A unified Framework for Robust Modelling of Financial Markets in discrete time

We unify and establish equivalence between the pathwise and the quasi-sure approaches to robust modelling of financial markets in discrete time. In particular, we prove a Fundamental Theorem of Asset Pricing and a Superhedging Theorem, which encompass the formulations of [Bouchard, B.,&Nutz, M. (2015). Arbitrage and duality in nondominated discrete-time models. The Annals of Applied Probability, 25(2), 823-859] and [Burzoni, M., Frittelli, M., Hou, Z., Maggis, M.,&Obloj, J. (2019). Pointwise arbitrage pricing theory in discrete time. Mathematics of Operations Research]. In bringing the two streams of literature together, we also examine and relate their many different notions of arbitrage. We also clarify the relation between robust and classical $\mathbb{P}$-specific results. Furthermore, we prove when a superhedging property w.r.t. the set of martingale measures supported on a set of paths $\Omega$ may be extended to a pathwise superhedging on $\Omega$ without changing the superhedging price.


Introduction
Mathematical models of financial markets are of great significance in economics and finance and have played a key role in the theory of pricing and hedging of derivatives and of risk management. Classical models, going back to (Samuelson, 1965) and (Black and Scholes, 1973) in continuous time, specify a fixed probability measure P to describe the asset price dynamics. They led to a powerful theory of complete, and later incomplete, financial markets. The original models have undergone a myriad of variations including, amongst others, local and stochastic volatility models and have been widely applied. However, they also faced important criticism for ignoring the issue of model uncertainty, particularly so in the wake of the 2007/08 financial crisis. Consequently, inspired by the theoretical developments going back to (Knight, 1921), new modelling approaches emerged which aim to address this fundamental issue. These can be broadly divided into two streams based on the so-called quasi-sure and pathwise approaches respectively. The quasi-sure approach introduces a set of priors P representing possible market scenarios. These priors can be very different and P typically contains measures which are mutually singular. This presents significant mathematical challenges and led to the theory of quasi-sure stochastic analysis (see, e.g., (Peng, 2004;Denis and Martini, 2006)). In discrete time, this framework was abstracted in (Bouchard and Nutz, 2015), which we call the quasi-sure formulation in the rest of this paper. By varying the set of probability measures P between the "extreme" cases of one fixed probability measure, P = {P}, and that of considering all probability measures, P = P(X), this formulation allows for widely different specifications of market dynamics. The quasi-sure approach has been employed to consider model uncertainty along market frictions and other related problems, see e.g. (Bayraktar and Zhou, 2017;Bayraktar and Zhang, 2016). The pathwise approach addresses Knightian uncertainty in market modelling by describing the set of market scenarios in absence of a probability measure or any similar relative weighting of such scenarios. It is also referred to as the pointwise, or ω by ω, approach and it bears similarity to the way central banks carry out stress tests using scenario generators. In discrete time a suitable theory was obtained in (Burzoni et al., 2019a), based on earlier developments in (Burzoni et al., 2017(Burzoni et al., , 2016. The methodology builds on the notion of prediction sets introduced in (Mykland et al., 2003) and used in continuous time in (Hou and Ob lój, 2018). The particular case of including all scenarios is often referred to as the model-independent framework and was pioneered in (Davis and Hobson, 2007) and (Acciaio et al., 2016). From here, a further model specification is carried out by including additional assumptions, which represent the different agents' beliefs. In this manner paths deemed impossible by all agents are eliminated. The remaining set of paths is then called the prediction set, or the model. Both approaches, the quasi-sure and the pathwise, allow thus to interpolate between the two ends of the modelling spectrum, as identified by (Merton, 1973): the modelindependent and the model-specific settings (see Figure 1). In doing so, they allow to capture how their outputs change in function of adding or removing modelling assumptions, thus allowing to quantify the impact and risk that a given set of assumptions bear on the problem at hand, see (Cont, 2006). Both approaches were Our main contribution is to unify these two approaches to model uncertainty. We show that, under mild technical assumptions, the pathwise and quasi-sure Fundamental Theorems of Asset Pricing and Superhedging Dualities can be inferred from one another and are thus equivalent. Our statements follow a meta-structure outlined below: Metatheorem. Suppose we are in the quasi-sure setting with a given set of priors P. Then, there exists a suitable selection of scenarios Ω P such that the pathwise result for Ω P implies the quasi-sure result for P. Conversely, suppose we are given a selection of scenarios Ω. Then, there is a set of priors P Ω such that the quasi-sure result for P Ω implies the pathwise result for Ω.
Establishing such equivalence allows us to gain significant additional insights into the core objects in both approaches, as well as clarify links to the classical modelspecific setting. In particular, when transposing the results from the pointwise to the quasi-sure setup, the key technical analytic product structure assumption in Bouchard and Nutz (2015), see Definition 2.1 below, is deduced naturally from the analyticity of the set of scenarios in (Burzoni et al., 2019a). When establishing the Superhedging Theorem, we not only show that the pathwise superhedging price of g is equal to the quasi-sure one, but we also show that both are equal to the model-specific P-superhedging price, where P depends on the setting, i.e., on P or equivalently on Ω, but also on the payoff g. Finally, the key implication in the proof of the robust Fundamental Theorem of Asset Pricing, i.e., (5) ⇒ (1) in Theorem 2.7 below, is obtained by carefully constructing a suitable P ∈ P which does not admit an arbitrage in the classical sense and hence admits an equivalent martingale measure. Furthermore, we survey and relate the concepts of arbitrage used in both approaches. We provide an extensive list of arbitrage notions introduced and used across the literature on robust finance and establish clear relations between them. We also investigate in detail the notion of pathwise superhedging. As noted in (Burzoni et al., 2017), the pathwise superhedging duality does not hold for general claims g when superhedging on a general set Ω is required. Instead, one has to consider hedging on a smaller "efficient" set Ω * (defined as the largest set supported by martingale measures and contained in Ω) to retain the pricing-hedging duality. We clarify when this is necessary and when one can extend the superhedging duality from Ω * to Ω. Intuitively, since there are arbitrage opportunities on Ω \ Ω * , one could try to superhedge the claim g on Ω \ Ω * without any additional cost by implementing an arbitrage strategy. We provide a number of counterexamples to show this idea is not feasible in general and link this to measurability constraints on arbitrage strategies, which were also encountered in (Burzoni et al., 2016). We then show that the above-mentioned intuition is only true for essentially uniformly continuous g under certain regularity conditions on Ω. The rest of the paper is organised as follows. Section 2 contains the main results. First, in Section 2.1, we introduce the general setup in which we work. We discuss different notions of (robust) arbitrage in Section 2.2. Then, in Section 2.3, we establish our version of the robust Fundamental Theorem of Asset Pricing which unifies the quasi-sure and pathwise perspectives. And in Section 2.4, we state a robust Superhedging Theorem. Section 3 presents complementary results on extending the superhedging duality from Ω * to Ω without additional cost and on relations between two strong notions of pathwise arbitrage. Finally, Section 4 contains technical results and most of the proofs. In particular, we give the proofs of Theorems 2.6 and 2.7 in Section 4.1 and of Theorem 2.9 in Section 4.2.

Unified Framework for Robust Modelling of Financial Markets
2.1. Trading strategies and pricing measures. We use notation similar to (Bouchard and Nutz, 2015) and work in their setting, so we only recall the main objects of interest here and refer to (Bouchard and Nutz, 2015) and (Bertsekas and Shreve, 1978, Chapter 7) for technical details. Let T ∈ N and X 1 be a Polish space. We define for t ∈ {1, . . . T } the Cartesian product X t := X t 1 and define X := X T , with the convention that X 0 is a singleton. We denote by B(X) the Borel sets on X, by P(X) the set of probability measures on B(X) and define the function proj t : X → X 1 which projects ω ∈ X to the t-th coordinate, i.e., proj t (ω) = ω t . Next we specify the financial market. Let d ∈ N, F an arbitrary filtration and let S t = (S 1 t , . . . , S d t ) : X t → R d be Borel-measurable, 0 ≤ t ≤ T , and adapted. All prices are given in units of a numeraire, S 0 , which itself is thus normalised, S 0 t ≡ 1, Trading strategies H(F) are defined as the set of F-predictable R d -valued processes. All trading is frictionless and self-financing. Given H ∈ H(F), we denote with H • S t representing the cashflow at time t from trading using H. Above, and throughout, H is a row vector, S is a column vector and 1 denotes either a scalar or a column vector (1, . . . , 1) T . We let Φ denote the vector of payoffs of the statically traded assets Φ = (φ λ : λ ∈ Λ), where Λ is some index set.
For notational convenience, we often identify Φ with the set of its elements. We assume that each φ ∈ Φ is Borel-measurable. When there are no statically traded assets we write Φ = 0. These assets, which we think of as options, can only be bought or sold at time zero (without loss of generality at zero cost) and are held until maturity T . A trading position h can only hold finitely many of these assets, h ∈ c 00 (Λ) the space of sequences of reals indexed by Λ with only finitely many nonzero elements, and generates the payoff h·Φ = λ∈Λ h λ φ λ at time T. We call a pair (h, H) ∈ c 00 (Λ) × H(F) a semistatic trading strategy. The class of such strategies is denoted A Φ (F) := c 00 (Λ) × H(F). For technical reasons we also introduce the level sets of S, which are denoted by ..,T the natural filtration generated by S and let F U t be the universal completion of F 0 t , t = 0, . . . , T . Furthermore we write (X, F U ) for (X T , F U T ) and often consider (X t , F U t ) as a subspace of (X, F U ). Within this setup, the literature on robust pricing and hedging adopts two approaches to model an agent's beliefs. One stream is scenario-based and proceeds by specifying a prediction set Ω ⊆ X, which describes the possible price trajectories. The other stream proceeds by specifying a set of probability measures P ⊆ P(X), which determines the set of negligible outcomes. We refer to the latter as the quasi-sure approach, while the former is usually called the pathwise, or pointwise, approach. In both cases, the model specification may depend on the agent's market information as well as on her specific modelling assumptions. Changing the sets Ω or P can be seen as a natural way to interpolate between different beliefs. One of the principal aims of this paper is to show that both model approaches are equivalent in terms of corresponding FTAPs and Superhedging prices.
In order to aggregate trading strategies on different level sets Σ ω t in a measurable way, we always assume in this paper that Ω is analytic and P has the following structure: where the sets P t (ω) ⊆ P(X 1 ) are nonempty, convex and graph(P t ) = {(ω, P) | ω ∈ X t , P ∈ P t (ω)} is analytic.
This structure facilitates a dynamic programming principle and allows to essentially paste together one-step results in order to establish their multistep counterparts. In order to formulate a Fundamental Theorem of Asset pricing we need to define the dual objects to trading strategies: the pricing (martingale) measures. Given a set of measures P, following (Bouchard and Nutz, 2015), we define Q P,Φ := {Q ∈ P(X) | S is an F U -martingale under Q, ∃P ∈ P s.t. Q P, which, in the model-specific case P = {P}, is simply the familiar set of all martingale measures equivalent to P. Within the pathwise approach, for a set Ω ⊆ X and a filtration F, we define where P f (X) denotes the finitely supported Borel probability measures on (X, B(X)). As a general convention, in this paper we interpret the above sub-and super-scripts as restrictions on the sets of measures. When we drop some of them it is to indicate that these conditions are not imposed, e.g., M Ω (F) denotes all F-martingale measures supported on Ω. Next let with the same convention regarding sub-and super-scripts as above. We also define All these filtrations generate the same martingale measures on Ω calibrated to Φ, which we denote by M Ω,Φ .
For P ∈ P(X), thus N P := N P (F U ) denotes the collection of its null sets. Likewise, given a family P ⊂ P(X), the collection of its polar sets if given by N P = P∈P N P . We say that a property holds P-q.s. if it holds outside a P-polar set. 2.2. Notions of Arbitrage. One of the most important underlying concepts in financial mathematics is the absence of arbitrage. In the literature on robust pricing and hedging many notions of arbitrage have been proposed to date. We present these here together in a unified manned and discuss their relative dependencies. To complement the picture, we establish some novel technical results. These are postponed to Section 3.2. Definition 2.3. Fix a filtration F, a set P, a set S of subsets of X and a set Ω. Recall that semistatic admissible trading strategies are given by (h, H) ∈ A Φ (F).

CA(P)
A Classical Arbitrage in P (see (Davis and Hobson, 2007)) is a family of strategies (h P , H P ) P∈P such that, for all P ∈ P, (h P , H P ) is a Parbitrage. WA(P) A Weak Arbitrage (see (Blanchard and Carassus, 2019)) is a strategy (h, H) ∈ A Φ (F) which is a P-arbitrage for some P ∈ P. IntA(P) An Interior Arbitrage (see (Bayraktar et al., 2014)) is a sequence of strategies (h n , H n ) ∈ A Φ (F) such that (h n , H n ) is a P-quasi-sure Arbitrage relative to option payoffs given by Φ + sign(h n )/n for all n large enough. WFLVR(Ω) A Weak Free Lunch With Vanishing Risk (see (Cox and Ob lój, 2011), (Cox et al., 2016)) is a sequence of strategies (h n , H n ) ∈ A Φ (F) such that there exists a constant c ≥ 0 and (h, locA(P t (ω)) A (t, ω)-local P-quasi-sure Arbitrage (see (Bartl, 2019)) is a strategy , . . . , T − 1} and ω ∈ X) and there exists P ∈ P t (ω) such that P(H∆S t+1 > 0) > 0. A(S) An Arbitrage de la Classe S (see (Burzoni et al., 2016)) is a strategy When we want to stress the role of the filtration we include it as an argument, e.g., we write, e.g., SA(Ω, F). When the filtration is not specified it is implicitly taken to be F U . We use a prefix N to indicate a negation of any of the above notions, e.g., we say that "NA(P) holds" when there does not exist a P-quasi-sure arbitrage strategy, likewise NUSA(Ω) denotes the absence of a uniformly strong arbitrage on Ω, etc. (2) SA(Ω) ⇒ WFLVR(Ω).
Proof. Items we have, for any ε > 0, where |h| 1 = λ∈Λ |h λ |. Absence of IntA(P) implies that there exists ε > 0 such that for any strategy as above we have so that h = 0 and absence of A(P) follows so (5) holds.
USA(X) was first discussed in (Davis and Hobson, 2007), see also (Cox and Ob lój, 2011) and (Cox et al., 2016) for a definition of USA(Ω) and WFLVR(Ω), where Ω ⊆ X. Note that if we take (h, H) = (0, 0) in the definition of WFLVR(Ω) and replace the pathwise inequalities by their P-a.s. counterparts for some fixed P ∈ P(X), we recover a discrete version of the NFLVR condition of (Delbaen and Schachermayer, 1994).  (Riedel, 2015) who introduced 1pA(Ω) and OA(Ω). OA(Ω) is furthermore defined in the setup of (Dolinsky and Soner, 2014). A(P) was introduced in the quasi-sure setting of (Bouchard and Nutz, 2015), where they prove a quasi-sure Fundamental Theorem of Asset pricing and Superhedging Theorem. From Lemma 2.4 above we see that the crucial distinction between CA(P) and A(P) is the aggregation of arbitrage strategies, which poses a fundamental technical difficulty overcome in Bouchard and Nutz (2015) by the specific (APS) structure of P. We also note that CA(P) was actually referred to as weak arbitrage in (Davis and Hobson, 2007). The notion of interior arbitrage IntA(P) was introduced, and called a robust arbitrage, by Bayraktar et al. (2014) in the context of transaction costs. Absence of IntA(P) is equivalent to absence of A(P) not only at the current prices of statically traded options Φ but also under all, sufficiently small, perturbations of their prices. This notion was also used in (Hou and Ob lój, 2018, Assumption 3.1). It is equivalent to saying that the prices of the options Φ are strictly inside the region of their P-q.s. no-arbitrage prices, thus avoiding the delicate issue of boundary classification. In general, IntA(P) does not imply A(P). To see this, take Φ = {(S T − K) + } for some K > S 0 and ∅ = P ⊆ {P ∈ P(X) | P(S T ≤ K) = 1}. Then there is no P-q.s. arbitrage, while for every ε > 0 we have (S T − K) + + ε ≥ ε > 0 and thus RA(P) holds. Throughout the remainder of this paper, unless otherwise stated, we take Λ = {1, . . . , k}, i.e., we have a finite Φ with k statically traded options.

Robust Fundamental Theorem of Asset Pricing. The first Fundamental
Theorem of Asset Pricing characterises absence of arbitrage in terms of existence of martingale (pricing) measures. In the classical discrete-time setting, this refers to the notion of P-arbitrage. However, in a robust setting, there are many possible notions of arbitrage one can consider. If we adopt a strong notion of arbitrage, its absence should be equivalent to a weak statement, e.g., M Ω,Φ = ∅. This is often done in the pathwise literature, see (Burzoni et al., 2019a), and leads to a robust (multi-prior) version of the familiar Dalang-Morton-Willinger theorem.
Theorem 2.5 (Robust DMW Theorem). Let P be a set of probability measures satisfying (APS). Then there exists a universally measurable set of scenarios Ω with P(Ω) = 1 for all P ∈ P and a filtrationF with F 0 ⊆F ⊆ F M , such that the following are equivalent: Conversely, for an analytic set Ω there exists a set P satisfying (APS) such that for all ω ∈ Ω there exists P ∈ P with P({ω}) > 0 and such that (1)-(5) are equivalent.
The above result follows from Theorem 2.7 below by setting S = {Ω}. To see its opposite twin we should adopt a weak notion of arbitrage, its absence thus being equivalent to a strong statement, e.g., for all P ∈ P there exists Q ∈ Q P,Φ such that P Q. This route is most often taken in the quasi-sure literature, see (Bouchard and Nutz, 2015), and leads to the following version of the robust FTAP.
Theorem 2.6. Let P be a set of probability measures satisfying (APS). Then there exists an analytic set of scenarios Ω with P(Ω) = 1 for all P ∈ P, such that the following are equivalent: (1) N1pA(Ω * Φ ) holds and Ω = Ω * Φ P-q.s.
Conversely, if Ω is an analytic set, then there exists a set P of probability measures satisfying (APS) such that for all ω ∈ Ω there exists P ∈ P with P({ω}) > 0 and such that the following are equivalent: (1) N1pA(Ω) holds and Ω = Ω * Φ .
Our proof of this theorem, given in Section 4.1, does not rely on the proof of (3) ⇒ (2) given in (Bouchard and Nutz, 2015). Instead we give pathwise arguments. In particular, given P ∈ P such that P(Ω \ Ω * Φ ) > 0 we explicitly construct a quasisure Arbitrage strategy using the Universal Arbitrage Aggregator of (Burzoni et al., 2019a). This strengthens the results of Theorem 2.7 below. Indeed, using the fact that P satisfies (APS), it is possible to select Q ∈ Q P,Φ for each P ∈ P such that P Q. Necessarily the support of each P is then concentrated on Ω * Φ . Finally, we give our main abstract result, which establishes a pathwise and probabilistic characterisation of the absence of Arbitrage de la Classe S. Its proof is presented in Section 4.1. As noted above, Arbitrage de la Classe S allows to consider many notions of arbitrage at once. Accordingly, the main result below implies Theorem 2.5 and can be strengthened to imply Theorem 2.6 as will be seen in Section 4.
Theorem 2.7. Assume that P satisfies (APS) and S ⊆ B(X) is such that Then there exists a co-analytic set of scenarios Ω such that P(Ω) = 1 for all P ∈ P and a filtrationF with F 0 ⊆F ⊆ F M , such that the following are equivalent: Conversely, for an analytic set Ω there exists a set P satisfying (APS), such that for all ω ∈ Ω there exists P ∈ P with P({ω}) > 0 and such that (1)-(5) are equivalent.
to hold, where we refer to Section 4.1 for a formal definition of Ω P . Conditions (2.2) and (2.3) are compatibility conditions on Ω, S, Φ and P. Indeed, they assert that the (likely uncountable) union of "inefficient" subsets of Ω P \ (Ω P ) * Φ (modulo P-polar sets), stays an "inefficient" subset (modulo P-polar sets). If this condition is not satisfied for some arbitrary P and S, then there is no reason why a set Ω for which (2) holds should exist. Take for example a collection P having densities and S a set of singletons in X. Then P(C) = 0 for any P ∈ P and any C ∈ S so the only Ω which could satisfy (2) is the empty set. We note that when S = {C ⊆ X | C open} then (2.2) is always satisfied and (2.3) is satisfied as soon as X is separable. However, in general, conditions (2.2) and (2.3) may be hard to verify, which is why we provide (2.1) as an easier sufficient condition. Lastly, we remark that it is not straightforward to show that S = {C | P(C) > 0 for some P ∈ P} corresponding to NA(P) satisfies (2.3), which is why we give a direct proof of Theorem 2.6 in Section 4.
We note that the set Ω can in general not be assumed to be analytic. The impli- follow directly from the definitions. Apart from measurability considerations regarding Ω, the equivalence of (3), (4) and (5) essentially follows from (Burzoni et al., 2016). Furthermore, given an analytic set Ω, we will simply define P as all the finitely supported probability measures on Ω. The analyticity of Ω then implies (APS) of P. We then also have Q P,Φ = M f Ω,Φ and equivalence of (1) and (3)-(5) follows from (Burzoni et al., 2019a). In this context, the essential connection we make is the combination of pathwise and quasi-sure criteria as stated in (2): for every C ∈ S, the pathwise efficient subset Ω * Φ ∩ C is required to be "seen" by at least one measure P in the set P. For a given P, the set Ω in Theorem 2.7 can be explicitly constructed as the concatenation of the quasi-sure supports of P t • ∆(S t+1 ) −1 . The main difficulty of the proof is then to show the implication (5) ⇒ (1), where one needs to establish existence of martingale measures Q ∈ Q P,Φ , which are compatible with Ω and S in the sense of (1). This, modulo measurable selection arguments, is achieved by finding an element P ∈ P t (ω) such that zero is in the relative interior of the support of P • ∆(S t+1 ) −1 . Indeed, let us explain the main idea of the proof of (5) ⇒ (1) based on the following example: assume T = 1, d = 3, Φ = 0, S = {Ω} and the set Ω * is given by the grey polyhydron in Figure 2. Assume that the support of P • ∆(S 1 ) −1 for a given measure P ∈ P 0 is given by the blue dot (see Figure 2). Then as 0 ∈ ri(Ω * ), we can find three additional points in Ω * , such that zero is in the relative interior of the convex hull of the four points. By definition of Ω, the three additional points are in the support of some measures in P 0 , which we call P 1 , P 2 , P 3 in P 0 . As P 0 is convex, it follows that P := P 1 + P 2 + P 3 + P 4 is an element of P 0 as well, as visualised in Figure 3. Since zero is in the relative interior of the support ofP, one can now use results from (Rokhlin, 2008) to find a martingale measure Q ∼P, in particular P Q. Note that this argument fundamentally relies on the convexity of P t . The analytic product structure assumption then grants suitable measurability for the concatenation procedure in the multiperiod case. For a set Ω ⊆ X we denote the pathwise superhedging price on Ω by and denote the P-q.s. superhedging price by Take an analytic set Ω such that for all P ∈ P we have P(Ω * Φ ) = 1. Using the Superhedging Theorems of (Bouchard and Nutz, 2015) and (Burzoni et al., 2019a) it is immediate that the following relationships hold for all upper semianalytic g: The above inequality is strict in general. An easy way to see this is to take d = Then Ω = Ω * = [0, 2] and the pathwise superhedging price is equal to 1/2, while the quasi-sure superhedging price is equal to zero. In fact, to link the super-hedging and pathwise formulations, we have to choose a specific set Ω P g which depends not only on P but also on g. We determine this set Ω P g by reducing to superhedging under a fixed measure P g as stated in the following theorem: Theorem 2.9. Let P be a set of probability measures satisfying (APS). Assume NA(P) holds and let g : X → R be upper semianalytic. Then there exists a measure P g = P g 0 ⊗ · · · ⊗ P g T −1 and an F U -measurable set Ω P g with P(Ω P g ) = 1 for all P ∈ P, such that Conversely, let Ω be an analytic subset of X with Ω * Φ = ∅ and let g : X → R be upper semianalytic. For any set P ⊆ P(X), which satisfies (APS) and In both cases, the value, if finite, is attained by a superhedging strategy The proof of this result is postponed to Section 4.2. In particular, Theorem 2.9 lets us interpret robust superhedging prices π P (g) as classical superreplication prices π P g (g) under an "extremal" measure P g . Determining such measures P g is not straightforward in general. In a one-period case and for a continuous g, we can use the arguments in the proof of Lemma 4.10 to see that any measure P which attains the one-step quasisure support {P • (∆S T (ω, ·)) −1 | P ∈ P T −1 (ω)} can be chosen.
To extend this result to the multiperiod-case, certain continuity properties of the maps ω → P t (ω) have to be guaranteed: we refer to (Carassus et al., 2019, Prop. 3.7) for a sufficient condition.

Complementary results on superhedging and arbitrage
3.1. Extension of Pathwise Superhedging from Ω * to Ω. The preceding results show that quasi-sure and pathwise superhedging are essentially equivalent. As P-q.s. superhedging strategies might be difficult to compute and implement in practice, it might be preferable to work on a prediction set Ω using pathwise arguments. Given that determining Ω * is computationally expensive as well, the quantity of interest is then the superhedging price on Ω and not on Ω * seen in the duality results in Section 2.4. Thus, we would like to find sufficient conditions under which the superhedging strategy associated with π Ω * Φ (g) can be extended to Ω without any additional cost. The intuition is that as Ω \ Ω * describes non-efficient beliefs, we should be able to superhedge g on this set implementing an arbitrage strategy. It turns out that this intuition is not true in general. Indeed, we run into problems regarding measurability of these arbitrage strategies, which means that this procedure only works in special cases. To simplify the analysis, throughout this section only, we assume that Φ = 0 and ω → S t (ω) is continuous. The latter is satisfied, e.g., when ω → S t (ω) is the coordinate mapping, i.e., S t (ω) = ω t . In order to give some intuition and to identify necessary conditions for the sets Ω, Ω * and the function g we first give two counterexamples:  {0})). We set S 0 = 2 and S 1 (ω) = 2 + ω. Then Ω * = ∅ and trivially Thus we have to assume that Ω * ∩ Σ ω t = ∅ in the remainder of this section. We also note that here SA(Ω) holds whilst USA(Ω) does not, see Section 3.2.
By assumption there existsδ > 0 such that dist(ṽ, span(∆S t+1 (Σ ω t ∩Ω * ))) >δ for all ,Ω * (g)(ṽ) for allṽ ∈ A ω t . This concludes the proof. 3.2. Comparison of Strong and Uniformly Strong Arbitrage. We take now a closer look at the notions SA(Ω) and USA(Ω) and establish their equivalence in specific market setups. Clearly every Uniformly Strong Arbitrage is a Strong Arbitrage. In general the opposite assertion is not true: take for example d = 1, S 0 = 1, S 1 (ω) = ω, Ω = (1, 2], then every H 1 > 0 is a Strong Arbitrage, but there do not exist any Uniformly Strong Arbitrages. This simple example can be generalised: a one-period market in the canonical setting with S 0 = 1 and an open convex set Ω such that {1} ∩ Ω = ∅ and 1 ∈Ω admits a Strong Arbitrage but exhibits no Uniformly Strong Arbitrages. On the level of superhedging prices a Uniformly Strong Arbitrage on Ω corresponds to π Ω (0) = −∞. For a financial market which exhibits a Strong Arbitrage but no uniformly Strong Arbitrages, the Pricing-Hedging duality cannot hold (as there are no martingale measures supported on Ω) but π Ω (0) = 0. In conclusion, the difference between Uniformly Strong Arbitrage and Strong Arbitrage can be seen as a property describing the boundary of the prediction set Ω and thus manifests itself in the boundary behaviour of the superhedging functional As it is an upper semicontinuous function, it takes the value zero on the boundary of Ω, while its lower semicontinuous version takes the value −∞. Nevertheless the two notions agree in specific cases, which we now explore. We assume the canonical setting X 1 = R d + , S 0 (ω) = s 0 and set F t = F 0 t for all 0 ≤ t ≤ T . In this section we allow for countably many statically traded options, Λ = N, but only of European type, Φ = {φ n = φ n (S T ) | n ∈ N}, with real-valued continuous payoffs and a common maturity T . We write c 00 = c 00 (N) for simplicity. We fix a closed subset Ω ⊆ (R d + ) T and recall that martingale measures on Ω calibrated to Φ are denoted by M Ω,Φ (F). We define |S(ω)| 1 := T t=1 d k=1 |S k t (ω)| and denote by C b |S|1 (Ω) the space of real-valued continuous functions f : Ω → R such that Finally, we define the calibrated supermartingale measures as The following theorem can be seen as a unification of (Acciaio et al.  (2) Assume φ n ∈ C b |S|1 (Ω), no short-selling in any of the assets and (3) As in (2) assume that φ n ∈ C b |S|1 (Ω). Furthermore assume that for every sequence (ω n ) n∈N with lim n→∞ |S(ω n )| 1 = ∞, there exists a sequence (h k , H k ) k∈N of trading strategies, a constant C > 0 and a sequence (p k ) k∈N such that • lim k→∞ lim n→∞ > 0 and in particular all three conditions in (3) are satisfied.
Proof. For simplicity of exposition we only give the proof for T = 1. This conveys the important ideas, while the multiperiod case extends these via a dynamic programming approach and can found in (Wiesel, 2020 . Then, as S 1 → h · Φ(S 1 ) + H(S 1 − S 0 ) is continuous and positive on a compact set Ω ∩ K, there exists ε > 0 such that Scaling (h, H) suitably we can without loss of generality assume take ε = 2|s 0 | 1 . Let e = (1, . . . , 1) be the row unit vector in R d . Then (2). Clearly the relation USA(Ω) ⇒ WFLVR(Ω) holds and by (1) also SA(Ω) ⇔ USA(Ω). Further, WFLVR(Ω) readily implies SM Ω,Φ (F) = ∅ since otherwise if Q ∈ SM Ω,Φ (F) then, by Fatou's lemma, We denote by c + 00 the subset of all non-negative sequences in c 00 . We define the set

Now we show
(Ω). Note that K is convex and non-empty. Furthermore denote the positive cone of By NUSA(Ω) we have K ∩ C ++ (Ω) = ∅. An application of Hahn-Banach theorem yields existence of a positive measure µ = µ r + µ s such that We now aim to show that the normalised measure Q given by is an element of SM Ω,Φ . For this let us first assume that µ r = 0. Then as µ is positive, which is a contradiction. As Ω (φn(S1)) − |S|1∨1 dµ s = 0, we conclude Furthermore and thus NUSA(Ω) ⇒ SM Ω,Φ = ∅ follows. Lastly we show (3). For this we follow the same construction as in (2). In particular redefining we note that again by NUSA(Ω) we have K ∩ C ++ (Ω) = ∅. Thus all that is left to show is µ s = 0. Let us assume towards a contradiction µ s = 0 and take (h k , H k ) k∈N such that Then by symmetry of K and the same reasoning as in (2) we have Note that for a sequence (p k ) n∈N with for all ω ∈ Ω we need to have by no WFLVR(Ω) that lim k→∞ p k = 0, so the LHS of (3.4) is equal to zero, a contradiction.
We now give a complete proof of the quasi-sure FTAP in (Bouchard and Nutz, 2015) using results from (Burzoni et al., 2019a). We first look at the case Φ = 0 and start with an auxiliary lemma: t . Also as intersections, projections and preimages of analytic sets are analytic (see (Bertsekas and Shreve, 1978, Prop. 7.35 & Prop. 7.40)), we find that {ω ∈ X | ψ t,Ω (ω) ∩ O = ∅} is analytic and in particular F U t−1 -measurable. Let S d be the unit sphere in R d , then by preservation of measurability also the multifunction Remark 4.3. We recall that this separator has the property that it aggregates all one-dimensional One-point Arbitrages on Σ ω t−1 ∩ Ω in the sense that {ω ∈ X | ξ(ω) · ∆S t (ω) > 0} ⊆ {ω ∈ X | ξ t,Ω (ω) · ∆S t (ω) > 0} for every measurable selector ξ of ψ * t,Ω . Proof of Theorem 2.6 for Φ = 0. We start by proving the first part of Theorem 2.6, i.e., we are given a set of measures P satisfying (AP S) and we need to construct Ω = Ω P such that (1)-(3) are equivalent. We define for ω ∈ X t−1 t−1 is closed valued and P(∆S t (ω, ·) ∈χ F 0 t−1 (ω)) = 1 for all P ∈ P t−1 (ω) and all ω ∈ X t−1 . Evidentlỹ Also it follows from (Bouchard and Nutz, 2015, Lemma 4.3, page 840), thatχ F 0 t−1 is analytically measurable. We quickly repeat their argument: let us define Then l is Borel measurable. Next we consider Since its graph is analytic, it follows that for We also note that for ε > 0 the function x → R(B ε (x)) is continuous, so (x, R) → R(B ε (x)) is Borel and ) is analytic and by Fubini's theorem P(U ) = 1 holds for all P ∈ P. We now set which is again analytic and P(Ω P ) = 1 for all P ∈ P.
Note that ω →P t−1 (ω) is universally measurable. We define the correspondence ρ : P(X 1 ) d+1 R d by ρ : (P 0 , P 1 , . . . , P d ) = supp Since P → P(O) is Borel measurable, we conclude that ρ is weakly measurable. Let us denote by S d the unit sphere in R d . By preservation of measurability (cf. (Rockafellar and Wets, 2009, Exercise 14.12, page 653)) it follows that the correspondence Ψ : is weakly measurable. Then also the correspondenceΨ : is weakly measurable and closed-valued. Let V be a countable base of R d . The set is Borel measurable. Note that for an arbitrary convex set C ⊆ R d the relationship
Before continuing the proof of Theorem 2.6 let us first give a short remark on the measurability of the arbitrage strategies involved in the proof of above: So in particular if P(Ω P \ (Ω P ) * ) > 0 for some P ∈ P, then H * is an H(F)-measurable P-q.s arbitrage. In general the inclusionF ⊆ F U does not hold. This is why we need to construct a new F U -measurable arbitrage strategy, which captures the arbitrages essential for P. More generally, in this paper we manage to avoid using projectively measurable sets, which were essential for the arguments in (Burzoni et al., 2019a). In fact, all our trading strategies are universally measurable without invoking the axiom of projective determinacy. Furthermore, we hope that by constructing an explicit arbitrage strategy in the proof of (3) ⇒ (1) we can clarify the proof of (Burzoni et al., 2016), Theorem 4.23, pp. 42-46 (in particular (Burzoni et al., 2016) [A.3]) by offering a similar to the above (but much simpler) reasoning for the case P = {P}. Introducing a measurable separator ξ it is apparent that j z in (Burzoni et al., 2016, p.44) can always be chosen equal to one in our setting. Also the resulting strategy H P therein can be chosen universally measurable.
To prove the first part of Theorem 2.6 for the case Φ = 0 we recall the following notion from (Burzoni et al., 2019a): for any i = 0, . . . , β,H i is an Arbitrage Aggregator for A i , (4) if β < k, then either A β = ∅ or for any α ∈ R k linearly independent from α 1 , . . . , α β there does not exist H such that  Proof of Theorem 2.6 for Φ = 0. The existence of Ω P and (1) ⇒ No Strong Arbitrage in A Φ (F) on Ω P , (2) ⇒ (3) follow exactly as before. We now argue that (1) ⇒ (2) holds in the spirit of (Bouchard and Nutz, 2015, Theorem 5.1, p. 850), by induction over the number e of options available for static trading. In particular we can assume without loss of generality that there exists a random variable ϕ ≥ 1 such that |φ j | ≤ ϕ for all j = 1, . . . , k and consider the set Q ϕ = {Q ∈ Q P | E Q [ϕ] < ∞} in order to avoid integrability issues. So let us assume there are e ≥ 0 traded options φ 1 , . . . , φ e , for which (1) ⇒ (2) holds. We introduce an additional option g = φ e+1 and assume P (Ω P ) * {φ1,...,φe+1} = 1 for all P ∈ P. Then clearly P (Ω P ) * {φ1,...,φe} = 1 for all P ∈ P and by the induction hypothesis there is no arbitrage in the market with options {φ 1 , . . . , φ e } available for static trading. Let P ∈ P. Then by exactly the same arguments as in (Bouchard and Nutz, 2015, proof of Theorem 5.1(a)) we can use convexity of Q ϕ and Theorem 2.9 to find a measure Q ∈ Q ϕ , such that P Q and Q ∈ Q P,{φ1,...,φe+1} , so (2) holds.
Proof of Theorem 2.7. We recall the analytic set Ω P from the proof of Theorem 2.6 for Φ = 0 and the sets {C n } n∈N from (2.1). Now we define We claim that (2.1) implies that In particular ω ∈ C for some C ∈ S such that C ∩ (Ω P ) * Φ ∈ N P . By By (2.1) there exists n 0 ∈ N such that C n0 ⊆ C and ω ∈ C n0 . This implies ω ∈ B and thus shows the claim. Let us now first assume that Φ = 0 and set Ω := Ω P \ ((Ω P ) * ∩ B) ∈ F U . (4.1) By assumption we have P(Ω) = 1 for all P ∈ P. By definition of the () * operation Ω * = ((Ω P ) * \ B) * Φ = (Ω P \ B) * follows. To see the above equality, take a martingale measure Q ∈ M Ω,Φ and assume that Q(Ω \ (Ω P \ B)) > 0. As Ω \ (Ω P \ B) = Ω P \ (Ω P ) * ∩ B we conclude that Q(Ω P \ (Ω P ) * ) > 0. Since any calibrated martingale measure supported on a subset of Ω is in M Ω P this leads to a contradiction to the definition of (Ω P ) * . Also, Ω P \ B = Ω P ∩ B c is the intersection of two analytic sets, so we conclude that Ω * is analytic. Lastly, by definition of Ω P we conclude Ω * = (Ω P ) * P-q.s.. The implications (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5) follow directly from the definition. Thus we only need to show (5) ⇒ (1). Let us fix C ∈ S such that C ⊆ Ω. No Arbitrage de la Classe S on Ω implies that Ω * ∩ C = ∅. From (4.1) we thus conclude that P((Ω P ) * ∩ C) > 0 for some P ∈ P. As Ω * = (Ω P ) * P-q.s. this implies P(Ω * ∩ C) > 0. Using a construction similar to the proof of Proposition 2.6 for the case Φ = 0, we can find a measureP ∈ P such thatP(C) > 0 and 0 is in the interior of the conditional support ofP(·|Ω * ). By (Rokhlin, 2008, Theorem 1), we conclude that there exists a martingale measure Q ∈ Q P equivalent toP(·|Ω * ), in particular Q(C) > 0. The case Φ = 0 can now be treated similarly: indeed, we define Ω P,Φ as in the proof of Theorem 2.6 for Φ = 0, but now including the statically traded options Φ in the definition of the quasi-sure support and follow the same arguments as above. This concludes the proof.

4.2.
Proof of Theorem 2.9. We first show that the quasi-sure superhedging theorem of (Bouchard and Nutz, 2015) implies the second part of Theorem 2.9. Proposition 4.9. Let Ω be an analytic subset of X and Ω * Φ = ∅. Let the set P satisfy (APS) and N P = N M f Ω,Φ Then N Q P,Φ = N M f Ω,Φ and for an upper semianalytic function g : Note that there is no M f Ω,Φ -q.s. arbitrage iff there is no P Ω -q.s arbitrage. We now show that Ω * Φ is analytic if Ω is analytic. Recall the set P Z,Φ from Lemma 5.4 of (Burzoni et al., 2017), page 13 defined by is analytic and the projection of the above set to the first coordinate is exactly Ω * Φ , which shows that Ω * Φ is analytic. We note ω → P Ω t (ω) = P f (proj t+1 (Σ ω t ∩ Ω * Φ )) has analytic graph by exactly the same argument as in the proof of Proposition 4.1 replacing Ω by Ω * Φ . The result now follows from the Superhedging Theorem of (Bouchard and Nutz, 2015) and the definition of M f Ω,Φ .
By (APS) and upper semianalyticity of g, every E t (g) is upper semianalytic. We show recursively that for every t = 0, . . . , T − 1 and for P-q.e. ω ∈ X t there exists a measure P ∈ P(X 1 ) such that NA(P) holds and sup Q∈Qt(ω) Note that by measurable selection arguments and construction of Ω P we conclude that for P-q.e. ω ∈ X t the properties NA(P t (ω)) and P(proj t+1 (Ω P ∩ Σ ω t )) = 1 hold for all P ∈ P t (ω). We now fix t ∈ {0, . . . , T − 1} and ω ∈ X t such that NA(P t (ω)) and P(proj t+1 (Ω P ∩ Σ ω t )) = 1 for all P ∈ P t (ω) holds. Note that there exists a sequence (P n ) n∈N such that P n ∈ P t (ω) for all n ∈ N and sup Q Pn, Q∈M X 1 We see from the proof of Theorem 2.6 for Φ = 0 in Section 4.1 that under NA(P t (ω)) and for a fixed P ∈ P t (ω), we can always findP ∈ P t (ω) such that P P and NA(P) holds. Thus we can assume without loss of generality that NA(P n ) holds for all n ∈ N. DefineP n := n k=1 2 −k /(1 − 2 −n )P k ∈ P t (ω) as well as P g t (ω) := ∞ k=1 2 −k P k and note that NA(P n ) as well as NA(P g t (ω)) hold for all n ∈ N. Furthermore Taking K n ↑ R d we have in particular a contradiction. Thus E t (g)(ω) = π Pt(ω) (g) = π P g t (ω) (g).
We now define Ω P g = Ω P ∩ ω ∈ X sup This concludes the proof.
Remark 4.12. By NA(P) Proposition 4.11 implies for g = 0 where we define E 0 = {g : X → (−∞, 0] F U -measurable |g = 0 P-q.s.}. In particular for everyg ∈ E 0 there exists Q ∈ M X,Φ such that E Q [g] = 0. A similar result was obtained by (Burzoni et al., 2019b) in a more general setup. Aggregating the martingale measures corresponding to allg (and thus to all P-polar sets) to achieve a result comparable to (Bouchard and Nutz, 2015) in a setup without using (APS) of P remains an open problem.
Data Availability Statement: Data sharing is not applicable to this article as no new data were created or analyzed in this study.