u}_{t}\)-measurable selector \(P_{t}\) of \(B_{t}\) and \(\delta _{\theta _{t+1}}\in \mathcal {L}^{0}(\mathcal {F}^{u}_{t+1};\delta _{\Theta _{t+1}})\), where \(\delta _{\Theta _{t+1}}\) is defined in Corollary 3.2. Since \(P_{t}[\omega ,\operatorname{dom}(\Theta _{t+1})]=1\) for any \(\omega \in (N'_{t})^{c}\), we can extend \(\delta _{\theta _{t+1}}\) arbitrarily on the complement of \(\operatorname{dom}(\Theta _{t+1})\), and with a slight abuse of notation, we still denote it by \(\delta _{\theta _{t+1}}\). The product measure \(P_{t}\otimes \delta _{\theta _{t+1}}\) belongs to \(\hat{\mathcal {P}}_{t}(\omega )\) for any \(\omega \in (N'_{t})^{c}\). This shows \((N'_{t})^{c}\subseteq (N_{t})^{c}\) and the thesis follows. □

### Corollary 3.5

*Assume*\(\operatorname{NA}^{\mathrm{s}}(\mathcal {P})\).

*For any*\(t\in \mathcal {I}_{-1}\),

*we have*\(\hat{S}_{t+1}\in \operatorname{int}(\tilde{K}_{t+1}^{*})\)\(\hat{\mathcal {P}}\)-

*q*.

*s*.

*and for any*\((\omega ,\theta )\in N_{t}^{c}\times \mathbb {R}^{d-1}\),

### Proof

This follows because \(P_{t}\otimes \delta _{\theta _{t+1}}\) belongs to \(\hat{\mathcal {P}}_{t}\) for any \(\delta _{\theta _{t+1}}\in \mathcal {L}^{0}(\mathcal {F}^{u}_{t+1};\delta _{\Theta _{t+1}})\) and \(P_{t}\) is a measurable selector of \(B_{t}\) in (3.6). □

### Definition 3.6

We call a process \(H\) an *admissible strategy* if \(H_{t+1}\in \mathcal {L}^{0}(\hat{\mathcal {F}}^{u}_{t};C_{t})\) and the *self-financing* condition \((H_{t+1}-H_{t})\cdot \hat{S}_{t}=0\)\(\hat{\mathcal {P}}\)-q.s. holds for \(0\le t\le T-1\). The class of admissible strategies is denoted by \(\hat {\mathcal {H}}^{r}\).

### Definition 3.7

The *no arbitrage condition*\(\mathrm{NA}(\hat{\mathcal {P}})\) holds if \((H\bullet \hat{S})_{T}\ge 0\)\(\hat{\mathcal {P}}\)-q.s. implies \((H\bullet \hat{S})_{T}= 0\)\(\hat{\mathcal {P}}\)-q.s. for any \(H\in \hat {\mathcal {H}}^{r}\).

### Theorem 3.8

*The following are equivalent*:

- (1)
\(\mathrm{NA}(\hat{\mathcal {P}})\).

- (2)
*For any*\(0\le t\le T-1\), \(N'_{t}:=\{(\omega ,\theta )\in \hat{\Omega }_{t}: \mathrm{NA}(\hat{\mathcal {P}}_{t}(\omega))\textit{ fails} \}\in \mathcal {F}^{u}\)*is a*\(\hat{\mathcal {P}}\)-*polar set*. - (3)
*For any*\(\mathbb {P}\in \hat{\mathcal {P}}\),*there exists*\(\mathbb {Q}\in \hat{\mathcal {Q}}\)*such that*\(\mathbb {P}\ll \mathbb {Q}\).

### Proof

The only difference from the proof of [8, Theorem 3.2] is that \(\hat{\mathcal {P}}_{t}(\omega )\) might have empty values on the \(\mathcal {P}\)-polar set \(N_{t}\in \mathcal {F}^{u}\). Recall that \(\operatorname{graph}(\hat{\mathcal {P}}_{t})\) is analytic by Proposition 3.3. Thus also \(\operatorname{dom}(\hat{\mathcal {P}}_{t})\) is an analytic set.

“\((1) \Rightarrow (2)\)” is proved in [8, Lemma 3.3]. It is shown there that \((N'_{t})^{c}\) is equal to the set \(\{\omega \in \Omega _{t}: (\Lambda ^{*}\cap C_{t})(\omega )\subseteq -\Lambda ^{*}( \omega )\}\), where we define the mapping \(\Lambda \) by \(\Lambda (\omega )=\operatorname{supp}_{\hat{\mathcal {P}_{t}}(\omega )}(\hat{S}_{t+1}( \omega ;\cdot )-\hat{S}_{t}(\omega ))\). In our framework, the above set must be intersected with \(\operatorname{dom}(\hat{\mathcal {P}}_{t})\) which is analytic, and therefore the intersection is again universally measurable. The same proof yields that \(N'_{t}\) is \(\mathcal {P}\)-polar.

“\((2) \Rightarrow (3)\)” is based on [8, Lemma 3.4]. The universally measurable kernels \(Q_{t}\) defining \(\mathbb {Q}\in \hat{\mathcal {Q}}\) are constructed outside a \(\mathcal {P}\)-polar set, and in particular, they are chosen as selectors of a set \(\Xi \) with \(\operatorname{dom}(\Xi )=(N'_{t})^{c}\). In our framework, the same \(\Xi \) satisfies \(\operatorname{dom}(\Xi )=(N_{t}\cup N'_{t})^{c}\), which is still universally measurable and \(\mathcal {P}\)-polar. The same proof allows us to conclude.

“\((3) \Rightarrow (1)\)” is standard. □

### Proposition 3.9

\(\operatorname{NA}^{\mathrm{s}}(\mathcal {P})\)*implies*\(\mathrm{NA}(\hat{\mathcal {P}})\).

### Proof

## 4 The superhedging duality

This section is devoted to the proof of Theorem 2.4. To this end, we compare both the primal and the dual problem with their randomised counterpart in the frictionless market induced by \(\hat{S}\) and constructed in Sect. 3. Using duality results known for the frictionless case, we obtain the result.

### 4.1 Equality of the primal problems

We first observe that using admissible strategies with respect to \(\mathbb {K}\) or with respect to \(\tilde{\mathbb {K}}\) yields the same superhedging price.

### Lemma 4.1

\(\pi _{\mathbb {K}}(G)=\pi _{\tilde{\mathbb {K}}}(G)\).

### Proof

We now consider the superhedging problem in the frictionless market defined by \(\hat{S}\). Note that a trading strategy in \(\hat {\mathcal {H}}^{r}\) (see Definition 3.6) could in principle depend on the variable \(\theta \). As this variable is only fictitious, a generic \(\hat{\mathbb {F}}^{u}\)-predictable process cannot consistently identify an element in \(\mathcal {H}^{K}\). Therefore we need to reduce the class of admissible strategies to those which only depend on the variable \(\omega \).

### Definition 4.2

*consistent strategy*\(H=(H_{t})_{t\in \mathcal {I}}\) is an \(\mathbb {R}^{d}\)-valued process satisfying \(H_{t+1}\in \mathcal {L}^{0}(\mathcal {F}^{u}_{t}\otimes \{\emptyset ,\mathbb {R}^{d}\};C_{t})\) for any \(0\le t\le T-1\) and the

*self-financing*condition

### Remark 4.3

Recall that the last component of \(\hat{S}\) serves as a numéraire. The self-financing condition for \(H\in \hat {\mathcal {H}}^{r}\) is standard, namely, it requires that \(-(H^{d}_{t+1}-H^{d}_{t})\) coincides with \(\sum _{i=1}^{d-1}(H^{i}_{t+1}-H^{i}_{t})\hat{S}^{i}_{t}\)\(\hat{\mathcal {P}}\)-q.s. On the other hand, a consistent strategy depends only on the \(\omega \)-variable and hence the position in the numéraire needs to be able to cover the worst-case scenario for the price of \(\hat{S}_{t}\), which explains (4.1). We show below that for any consistent strategy, the left-hand side of (4.1) is measurable.

### Lemma 4.4

*Let*\(\mathbb {P}\in \mathcal {P}\), \(H\in \hat {\mathcal {H}}\).

- (1)
*Suppose that*\(\mathbb {P}[A^{\infty }]>0\).*Then for any*\(n\in \mathbb {N}\),*there exists*\(\hat{\mathbb {P}}^{n}\in \hat{\mathcal {P}}\)*such that*\(\hat{\mathbb {P}}^{n}|_{\Omega }=\mathbb {P}\)*and*\(\hat{\mathbb {P}}^{n}[\sum _{i=1}^{d-1}\Delta H^{i}_{t}\ \hat{S}^{i}_{t}\ge n]>0\)*for some*\(0\le t\le T-1\). - (2)
*Suppose that*\(\mathbb {P}[A^{\infty }]=0\).*Then for any*\(n\in \mathbb {N}\),*there exists*\(\hat{\mathbb {P}}^{n}\in \hat{\mathcal {P}}\)*such that*\(\hat{\mathbb {P}}^{n}|_{\Omega }=\mathbb {P}\)*and*\(\sum _{i=1}^{d-1}\Delta H^{i}_{t}\ \hat{S}^{i}_{t}\ge -\Delta H^{d}_{t}- \frac{1}{n}\)\(\hat{\mathbb {P}}^{n}\)-*a*.*s*.*for any*\(0\le t\le T-1\).

### Proof

### Lemma 4.5

\(\pi _{\tilde{\mathbb {K}}}(G)=\hat {\pi }(G\cdot \hat{S}_{T})\).

### Proof

### 4.2 Equality of the dual problems

### Proposition 4.6

*For any random vector*\(G\in \mathcal {F}^{u}\),

### Proof

Conversely, suppose \(\hat{\mathbb {Q}}\in \hat{\mathcal {Q}}\). We define \((Z,\mathbb {Q})\) via \(\mathbb {Q}:=\hat{\mathbb {Q}}_{\mid _{\Omega }}\) and \(Z_{t}:=\mathbb {E}_{\hat{\mathbb {Q}}}[\hat{S}_{t}|\mathcal {F}_{t}]\) for every \(t\in \mathcal {I}\). Denote by \((\hat{Q}_{t})_{t\in \mathcal {I}_{-1}}\) (respectively \((Q_{t})_{t\in \mathcal {I}_{-1}}\)) the disintegration of \(\hat{\mathbb {Q}}\) (respectively of ℚ). From \(\mathbb {E}_{\hat{Q}_{t}}[y\cdot \Delta \hat{S}_{t+1}(\hat{\omega };\cdot )] \le 0\) for any \(y\in C_{t}\) and from the fact that \(C_{t}\) is \(\mathcal {F}_{t}\)-measurable, we deduce that \(Q_{t}\) satisfies (4.8) for any \(t=0,\ldots , T-1\). Moreover, since \(\hat{\mathbb {Q}}\ll \hat{\mathcal {P}}\), we obtain by the definition of \(\hat{\mathcal {P}}\) that \(\mathbb {Q}\ll \mathcal {P}\) and \(Z_{t}\) takes values in \(\operatorname{int}(\tilde{K}^{*}_{t})\) ℚ-a.s. for any \(t\in \mathcal {I}\). We conclude that \((Z,\mathbb {Q})\in \tilde{\mathcal {S}}^{0}\). Moreover, we obviously have \(\mathbb {E}_{\mathbb {Q}} [G\cdot Z_{T}]=\mathbb {E}_{\hat{\mathbb {Q}}}[G\cdot \hat{S}_{T}]\). □

### 4.3 Proof of Theorem 2.4

*consistent*trading in the enlarged market (compare with (4.3)). It remains to show that the same price is obtained with

*randomised*strategies as defined in (4.2); in other words, we need to prove that \(\hat {\pi }(G\cdot \hat{S}_{T})=\hat {\pi }^{r}(G\cdot \hat{S}_{T})\). Denote by \(USA(\hat{\Omega }_{t},t)\) the class of upper semianalytic functions \(g:\hat{\Omega }_{t}\rightarrow \mathbb {R}\) which depend on \(\theta \) only through \(\theta _{t}\), i.e.,

### Proposition 4.7

*Suppose*\(T=1\)*and*\(g\in USA(\hat{\Omega }_{T},T)\). *If*\(\mathrm{NA}(\hat{\mathcal {P}})\)*holds true*, *then*\(\hat {\pi }(g)= \hat {\pi }^{r}(g)\).

### Proof

^{2}We define

Note that if \(G:\Omega \rightarrow \mathbb {R}^{d}\) is a Borel-measurable vector, \(G\cdot \hat{S}_{T}:\hat{\Omega }\rightarrow \mathbb {R}\) is a Borel-measurable function which depends on \(\theta \) only through \(\theta _{T}\). In particular, Proposition 4.7 together with Lemmas 4.1 and 4.5 yields \(\pi _{K}(G)= \hat {\pi }^{r}(G\cdot \hat{S}_{T})\).

### Proof of Theorem 2.4 for \(T=1\)

### Lemma 4.8

*For any*\(0\le t\le T-1\),

*the function*\(f:\Omega _{t}\times \mathbb {R}^{d}\times \mathbb {R}^{d}\rightarrow \overline{\mathbb {R}}\)

*defined as*

*is a universally measurable normal integrand*.

### Proof

Denote by \(f^{\tilde{\mathbb {P}}}(\omega ,h,x)\) the functions on the right-hand side of (4.13) over which the supremum is taken. From [24, Corollary 14.41], we need to check that

(1) for any \((\omega ,h)\in \Omega _{t}\times \mathbb {R}^{d}\), the function \(f(\omega ,h,\cdot )\) is lower semicontinuous;

### Remark 4.9

Note that for any \(\omega \in \Omega _{t}\), the right-hand side of (4.13) is equal to \({\inf \{K\in \mathbb {R}: X\le K\ \hat{\mathcal {P}}_{t}(\omega )\text{-q.s.}\}}\), where \(X\) is the random variable inside the expectation. In particular, this is equal to the minimal amount at time \(t\) for which the strategy \(x\) is a superhedge for \(g_{t+1}\), given that \(h\) is the strategy used at time \(t-1\). Moreover, by the construction of \(\tilde{\mathcal {P}}_{t}\), the strategy \(x\) with the initial amount \(f(\omega ,h,x)\) is a (conditional) superhedging strategy which depends only on the event \(\omega \) and not on the event \((\omega ,\theta )\). In the terminology of Definition 4.2, this construction provides *consistent strategies*.

Recall that \(\mathrm{NA}(\hat{\mathcal {P}}_{t}(\omega))\) is the conditional version of \(\mathrm{NA}(\hat{\mathcal {P}})\) (see Theorem 3.8).

### Proposition 4.10

*Let*\(0\le t\le T-1\)

*and assume*\(\mathrm{NA}(\hat{\mathcal {P}}_{t}(\omega))\).

*There exist a universally measurable map*\(\varphi :\Omega _{t}\times \mathbb {R}^{d}\rightarrow \mathbb {R}^{d}\)

*and a*\(\mathcal {P}\)-

*polar set*\(N\)

*such that for any*\((\omega ,h)\in N^{c} \times \mathbb {R}^{d}\),

*we have*

*and*\(g'_{t}(\omega ,h)>-\infty \).

### Proof

*consistent*conditional superhedging price of \(g_{t+1}\) given \((\omega ,h)\) as the map

### Proof of Theorem 2.4 for \(T>1\)

### 4.4 The case with options

We now consider the case where a finite number of options \(\varphi _{1},\ldots ,\varphi _{e}\) are available for semi-static trading. In this section, we show that this case can be embedded in the previous one. For any \(k=1,\ldots ,e\), we assume that \(\varphi _{k}:\Omega \to \mathbb {R}^{d}\) is a Borel-measurable function representing the terminal payoff of an option, in terms of physical units of an underlying \(d\)-dimensional asset. Any \(\varphi _{k}\) has bid and ask prices at time 0 denoted, respectively, by \(b_{k}\) and \(a_{k}\). We set \(\Phi := (\varphi _{1},\dots ,\varphi _{e},-\varphi _{1},\dots ,- \varphi _{e})\) with corresponding prices \(p:=(a_{1},\ldots ,a_{e},-b_{1},\ldots ,-b_{e})^{T}\). \(\Phi \) takes values in \(\mathbb {R}^{d\times m}\) and \(p\in \mathbb {R}^{m}\) with \(m:=2e\). For ease of notation, we relabel the options and incorporate their price in the payoff so that \(\Phi = (\phi _{1}-p_{1}e_{d},\dots ,\phi _{m}-p_{m}e_{d})\). In addition, we suppose that we are given a dynamic trading market \((\mathbb {K},C)\) satisfying all the hypotheses of Sect. 2. An admissible strategy has the form \(\bar{\eta }:=(\eta ,\alpha )\), where \(\eta \in \mathcal {H}^{K}\) is a dynamic strategy and \(\alpha \in \mathbb {R}^{m}_{+}\).

### Definition 4.11

We say that \(\operatorname{NA}_{\Phi}^{\mathrm{s}}(\mathcal {P})\) holds if \(\operatorname{NA}^{\mathrm{s}}(\mathcal {P})\) holds for the dynamic trading market \((\mathbb {K},C)\) and \(\eta _{T}+\Phi \alpha \in K_{T}\)\(\mathcal {P}\)-q.s. implies \(\alpha =0\).

### Theorem 4.12

*Assume*\(\operatorname{NA}_{\Phi}^{\mathrm{s}}(\mathcal {P})\).

*For any Borel*-

*measurable random vector*\(G\),

*Moreover*,

*the superhedging price is attained when*\(\pi _{\mathbb {K},\Phi }(G)<\infty \).

in the first \(d\) components, \(\bar{S}_{t}(\omega ,\theta ,x)=\hat{S}_{t}(\omega ,\theta )\), and

- in the last \(m\) components, \(\bar{S}_{t}(\omega ,\theta ,x)=x\) for \(1\le t\le T-1\), and$$ \bar{S}^{k}_{0}=p_{k},\quad \bar{S}^{k}_{T}(\omega ,\theta ,x)=\phi _{k}( \omega )\cdot \hat{S}_{T}(\omega ,\theta ),\qquad k=d+1,\ldots , m. $$

### Lemma 4.13

\(\bar{\pi }(G\cdot \hat{S}_{T})=\bar{\pi }_{\Phi }(G\cdot \hat{S}_{T})\).

### Proof

### Proof of Theorem 4.12

## Footnotes

## Notes

## References

- 1.Acciaio, B., Beiglböck, M., Penkner, F., Schachermayer, W.: A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance
**26**, 233–251 (2016) MathSciNetCrossRefGoogle Scholar - 2.Aksamit, A., Deng, S., Obłój, J., Tan, X.: The robust pricing-hedging duality for American options in discrete time financial markets. Math. Finance
**29**, 861–897 (2019) MathSciNetCrossRefGoogle Scholar - 3.Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer, Berlin (2006) zbMATHGoogle Scholar
- 4.Avellaneda, M., Lévy, A., Paras, A.: Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance
**2**, 73–88 (1995) CrossRefGoogle Scholar - 5.Bartl, D., Cheridito, P., Kupper, M., Tangpi, L.: Duality for increasing convex functionals with countably many marginal constraints. Banach J. Math. Anal.
**11**, 72–89 (2017) MathSciNetCrossRefGoogle Scholar - 6.Bayraktar, E., Munk, A.: High-roller impact: a large generalized game model of parimutuel wagering. Mark. Microstruct. Liq.
**03**, 1750006 (2017) CrossRefGoogle Scholar - 7.Bayraktar, E., Zhang, Y.: Fundamental theorem of asset pricing under transaction costs and model uncertainty. Math. Oper. Res.
**41**, 1039–1054 (2016) MathSciNetCrossRefGoogle Scholar - 8.Bayraktar, E., Zhou, Z.: On arbitrage and duality under model uncertainty and portfolio constraints. Math. Finance
**27**, 988–1012 (2017) MathSciNetCrossRefGoogle Scholar - 9.Bayraktar, E., Zhou, Z.: No-arbitrage and hedging with liquid American options. Math. Oper. Res.
**44**, 468–486 (2019) MathSciNetCrossRefGoogle Scholar - 10.Bertsekas, D.P., Shreve, S.E.: Stochastic Optimal Control: The Discrete Time Case. Academic Press, New York (1978) zbMATHGoogle Scholar
- 11.Bouchard, B., Deng, S., Tan, X.: Superreplication with proportional transaction cost under model uncertainty. Math. Finance
**29**, 837–860 (2019) MathSciNetCrossRefGoogle Scholar - 12.Bouchard, B., Nutz, M.: Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab.
**25**, 823–859 (2015) MathSciNetCrossRefGoogle Scholar - 13.Bouchard, B., Nutz, M.: Consistent price systems under model uncertainty. Finance Stoch.
**20**, 83–98 (2016) MathSciNetCrossRefGoogle Scholar - 14.Burzoni, M.: Arbitrage and hedging in model-independent markets with frictions. SIAM J. Financ. Math.
**7**, 812–844 (2016) MathSciNetCrossRefGoogle Scholar - 15.Burzoni, M., Frittelli, M., Hou, Z., Maggis, M., Obłój, J.: Pointwise arbitrage pricing theory in discrete time. Math. Oper. Res.
**44**, 1034–1057 (2019) MathSciNetCrossRefGoogle Scholar - 16.Burzoni, M., Šikić, M.: Robust martingale selection problem and its connections to the no-arbitrage theory. Math. Finance (2019). Forthcoming, available online at https://doi.org/10.1111/mafi.12225 CrossRefGoogle Scholar
- 17.Cheridito, P., Kupper, M., Tangpi, L.: Duality formulas for robust pricing and hedging in discrete time. SIAM J. Financ. Math.
**8**, 738–765 (2017) MathSciNetCrossRefGoogle Scholar - 18.Deng, S., Tan, X., Yang, X.: Utility maximization with proportional transaction costs under model uncertainty. Math. Oper. Res. (2019). To appear, available online at arXiv:1805.06498
- 19.Dolinsky, Y., Soner, H.M.: Robust hedging with proportional transaction costs. Finance Stoch.
**18**, 327–347 (2014) MathSciNetCrossRefGoogle Scholar - 20.Kabanov, Yu.M.: Hedging and liquidation under transaction costs in currency markets. Finance Stoch.
**3**, 237–248 (1999) CrossRefGoogle Scholar - 21.Kabanov, Yu.M., Rásonyi, M., Stricker, C.: On the closedness of sums of convex cones in \(L^{0}\) and the robust no-arbitrage property. Finance Stoch.
**7**, 403–411 (2003) MathSciNetCrossRefGoogle Scholar - 22.Knight, F.: Risk, Uncertainty and Profit. Houghton Mifflin, Boston (1921) Google Scholar
- 23.Kühn, C., Molitor, A.: Prospective strict no-arbitrage and the fundamental theorem of asset pricing under transaction costs. Finance Stoch.
**23**, 1049–1077 (2019) MathSciNetCrossRefGoogle Scholar - 24.Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998) CrossRefGoogle Scholar
- 25.Terkelsen, F.: Some minimax theorems. Math. Scand.
**31**, 405–413 (1972) MathSciNetCrossRefGoogle Scholar

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