Duality for pathwise superhedging in continuous time

We provide a model-free pricing–hedging duality in continuous time. For a frictionless market consisting of d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d$\end{document} risky assets with continuous price trajectories, we show that the purely analytic problem of finding the minimal superhedging price of a path-dependent European option has the same value as the purely probabilistic problem of finding the supremum of the expectations of the option over all martingale measures. The superhedging problem is formulated with simple trading strategies, the claim is the limit inferior of continuous functions, which allows upper and lower semi-continuous claims, and superhedging is required in the pathwise sense on a σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma $\end{document}-compact sample space of price trajectories. If the sample space is stable under stopping, the probabilistic problem reduces to finding the supremum over all martingale measures with compact support. As an application of the general results, we deduce dualities for Vovk’s outer measure and semi-static superhedging with finitely many securities.


Introduction
Given the space C([0, T ], R d ) of all continuous price trajectories, the superhedging problem of a contingent claim X : C([0, T ], R d ) → R consists of finding the infimum over all λ ∈ R such that there exists a trading strategy H which satisfies where (H · S) T (ω) denotes the capital gain by trading according to the strategy H in the underlying assets S t (ω) := ω(t).
In the classical framework of mathematical finance, one commonly postulates a model for the price evolution by fixing a probability measure P such that S is a semimartingale and defines (H · S) T as the stochastic integral T 0 H t dS t . Then a consequence of the fundamental theorem of asset pricing states that the infimum over all λ such that there are admissible predictable integrands H fulfilling inequality (1.1) is equal to the supremum of E Q [X] over all absolutely continuous local martingale measures Q; see Delbaen and Schachermayer [19,Sect. 9.5]. Here, the superhedging (i.e., inequality (1.1)) is assumed to hold P -almost surely and the set of absolutely continuous local martingale measures is non-empty, which is guaranteed by the exclusion of some form of arbitrage; see [19,Corollary 9.1.2] for the precise formulation.
More recently, alternative possibilities to specify the superhedging requirement without referring to a fixed model have been proposed. For instance, if an investor takes into account a class P of probabilistic models, then superhedging is naturally required to hold P-quasi surely, i.e., P -almost surely for all considered models P ∈ P. The pioneering works of Lyons [33] and Avellaneda et al. [4] on Knightian uncertainty in mathematical finance consider models with uncertain volatility in continuous time. The study of the pricing-hedging duality in this setting has given rise to a rich literature starting with the capacity-theoretic approach of Denis and Martini [20]. Further, Peng [39,Theorem 2.4] obtains the duality using stochastic control techniques, whereas Soner et al. [47,46,45] rely on supermartingale decomposition results under individual models and eventually build on aggregation results to derive the duality under model uncertainty. This approach has been extended by Neufeld and Nutz [37] to cover measurable claims using the theory of analytic sets; see also Biagini et al. [15] for a robust fundamental theorem of asset pricing under a model ambiguity version of the no-arbitrage-of-the-first-kind condition NA 1 (P), and Nutz [38] for the case of jump diffusions.
In the present work, we focus on the pathwise/model-free approach and assume that the superhedging requirement (1.1) has to hold pointwise for all price trajectories in a given set ⊆ C([0, T ], R d ). In this pathwise setting, finding the minimal superhedging price turns out to be a purely analytic problem whose formulation is independent of any probabilistic assumptions. This is in contrast to the abovementioned approaches working with a fixed model, under Knightian uncertainty or in a quasi-sure setting. Notice that the pathwise approach corresponds to the quasi-sure approach when P contains all Dirac measures, which in continuous time is excluded; see e.g. [15,Corollary 3.5].
In the now classical paper [30], Hobson first addressed the problem of pathwise superhedging for the lookback option. His analysis was based on some sharp pathwise martingale inequalities and has motivated Beiglböck et al. [11] to introduce the martingale optimal transport problem in discrete time. Here, the investor takes static positions in some liquidly traded vanilla options and dynamic positions in the stocks. The rationale is that information on the price of options translates into the knowledge of some marginals of the martingale measures; see also [1,7,18,17,16,12] for further developments in this direction. In continuous time, the duality for the martingale optimal transport has been obtained by Galichon et al. [26] and Possamaï et al. [41] in the quasi-sure setting. The pathwise formulation was studied by Dolinsky and Soner [21] using a discretisation of the sample space. These results have been extended by Hou and Obłój [31], who in particular allow incorporation of investor's beliefs (of possible price paths) by relying on the notion of "prediction set" due to Mykland [36].
Following this consideration in our analysis, we also assume that the investor does not deem every continuous path plausible, but focuses instead on a prediction set ⊆ C([0, T ], R d ) that is required to be σ -compact (i.e., at most a countable union of compact sets), and define the pathwise superhedging problem on the sample space . Moreover, restricting the set of possible price paths has the financially desirable effect of reducing the superhedging price. See also Aksamit et al. [3] and Acciaio and Larsson [2] for other treatments of belief and information in robust superhedging, and Dolinsky and Soner [22] and Guo et al. [29] for extensions of the pathwise formulation to the Skorokhod space.
In the continuous-time setting, already the definition of a pathwise "stochastic integral" is a non-trivial issue. We circumvent this problem by working with simple strategies and consider as "stochastic" integral the pointwise limit inferior of pathwise integrals against simple strategies, an approach that was proposed by Perkowski and Prömel [40] to define an outer measure allowing to study stochastic integration under model ambiguity. This outer measure is very similar in spirit to that of Vovk [48] and can be seen as the value of a pathwise superhedging problem; cf. Sect. 2.1 for details and Beiglböck et al. [10] and Vovk [49] for existing duality results in this setting.
Formally, the superhedging price of a contingent claim X : → [−∞, +∞] is defined as the infimum over all λ ∈ R such that there exists a sequence (H n ) of simple strategies which satisfies for all ω ∈ and the admissibility condition λ + (H n · S) t (ω) ≥ 0 for all n ∈ N, ω ∈ and t ∈ [0, T ]. If X is the limit inferior of a sequence of continuous functions, then under the assumptions that is σ -compact and contains all its stopped paths, we show that the superhedging price coincides with the supremum of E Q [X] over all martingale measures Q. Furthermore, this duality is generalised to the case when X is unbounded from above and when does not contain all its stopped paths. In addition to providing a way around the technical difficulty posed by the definition of pathwise stochastic integrals, the superhedging in terms of limit inferior turns out to be necessary to guarantee the duality on a sufficiently large space; see Remark 2.6 for a counterexample.
Our main contributions to the pathwise pricing-hedging duality in continuous time and with finitely many risky assets are as follows. While in the current literature (see e.g. [31,21,29]) pathwise duality results hold for uniformly continuous options, the proposed method allows much less regular claims (including for example European options, spread options, continuously and discretely monitored Asian options, lookback options, certain types of barrier options, and options on realised variance). In particular, this implies a duality for Vovk's outer measure on closed sets. A related duality result was given by Vovk [49], however, under an additional closedness assumption on the set of attainable outcomes. Moreover, our pricing-hedging duality holds for every prediction set which is σ -compact. Let us remark that the assumption of σ -compactness is an essential ingredient of the presented method to get the pricing-hedging duality. We show in Sect. 3.1 that typical price trajectories for various popular financial models such as local, stochastic or even rough volatility models belong to the σ -compact space of Hölder-continuous functions. In the related work [31], the pricing-hedging duality holds for an approximate version of the superhedging price which requires the superhedging on an enlarged prediction set The article is organised as follows. In Sect. 2, we present the main results (Theorems 2.1 and 2.7) and some direct applications. Section 3 contains a detailed discussion of feasible choices for the underlying sample space. The proofs of the main results are carried out in Sect. 4. A criterion for the sample path regularity of stochastic processes and the construction of a counterexample are given in the Appendix.

Main results
Let C([0, T ], R d ) be the space of continuous functions ω : [0, T ] → R d , where T > 0 is a finite time horizon and d ∈ N. Throughout the entire paper, ⊆ C([0, T ], R d ) is a non-empty metric space, that is, = ∅ and there is a fixed metric d on . We consider on the topology T which is induced by d and the Borel σ -algebra which is generated by the open sets with respect to d. The metric space is called σ -compact if there exists a countable sequence of compact (with respect to d) sets K n ⊆ such that = n K n . A map X : → R is said to be continuous if X is continuous with respect to d and the Euclidean distance on R.
The canonical process S : X = lim inf n X n for a sequence (X n ) such that X n : → R is bounded and continuous .
Note that C δσ contains all upper and lower semicontinuous functions from to R.
where N ∈ N, 0 ≤ τ 1 ≤ · · · ≤ τ N +1 ≤ T are stopping times with respect to the filtration (F t ), and h n : → R d are bounded F τ n -measurable functions. The set of all simple predictable processes is denoted by is well defined for all t ∈ [0, T ] and all ω ∈ . Similarly, the pathwise stochastic integral H · S is also well defined for every H : where 0 ≤ τ 1 ≤ τ 2 ≤ · · · are stopping times such that for each ω ∈ there exists an N(ω) ∈ N with τ k (ω) = T for all k ≥ N(ω), and h n : → R are bounded F τ n -measurable functions.
We introduce the following two assumptions, which we shall use frequently.
(A1) is σ -compact, the metric on induces a topology finer than (or equal to) that induced by the maximum norm ω ∞ := max t∈[0,T ] |ω(t)|, and for each Borel probability Q on and every bounded F 0 t -measurable function h, there exists a sequence of F 0 t -measurable continuous functions (h n ) which converges Q-almost surely to h.
If is a σ -compact metric space for the metric and the topology induced by the maximum norm, then (A1) is always satisfied; see Remark 4.1. Now we are ready to state the main results of this paper. The proofs are given in Sect. 4. Theorem 2.1 Suppose that (A1) and (A2) hold and let Z : → [0, +∞) be a continuous function such that Z(ω s ) ≤ Z(ω t ) for all ω ∈ and 0 ≤ s ≤ t ≤ T . Then for every X ∈ C δσ which satisfies X(ω) ≥ −Z(ω) for all ω ∈ , one has < +∞ for every Borel probability measure Q which integrates Z. Hence the set of all local martingale measures which integrate Z coincides with M Z ( ).
In particular, for Z = 0 the previous theorem reads as follows.
Hence the pathwise pricing-hedging duality in Theorem 2.1 holds for this claim. This shows that the class C δσ includes in particular the financial derivatives in the scope of [10], i.e., options on realised variance, among many others.

Remark 2.6
While the pathwise pricing-hedging duality results in [21,31] hold for sufficiently regular claims when trading is limited to simple predictable processes (i.e., without the "lim inf" as in our definition), the following example shows the necessity of "lim inf" for claims in C δσ . Let be the set of all Hölder-continuous functions starting at zero with values in [0, 1] and equipped with the distance induced by the maximum norm. There exist a refining deterministic sequence (π n ) n∈N of partitions with mesh size going to zero and a functionω ∈ such that -0 ≤ω(t) ≤ 1 for all t ∈ [0, T ], -ω t := lim n ω n t exists for all t ∈ [0, T ) and lim t→T ω t = +∞, where ω n t is defined as in (2.2). For the existence of such a functionω, we refer to Lemma A.3. We fix now the above sequence (π n ) n∈N and denote by ω t the corresponding quadratic variation along (π n ) n∈N defined as in (2.2) for all ω ∈ . Furthermore, let us consider the option X(·) := · T ∈ C δσ . Firstly, we get by Itô's formula and Fatou's lemma that Secondly, we observe that Indeed, assume that there exist (even more generally) a predictable process H of bounded variation and a constant λ 0 > 0 such that where (H · S) T (ω) denotes the classical Riemann-Stieltjes integral defined using the integration by parts formula. Forω we get where H (ω) 1-var;[0,T ] denotes the bounded variation seminorm of H . Because X(ω) = +∞, this implies that (2.4) cannot hold for any λ 0 ∈ R and therefore establishes (2.3).
Hence, there exists a duality gap if the superhedging is restricted to trading strategies of bounded variation as in [21,31]. However, the pricing-hedging duality using the limit inferior of simple predictable processes holds true since and X satisfy all assumptions of Theorem 2.1; see Sect. 3 below.
If does not contain all its stopped paths, then the following version of Theorem 2.1 holds true.

Theorem 2.7
Let Z : → [1, +∞) be a function with compact sublevel sets {Z ≤ c} for all c ∈ R and such that Z(ω) ≥ ω ∞ for all ω ∈ . If (A1) holds true and there are c ≥ 0 and a sequence (H n for every X ∈ C δσ which is bounded from below.

Relation to Vovk's outer measure
In recent years (see e.g. [48,49] and the references therein), Vovk introduced on different path spaces an outer measure, defined as the minimal superhedging price, which allows quantifying the path behaviour of "typical price paths" in frictionless financial markets without any reference measure. In order to recall Vovk's outer measure on a set ⊆ C([0, T ], R d ) endowed with the maximum norm, we write H λ for the set of λ-admissible strategies, i.e., the set of for an initial capital λ ∈ (0, +∞). Note that for every H = (H k ) ∈ V λ , all ω ∈ and all t ∈ [0, T ], the corresponding capital process is well defined and takes values in [−λ, +∞]. Then Vovk's outer measure on is given by A slight modification of Q was introduced in Perkowski and Prömel [40], namely for A ⊆ . The latter definition seems to be more in the spirit of superhedging prices in semimartingale models as discussed in [  By restricting the outer measure P to a σ -compact space satisfying assumptions (A1) and (A2), we get the following duality result for the slightly modified version of Vovk's outer measure as a direct application of Theorem 2.1.

Proposition 2.9 Under the assumptions on of Theorem 2.1, one has
for all closed subsets A ⊆ .
Proof For every closed subset A ⊆ , it follows from Corollary 2.3 that

Semi-static superhedging
Let us fix a continuous function Z : → [1, +∞) such that Z(ω s ) ≤ Z(ω t ) for all ω ∈ and 0 ≤ s ≤ t ≤ T , and consider a finite number of securities with (discounted) continuous payoffs G 1 , . . . , G K such that |G i | ≤ cZ for i = 1, . . . , K and some c ≥ 0. We assume that these securities can be bought and sold at prices g k ∈ R and satisfy the no-arbitrage condition where ri denotes the relative interior. Then the following semi-static hedging duality holds.

Proposition 2.11
Suppose the assumptions (A1) and (A2) are satisfied and the securities with payoffs G 1 , . . . , G K satisfy the static no-arbitrage condition (2.6). Then for every upper semicontinuous function X : there are c ≥ 0 and a sequence ( and we remark that by interchanging two infima, the left-hand side of (2.7) can be expressed as inf α∈R K φ(X − K k=1 α k (G k − g k )). Further, Theorem 2.1 yields for every α ∈ R K . Now define the function It is immediate that J (Q, ·) is convex for every Q ∈ M c ( ) and that J (·, α) is concave for each α ∈ R K since M c ( ) is convex. Therefore, it follows exactly as in step (a) of the proof of [5, Theorem 2.1] that the assumption (2.6) of 0 being in the relative interior of where the first equality follows from Theorem 2.1 and the last by The proof is complete.

Discussion of σ -compact spaces
By definition, the σ -compactness of the metric space ⊆ C([0, T ], R d ) with metric d requires to find a covering of by compact sets K m , m ∈ N. It is an easy consequence of the Arzelà-Ascoli theorem (see e.g. [25,Theorem 1.4]) that these K m have to be bounded, closed and equicontinuous.
In the next lemma, we provide an easy-to-check criterion for a set of continuous functions to be σ -compact. This leads to many interesting examples of such ⊆ C([0, T ], R d ) appearing in the context of (classical) financial modelling; see Sect. 3.1.
with α ∈ (0, 1] and the convention 0 0 := 0. Then the spaces are σ -compact with respect to the norm · c n ,α for α ∈ (0, 1) and in particular with respect to the maximum norm · ∞ . Moreover, the set := n∈N n is σ -compact with respect to the maximum norm · ∞ .
Proof For m, n ∈ N, we observe that In order to show the σ -compactness of n with respect to · ∞ , we show that each K m n is compact. Due to the Arzelà-Ascoli theorem, it is sufficient to show that each K m n is bounded, equicontinuous and closed. As to boundedness, for every ω ∈ K m n , we have Next, because c n is continuous on a compact set and c n (t, t) = 0 for t ∈ [0, T ], there exists for every ε > 0 a δ > 0 such that |c n (s, t)| < ε/m for |t − s| ≤ δ. Hence for every ω ∈ K m n and s, t ∈ [0, T ] with |t − s| ≤ δ, we get |ω(t) − ω(s)| ≤ ε, which yields equicontinuity. Finally, for closedness, we show that if (ω k ) ⊆ K m n converges uniformly to ω, then ω ∈ K m n . Indeed, this can be seen by The σ -compactness of n with respect to · c n ,α for α ∈ (0, 1) follows by the fact that uniform convergence in each K m n implies convergence with respect to · c n ,α , which is a consequence of the interpolation inequality, for s, t ∈ [0, T ], Finally, is σ -compact (with respect to · ∞ ) as a countable union of σ -compact sets.
From the previous lemma, it is easy to deduce that many well-known function spaces ⊆ C([0, T ], R d ) are σ -compact spaces. To state the next corollary, we recall that a function c : , is σ -compact with respect to · ∞ and for β ∈ (0, α) with respect to the Hölder norm · β defined by for δ ∈ (0, 1) and p ∈ [1, +∞), is σ -compact with respect to · ∞ .
(iv) The space C p-var,c ([0, T ], R d ), which is a subspace of continuous functions with finite p-variation, given by for p ∈ [1, +∞) and a control function c, is σ -compact with respect to · ∞ and for p ∈ (p, +∞) with respect to the p -variation norm · p -var defined by Proof (i) and (ii) follow directly by Lemma 3.1 and the fact that (iii) Classical Sobolev embedding results, see e.g. [25,Corollary A.2], imply that  ) with ω W δ,p ≤ K for some constant K > 0 converges uniformly to a function ω, then ω W δ,p ≤ K. However, this is a simple consequence of Fatou's lemma.
(iv) The σ -compactness with respect to · ∞ and · c,α for α ∈ (0, 1) follows again by Lemma 3.1. The σ -compactness with respect to · p -var can be deduced from the inequality

Examples from mathematical finance
As mentioned in the introduction, the prediction set can be interpreted to contain all the price paths that an investor believes could possibly appear in a financial market. Hence it is natural to choose in a way that it includes those price paths coming from financial models which have been proved to provide fairly reasonable underlying price processes.
Example 3.4 A natural assumption coming from semimartingale models is to consider a prediction set QV of continuous paths possessing pathwise quadratic variation in the sense of Föllmer [24]. We refer e.g. to the work [43] (and the references therein) for such frameworks. To be more precise, fix a refining sequence of partitions (π n ) n∈N with mesh size going to zero and consider the prediction set : ω(0) = 0 and ω QV < C} for α ∈ (0, 1) and some constant C > 0, where Note that QV is σ -compact with respect to the norm · ∞ . Indeed, we have QV = n∈N n with n := {ω ∈ C α ([0, T ], R) : ω α ≤ n and ω QV ≤ C − 1/n}, where n is a compact set for each n ∈ N. In order to see the compactness of n , we observe that the condition ω α ≤ n ensures that the set n is equicontinuous and uniformly bounded, and furthermore, every sequence (ω m ) ⊆ n possesses a subsequence which converges in the maximum norm to a function ω ∈ C α ([0, T ], R) with ω α ≤ n; cf. Lemma 3.1. The required bound ω QV ≤ C − 1/n follows by the same estimates as used for the proof of [25,Proposition 5.28].
Let us consider for instance a simple lookback option on the market QV . Using a pathwise version of the Burkholder-Davis-Gundy inequality (see [13, Theorem 2.1]), we get for all ω ∈ QV and some sequence (H n ) of simple predictable processes. From this, we can conclude that the superhedging price is less than or equal to 6 √ C, using the definitions from Theorem 2.1. Note that the superhedging price on the entire space C([0, T ], R) has to be +∞ if we aim to have the duality between the superhedging price and the supremum of E Q [X] over all martingale measures Q.  H ∈ (0, 1). If H > 1/2, for every upper semicontinuous claim X : → [0, +∞], we can apply our pathwise pricing-hedging duality (Theorem 2.1) to see that the superhedging price is given by where 0 stands for the constant path equal to 0, since the Dirac measure at 0 is the only martingale measure in M c ( ). Notice that the pathwise superhedging price considering the entire space C([0, T ], R) is sup ω∈C([0,T ],R) |X(ω)| for many options X. Note that it is actually a delicate question under which conditions non-semimartingale models are almost surely arbitrage-free. However, even for prediction sets like , on which one expects arbitrage in a probabilistic sense, the pathwise superreplication price turns out to be finite. However, prediction sets of this form are not σ -compact in general and the duality results of this paper do not apply. As shown in Bartl et al. [8], a pathwise pricinghedging duality on such prediction sets can still be obtained, but it requires a modified superhedging price which allows investing directly in the quadratic variation. This new superhedging price of a contingent claim X is defined as the infimum over all λ ∈ R for which there exist sequences (H n ) and (G n ) of simple predictable strategies satisfying for all ω ∈ and the admissibility condition λ + (H n · S) t (ω) + (G n · S dS) t (ω) ≥ 0 for all n ∈ N, ω ∈ and t ∈ [0, T ]. The key idea is to extend the market model, consider a two-dimensional price process (S, S dS) on the product space C([0, T ], R) × C([0, T ], R) and adapt the duality results (and their proofs) of the present paper accordingly. For a detailed discussion on prediction sets depending on pathwise quadratic variation, we refer to [8].
In the following, we present several examples coming from the modelling of financial markets which satisfy the assumptions (A1) and (A2) and are concentrated on a σ -compact metric spaces ⊆ C([0, T ], R d ). For simplicity, we consider onedimensional processes and denote by W a one-dimensional Brownian motion on a probability space (˜ , F, P ). However, all arguments extend straightforwardly to multidimensional settings.
Example 3.7 A classical example from mathematical finance is the famous Black-Scholes model, which is given by for μ ∈ R and σ > 0. In this case, the price process S is a so-called geometric Brownian motion, which possesses the same sample path regularity as a Brownian motion. Hence, one has almost surely S ∈ C α ([0, T ], R) and S ∈ W α− 1 q ,q ([0, T ], R) for every α ∈ (0, 1/2) and q > 2; cf. Corollary A.2.

Example 3.9 A frequently applied generalisation of the Black-Scholes model is given by stochastic volatility models
for s 0 ∈ R and predictable real-valued processes μ and σ . This type of linear stochastic differential equation can be explicitly solved by Based on Corollary A.2, one can easily deduce the sample path regularity of the price process S. Indeed, for q ∈ (2, +∞), α ∈ (0, 1/2 − 1/(2q)) and δ : For example, the Heston model is a stochastic volatility model in which the volatility process σ satisfies such a bound.
In the context of stochastic volatility modelling with Knightian uncertainty, one usually replaces the fixed volatility process σ by a class of volatility processes. For example, the seminal works [4] and [33] on volatility uncertainty require the volatility processes σ to be such that σ t ∈ [σ min , σ max ] for all t ∈ [0, T ] and some constants σ min , σ max > 0 with σ min < σ max . Therefore, due to the bounds on the volatility, all possible price paths considered in [4] and [33] belong to the function spaces as stated in (3.2). Example 3.10 (Rough volatility models) Recently, investigating time series of volatility using high-frequency data, Gatheral et al. [27] showed that the log-volatility behaves essentially like a fractional Brownian motion with Hurst exponent H close to 0.1. This new insight has led to various fractional extensions of classical volatility models (see e.g. [27,9,14,23]) which nicely lead to price paths belonging to the σ -compact metric space of Hölder-continuous functions. Indeed, if the stochastic volatility σ fulfils for some M > 0 and q > r ≥ 1 the bound

Example 3.11
The most general case of volatility uncertainty is usually provided by simultaneously considering all processes of the type for strictly positive and predictable processes σ ; see [37,41]. While they can deal with all σ such that T 0 σ s ds < +∞ a.s., we have seen in Example 3.9 that we can deal with all volatility processes σ such that E P [ T 0 σ q s ds] < +∞ for q ∈ (1, +∞). Another subclass of price processes S leading to σ -compact sets of price paths is given by all processes S with corresponding volatility process σ such that σ ≤ f for some deterministic integrable function f : [0, T ] → (0, +∞). Indeed, defining the quadratic variation of S by S t =

Proofs of the main results
Denote by C b := C b (( , d), R) the set of all bounded continuous (with respect to d) functions X : → R.

Remark 4.1
If is a σ -compact space endowed with the maximum norm, then (A1) is always satisfied.
Proof Fix t ∈ [0, T ], a bounded F 0 t -measurable function h and a Borel probability Q. Define π : → C([0, t], R d ), π(ω)(s) := ω(s), and set t := π( ) endowed with the maximum norm ω ∞ := max s∈[0,t] |ω(s)|. By σ -compactness, there exist compact sets K n , n ∈ N, such that = n K n . Further, since t = n π(K n ) and π(K n ) is compact by continuity of π , it follows that t is σ -compact and therefore separable. Standard arguments show that F 0 t = {π −1 (B) : B ∈ B( t )}, where B( t ) denotes the Borel sets of t . Hence, h =h • π for some Borel functionh : t → R. Again by σ -compactness of t , the probability measurẽ Q := Q • π −1 is tight and thus regular, i.e., Borel sets can be approximated from inside inQ-measure by compact subsets. In particular, there exists a sequence of continuous functionsh n : t → R such thath n →hQ-almost surely, which in turn implies h n :=h n • π →h • π = h Q-almost surely.
The following lemma is an immediate consequence of standard results about discrete-time local martingales (see [32, Theorems 1 and 2]), which we recall for later reference. Next we need to establish some auxiliary results.
If τ (ω) > t, then again τ − (ω n ) > t for eventually all n so that the same argument shows that lim inf n S τ t (ω n ) ≥ S τ t (ω).
(d) If S τ t (ω) = −m, then ω(s) ≥ −m. Assume that lim inf n S τ t (ω n ) < −m. Then there is a subsequence still denoted by (ω n ) such that τ (ω n ) = τ − (ω n ) = s. However, this contradicts lim inf n S τ t (ω n ) = lim n ω n (s) = ω(s) ≥ −m. The fact that σ ε increases to σ as ε tends to 0 (and therefore S σ ε t → S σ t by continuity of S) shows that Furthermore, notice that σ = τ on {τ ≥ s} so that 1 {τ ≥s} (S σ t − S s ) = S τ t − S τ s . Since τ is the hitting time of a closed set, it is also a stopping time with respect to the raw filtration (F 0 t ), so that h1 {τ ≥s} : → [0, 1] is F 0 s -measurable. This shows that , S τ is a supermartingale with respect to the raw filtration (F 0 t ). Finally, using that S τ is bounded and F s ⊆ F 0 s+ε yields which shows that S τ is a supermartingale. By similar arguments, one can also show that S τ is a submartingale (and thus a martingale). Indeed, replace h by a continuous F 0 s -measurable functionh : → [−1, 0] and the stopping times σ ε byσ ε := inf{r ≥ s : S r ≥ m − ε or S r < ε − m} ∧ T for ε > 0. The same arguments as in Lemma 4.3 show that ω → S t∧σ ε (ω) (ω) is upper semicontinuous, which implies that (H · S) T is lower semicontinuous for H :=h1 ]]s,σ ε ∧t]] ∈ H f . The rest follows in the same way as before.

Lemma 4.5
Assume that (A1) and (A2) hold true. Then there exists an increasing sequence of non-empty compacts (K n ) such that = n K n and ω t ∈ K n for every (t, ω) ∈ [0, T ] × K n .
We have now all ingredients at hand to prove the main results of the present paper.
Proof of Theorem 2.1 Fix a continuous function Z : → [0, +∞) and a sequence (K n ) of compact sets as in Lemma 4.5.
Step (a). Fix n ∈ N and define φ n (X) := inf λ ∈ R : there are H ∈ H f and c ∈ R such that (H · S) T ≥ c on and λ + (H · S) T ≥ X on K n for X : → R. By Lemma 4.2, it follows that for every Borel measurable X which is bounded from below on K n . Letω ∈ K n be the constant path t →ω(t) := ω(0) for some ω ∈ K n . Since the Dirac measure δω assigning probability 1 toω belongs to M(K n ), it follows that φ n is real-valued on C b and φ n (m) = m for every m ∈ R. Further, it is straightforward to check that φ n is convex and increasing in the sense that φ n (X) ≤ φ n (Y ) whenever X ≤ Y . Moreover, φ n is continuous from above on C b , i.e., φ n (X k ) ↓ φ n (0) for every sequence (X k ) in C b such that X k ↓ 0. To see this, fix such a sequence (X k ) and let ε > 0 be arbitrary. By Dini's lemma, one has X k ≤ ε on K n for all k large enough so that φ n (X k ) ≤ ε for all such k, which shows that φ n (X k ) ↓ 0. It follows from [6, Theorem 2.2] that φ n (X) = max − φ n (X)) and ca + ( ) denotes the set of non-negative countably additive Borel measures on . We claim that for all Q ∈ ca + ( ). First notice that (4.1) implies φ * n (Q) ≤ 0 whenever Q ∈ M(K n ).
It remains to show that if Q is a probability with Q[K n ] = 1 but not a martingale measure, then φ * n (Q) = +∞. Note that compactness of K n implies boundedness of K n with respect to · ∞ , and therefore Q is also not a local martingale measure. Thus Proposition 4.4 yields the existence of X ∈ C b and H ∈ H f such that X ≤ (H · S) T and E Q [X] > 0. Since φ n (mX) ≤ 0 for all m > 0, it follows that φ * n (Q) ≥ sup m>0 (E Q [mX] − φ n (mX)) = +∞. Now fix some upper semicontinuous X which is bounded from above (i.e., we have X = X ∧ m for some m > 0) and satisfies X ≥ −Z. We claim that To see this, let (X k ) be a sequence in C b such that X k ↓ X. By (4.2) and (4. Since M(K n ) is (sequentially) compact in the weak topology induced by the continuous bounded functions, we may assume, possibly after passing to a subsequence, that Q k → Q for some Q ∈ M(K n ).
Step (b). For X : → (−∞, +∞], define Let X ∈ C δσ be such that X ≥ −Z, and let (Y n ) be a sequence of upper semicontinuous functions which increases pointwise to X. Define X n := (Y n ∧ n) ∨ (−Z) which is still upper semicontinuous and increases to X. We claim that sup n φ n (X n ) = φ(X). First observe that for every Q ∈ M c ( ), Fatou's lemma and Lemma 4.2 imply for every λ ∈ R and every sequence (H n ) in H f such that λ + lim inf n (H n · S) T ≥ X and λ + (H n · S) T ≥ −mZ for all n and some m ≥ 0. Hence, one gets where the last equality follows from (4.4).
On the other hand, let m > sup n φ n (X n ) so that by definition, for each n, there exists H n ∈ H f such that m + (H n · S) T ≥ X n ≥ −Z on K n . Thus it follows from Lemma 4.6 that for all (t, ω) ∈ [0, T ] × K n . forH Fix ω ∈ . Then ω ∈ K j for some j ∈ N and therefore it follows by (4.6) that σ n (ω) = T whenever n ≥ j . Hence, we have for n ≥ j.
As ω was arbitrary, it follows that lim inf n (m + ε + (H n · S) T ) ≥ X. Moreover, it follows from (4.7) that which shows that φ(X) ≤ m + ε. Finally, since m > sup n φ n (X n ) and ε > 0 was arbitrary, we conclude that φ(X) ≤ sup n φ n (X n ), which shows that all inequalities in (4.5) are equalities. In particular, φ(X) = sup Q∈M c ( ) E Q [X] which shows (2.1).
Step (c). We finally show that M c ( ) can be replaced by the set M Z ( ), and H f by H. To that end, fix X : → (−∞, +∞] satisfying X ≥ −Z for some λ ∈ R, Q ∈ M Z ( ), and (H n ) in H such that λ + (H n · S) t (ω) ≥ −Z(ω t ) for all (t, ω) ∈ [0, T ] × and λ + lim inf n (H n · S) T ≥ X. Define Then we obtain where the last inequality holds by assumption. Hence, by Lemma 4.2 and Fatou's lemma, it follows that This shows that there is a sequence (H n ) in H f such that λ + (H n · S) t (ω) ≥ −Z(ω t ) for all (t, ω) ∈ [0, T ] × and λ + lim inf n (H n · S) T there is a sequence (H n ) in H such that λ + (H n · S) t (ω) ≥ −Z(ω t ) for all (t, ω) ∈ [0, T ] × and λ + lim inf n (H n · S) T where the first and last terms coincide by the previous steps (a) and (b).
The proof of Corollary 2.4 is a consequence of the following lemma. Proof If X = 0, there is nothing to prove. Otherwise, since is a Polish space, there exists K ⊆ compact such that Q(K c ) ≤ ε/ X ∞ . By an Arzelà-Ascoli type theorem [25,Theorem 1.4], there exist a ∈ R and a continuous increasing function f : [0, +∞) → [0, +∞) such that K ⊆K := ω ∈ : ω ∞ ≤ a and |ω(t) − ω(s)| ≤ f (|t − s|) for s, t ∈ [0, T ] andK is compact. Now define the stopping time Proof of Corollary 2.4 Denote by K the set of all compact subsets K ⊆ . For K ∈ K, defineK := {ω t : t ∈ [0, T ] and ω ∈ K} which is compact due to (the proof of) Lemma 4.5. For K ∈ K and every bounded upper semicontinuous function X : → R, define Then we have The first and third equalities follow from K ⊆K, the second one follows from φK (X) = max Q∈M(K) E Q [X] as in (4.4) for every K ∈ K and the last equality follows by the definition of M c ( ). Now use Lemma 4.7 to conclude.
Since {Z ≤cn} is compact, it follows from Dini's lemma that X k 1 {Z≤cn} ≤ ε for k large enough. Hence we get so that φ n (X k ) ≤ φ n (0) + 2ε for k large enough, which shows that φ n (X k ) ↓ φ n (0). Now a computation similar to the one in the proof of Theorem 2.1 shows that for every bounded upper semicontinuous function X : → R. Indeed, first notice that since Z ≥ · ∞ by assumption, the set M Z ( ) coincides with the set of all local martingale measures which integrate Z. Therefore, the same arguments as in the proof of Theorem 2.1 show that and thus that (4.8) is true, at least whenever X ∈ C b . For the extension to upper semicontinuous functions, notice that φ( Using the fact that Z has compact sublevel sets and Proposition 4.4, it follows that Λ c is (sequentially) compact. The rest follows analogously to the proof of Theorem 2.1.
Step (b). For X ∈ C δσ , define there are (H n ) in H f and c ≥ 0 such that (H n · S) T ≥ −cZ for all n and λ + lim inf n (H n · S) T ≥ X ⎫ ⎬ ⎭ .
Fix X ∈ C δσ bounded from below and X n upper semicontinuous bounded from below such that X = sup n X n . Then it follows from Fatou's lemma and Lemma 4.2 that On the other hand, if m > sup n φ n (X n ), then for every n, there exists H n ∈ H f such that m + (H n · S) T ≥ X n − Z/n. Hence (H n · S) T ≥ −cZ for c := X 1 ∧ 0 ∞ + m + 1 and m + lim inf n (H n · S) T ≥ lim inf n (X n − Z/n) = X, which completes the proof.
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A.1 Kolmogorov continuity criterion
In this section, we briefly recall a version of the so-called Kolmogorov continuity criterion, which provides a sufficient condition for Hölder and Sobolev regularity of stochastic processes. The presented version is a slight reformulation of [25, Theorem A.10]. Let (˜ , F, P ) be a probability space, X : [0, T ] ×˜ → R d a stochastic process, T ∈ (0, +∞), (R d , | · |) the Euclidean space and W a d-dimensional Brownian motion. Then for any α ∈ [0, 1/r − 1/q) and with δ := α + 1/q, there exists a constant C = C(r, q, α, T ) such that Therefore Theorem A.1 implies the assertion.

A.2 Construction of counterexample
The example (see Remark 2.6) showing that bounded variation strategies, and in particular simple trading strategies, are not rich enough to obtain the pathwise pricinghedging duality was based on a Hölder-continuous function with exploding quadratic variation. The existence of such a function is ensured by the following lemma. Recalling the properties of a Brownian motion W , we know that the event of a Brownian motion W starting at 0, τ ≥ 1 and W t ∈ (0, 1) for t ∈ (0, τ ) has a strictly positive probability. This fact ensures the existence of a constant T 0 > 1 and a (nowhere constant) function f ∈ C β ([0, T 0 ], R) for every β ∈ (0, 1/2) such that (i) f (0) = f (T 0 ) = 0; (ii) 0 ≤ f (t) ≤ 1 for all t ∈ [0, T 0 ]; (iii) the pathwise quadratic variation given by f t := lim n f n t exists along (π n ) for every t ∈ [0, T 0 ] (as limit in uniform convergence), and f T 0 > 0.
Without loss of generality, we may assume T 0 = 1 since it is always possible to modify f to ensure this without losing the other properties.
To obtain the desired properties of the quadratic variation, we define the partitioñ π m for m ∈ N as follows. For n ≤ m,π m restricted to [t n−1 , t n ] consists of the point for n ≥ m, chooseπ m restricted to [t n−1 , t n ] to be empty, and T is included inπ m . Note that (π m ) is a refining sequence of partitions. Furthermore, by the construction of (π m ), the pathwise quadratic variation ofω exists along (π m ) for all t ∈ [0, T ), and for t n , we observe that ω t n = n k=1 f T 0 n which goes to infinity as t n → T or in other words as n → ∞.