Gaoyue Guo
email: guo@cmap.polytechnique.fr

Xiaolu Tan
email: tan@ceremade.dauphine.fr

Nizar Touzi
email: nizar.touzi@polytechnique.edu

Tightness and duality of martingale transport on the Skorokhod space *

Keywords: S-topology, dynamic programming principle, robust superhedging

The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the continuoustime martingale transport on the Skorokhod space of càdlàg paths. Similar to the classical setting of optimal transport, we introduce different dual problems and establish the corresponding dualities by a crucial use of the S-topology and the dynamic programming principle 1 .

Introduction

Initialed by the famous work of Monge and Kantorovich, the optimal transport problem concerns the optimal transfer of mass from one location to another. Namely, let P(R d ) be the space of probability measures on the Euclidean space R d . For any given measures µ, ν ∈ P(R d ), put

P(µ, ν) := P ∈ P(R d × R d ) : P • X -1 = µ and P • Y -1 = ν , (1.1)
where (X, Y ) denotes the canonical process on R d × R d , i.e. X(x, y) = x and Y (x, y) = y for all (x, y) ∈ R d × R d . Then the optimal transport problem consists in optimizing

The last dual formulation has the interpretation of minimal robust superhedging cost of derivative security defined by the payoff ξ by trading the underlying security and any possible Vanilla option. As observed by Breeden and Litzenberger [START_REF] Breeden | Prices of state-contingent claims implicit in options prices[END_REF], the marginal distributions of the underlying asset is recovered by the market prices of calls for all strikes, and any Vanilla option has a non-ambiguous price as the integral of its payoff function with respect to the marginal. Therefore, the inequality (1.2) represents a super-replication of ξ, which consist of the trading of the underlying and Vanilla options at different maturities. Since there is no specific model imposed on the process (X, Y ), the dual problem may be interpreted as the robust superhedging cost, i.e. the minimum cost to construct super-replications. Similar to the classical setting, the duality P(µ, ν) = D(µ, ν) holds under quite general conditions. The present paper considers the continuous-time martingale optimal transport problem. Let X := ω = (ω t ) 0≤t≤1 : ω t ∈ R d for all t ∈ [0, 1] , where X is either the space of continuous functions or the Skorokhod space of càdlàg functions. Denote by X = (X t ) 0≤t≤1 the canonical process and by M the set of all martingale measures P, i.e. X is a martingale under P. For a given family of probability measures µ = (µ t ) t∈T , where T ⊆ [0, 1] is a subset, define by M(µ) the subset of transport plans P, i.e. P • X -1 t = µ t for all t ∈ T. Then for a measurable function ξ : X → R, the problem is defined by

P(µ) := sup P∈M(µ) E P [ξ(X)]. (1.3)
In contrast with the discrete-time case, the set M(µ) is generally not tight with respect to the usual topologies. Without the crucial compactness, the arguments in the classical setting fail to be adapted to handle the related issues.

In the existing literature, there are two dual formulations for the problem (1.3), Galichon, Henry-Labordère & Touzi studied a class of transport plans defined by stochastic differential equations in [START_REF] Galichon | A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options[END_REF] and introduced a quasi-sure dual problem. They applied a stochastic control approach and deduced the duality. Another important contribution is due to Dolinsky & Soner [START_REF] Dolinsky | Robust hedging and martingale optimal transport in continuous time, Probability Theory and Related Fields[END_REF][START_REF] Dolinsky | Martingale optimal transport in the Skorokhod space[END_REF], see also Hou & Oblój [START_REF] Hou | On robust pricing-hedging duality in continuous time[END_REF], where the dual problem is still pathwisely formulated as in (1.2). By discretizing the paths and a technical construction of approximated martingale measures, they avoid the compactness issue and derive the duality.

In addition, the martingale optimal transport problem is studied by the approach of Skorokhod embedding problem. Following the seminal paper of Hobson [START_REF] Hobson | Robust hedging of the lookback option[END_REF], this methodology generated developments in many directions, see e.g. Brown, Hobson & Rogers [START_REF] Brown | Robust hedging of barrier options[END_REF], Cox & Oblój [START_REF] Cox | Robust pricing and hedging of double no-touch options[END_REF][START_REF] Cox | Robust hedging of double touch barrier options[END_REF], Cox, Hobson & Oblój [START_REF] Cox | Pathwise inequalities for local time: applications to Skorokhod embeddings and optimal stopping[END_REF], Cox, Oblój & Touzi [START_REF] Cox | The Root solution to the multi-marginal embedding problem: an optimal stopping and time-reveral approach[END_REF], Cox & Wang [START_REF] Cox | Root's barrier: Construction, optimality and applications to variance options[END_REF], Davis, Oblój & Raval [START_REF] Davis | Arbitrage Bounds for Weighted Variance Swap Prices[END_REF], Gassiat, Oberhauser & dos Reis [START_REF] Gassiat | Root's barrier, viscosity solutions of obstacle problems and reflected FBSDEs[END_REF],

Hobson & Klimmek [START_REF] Hobson | Model independent hedging strategies for variance swaps[END_REF][START_REF] Hobson | Robust price bounds for the forward starting straddle[END_REF][START_REF] Hobson | Maximising functionals of the joint law of the maximum and terminal value in the Skorokhod embedding problem[END_REF], Hobson & Neuberger [START_REF] Hobson | Robust bounds for forward start options[END_REF] and Madan & Yor [START_REF] Madan | Making Markov martingales meet marginals: with explicit constructions[END_REF]. A thorough literature is provided in the survey papers Hobson [START_REF] Hobson | The Skorokhod Embedding Problem and Model-Independent Bounds for Option Prices[END_REF] and Oblój [START_REF] Oblój | The Skorokhod embedding problem and its offspring[END_REF].

Our main contribution in the paper is to study systematically the tightness of the set M(µ) by means of the S-topology introduced in Jakubowski [START_REF] Jakubowski | A non-Skorohod topology on the Skorohod space[END_REF]. Endowing properly the space of marginal laws with a Wasserstein kind topology, the tightness yields the upper semicontinuity of the map µ → P(µ) and further the first duality, obtained by penalizing the marginal constraints. Based on the first duality and using respectively the dynamic programming principle and the discretization argument of path-space, the dualities are established for both quasi-sure and pathwise dual formulations.

The above analysis immediately gives rise to a stability consequence. Denote P := P and P(µ) := inf P∈M(µ) E P [ξ(X)], then it is shown that the map µ → P(µ) (resp. µ → P(µ)) is upper (resp. lower) semicontinuous, which yields the stability, i.e. for any sequence (µ n ) n≥1 convergent to µ, there exists a sequence (ε n ) n≥1 ⊆ R + convergent to zero such that P(µ n ), P(µ n ) ⊆ P(µ) -ε n , P(µ) + ε n for all n ≥ 1, which implies that the interval of model-free prices is stable with respect to the market.

The paper is organized as follows. We formulate the martingale optimal transport problem and provide the dual problems in Section 2. In Section 3, the duality results are presented and we reduce the infinitely-many marginal constraints to the finitelymany marginal constraints. In Sections 4, 5 we focus on the finitely-many marginal case and provide all related proofs.

Martingale optimal transport

For all 0 ≤ s < t, denote by D([s, t], R d ) the space of càdlàg functions defined on [s, t] taking values in R d . Let Ω := D([0, 1], R d ) with generic element denoted by ω. Denote further by X := (X t ) 0≤t≤1 the canonical process, i.e. X t (ω) = ω t and by F := (F t ) 0≤t≤1 its natural filtration, i.e. F t = σ(X u , u ≤ t). Let P := P(Ω, F 1 ) be the set of probability measures on Ω. A probability measure P ∈ P is called a martingale measure if the canonical process X is a martingale under P. Denote by M the collection of all martingale measures.

Peacock and martingale optimal transport

Let P := P(R d ) be the space of all probability measures µ on R d with finite first moment. A pair (µ, ν) ∈ P × P is said to be increasing in convex ordering if

R d λ(x)µ(dx) := µ(λ) ≤ ν(λ) := R d λ(x)ν(dx)
holds for every convex function λ : R d → R. This relation is denoted by µ ν. Let T ⊆ [0, 1] be some subset containing 1 and define the T-product of P by

P T := µ := (µ t ) t∈T : µ t ∈ P for all t ∈ T .
Definition 2.1. A family of probability measures µ = (µ t ) t∈T ∈ P T is called a peacock (T-peacock) if µ s µ t holds for all s, t ∈ T such that s ≤ t. A peacock µ is said to be càdlàg if the map t → µ t is càdlàg on T with respect to the weak convergence. Denote by P T the set of all càdlàg peacocks.

For each peacock µ ∈ P T , define the set of transport plans

M(µ) := P ∈ M : P • X -1 t = µ t for all t ∈ T . (2.1)
We may assume without loss of generality that T is closed under the lower limit topology, i.e. the topology generated by all half-open intervals [s, t) ⊆ [0, 1], see e.g. Steen & Seebach [START_REF] Seebach | and Lynn Arthur Steen Counterexamples in Topology[END_REF]. Indeed, denote by T the closure of T under the lower limit topology, then it follows that the law of X t for t ∈ T is uniquely determined by the right continuity of X. This implies that M(μ) = M(µ), where μ := (μ t ) t∈ T is defined by

μt := lim n→∞ µ tn for any sequence (t n ) n≥1 ⊆ T decreasing to t. (2.2)
Remark 2.2. (i) Since µ tn µ 1 for all n, we have

µ tn (x i -K) + ≤ µ 1 (x i -K) + for all i = 1, • • • , d,
thus showing that the sequence (µ tn ) n≥1 is uniformly integrable. In particular, (µ tn ) n≥1 is tight, and we may verify immediately by a direct density argument that any two possible accumulation points μt and μ′ t coincides, i.e. μt = μ′ t . Hence the sequence (µ tn ) n≥1 converges weakly, justifying the convergence in (2.2) is well defined.

(ii) When T = [0, 1], M(µ) is nonempty by Kellerer's theorem, see e.g. Hirsch & Roynette [START_REF] Hirsch | A new proof of Kellerer's theorem[END_REF][START_REF] Hirsch | On R d -valued peacocks[END_REF] and Kellerer [START_REF] Hans | Markov-Komposition und eine Anwendung auf Martingale[END_REF]. For a general closed T, we may extend µ to some μ = (μ t ) 0≤t≤1 by μt := μt with t := inf{s ≥ t : s ∈ T}. Clearly, μ ∈ P [0,1] and μt = µ t for all t ∈ T. Hence M(µ) ⊇ M(μ) is again nonempty.

Let ξ : Ω → R be a measurable function. For every peacock µ ∈ P T , define the martingale optimal transport problem by

P(µ) := sup P∈M(µ) E P ξ(X) , (2.3)
where

E P [ξ] := E P [ξ + ] -E P [ξ -] with the convention +∞ -∞ = -∞.

Dual problems

First dual problem Let Λ be the set of continuous functions λ :

R d → R with linear growth, i.e. sup x∈R d |λ(x)|/(1 + |x|) < +∞. Define Λ T := λ := (λ t i ) 1≤i≤m : t i ∈ T, λ t i ∈ Λ for all i = 1, • • • , m, m ∈ N . For every λ = (λ t i ) 1≤i≤m ∈ Λ T , µ = (µ t ) t∈T ∈ P T and ω ∈ Ω, denote λ(ω) := m i=1 λ t i (ω t i ) and µ(λ) := m i=1 µ t i (λ t i ).
Next, we introduce three dual formulations. Roughly speaking, as X is required to be a martingale and has the given marginal laws in problem (2.3), then we dualize respectively these two constraints. The first dual problem is defined by

D 1 (µ) := inf λ∈Λ T µ(λ) + sup P∈M E P ξ(X) -λ(X) .
(2.4)

The dual problem D 1 is the Kuhn-Tucker formulation in convex optimization, where the marginal constraints µ are penalized by the Lagrange multipliers λ.

Second dual problem

The second dual problem dualizes further the martingale constraint and has close analogues in the mathematical finance literature in the context of a financial market with d risky assets, where the price process is modeled by the canonical process X = (X t ) 0≤t≤1 . For technical reasons, the underlying process X is assumed to be non-negative and start at some fixed price that may be normalized to be

1 := (1, • • • , 1) ∈ R d .
Namely, define the set of market scenarios

Ω + := ω ∈ Ω : ω 0 = 1 and ω t ∈ R d + for all t ∈ [0, 1]
and the set of all possible models M + := P ∈ M : supp(P) ⊆ Ω + . Consequently, the market calibration µ should satisfy µ 0 (dx) = δ 1 (dx) and supp(µ 1 ) ⊆ R d + .

(2.5)

Moreover, let us denote by F U = (F U t ) 0≤t≤1 the universally completed filtration, i.e. F U t := ∩ P∈P F P t , where F P t is the completed σ-field of F t under P.

Definition 2.3. A process S = (S t ) 0≤t≤1 is called a M + -supermartingale if it is F U -adapted and is a P-supermartingale for all P ∈ M + . Denote by S the collection of all M + -supermartingales and by S 0 ⊆ S the subset of processes starting at 0. Denote further

D 2 (ξ) := (λ, S) ∈ Λ T × S 0 : λ(ω) + S 1 (ω) ≥ ξ(ω) for all ω ∈ Ω + .
For a peacock µ ∈ P T satisfying (2.5) , the second dual problem is defined by

D 2 (µ) := inf (λ,S)∈D 2 (ξ)
µ(λ).

(2.6)

Remark 2.4. (i) Notice that the supermartingale S ∈ S is not required to have any regularity. If it were càdlàg then it would follow from Theorem 2.1 in Kramkov [START_REF] Kramkov | Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets[END_REF] that, for every P ∈ M + there exist a predictable process H P = (H P t ) 0≤t≤1 and an optional non-decreasing process A P = (A P t ) 0≤t≤1 such that

S t = S 0 + t 0 H P s dX s -A P t for all t ∈ [0, 1], P-a.s.
However, it is not clear whether one can aggregate the last representation, i.e. find predictable processes H and A such that (H, A) = (H P , A P ), P-almost surely. See also Nutz [START_REF] Nutz | Robust Superhedging with Jumps and Diffusion[END_REF] for a partial result of this direction.

(ii) In financial mathematics, the pair (λ, H P ) has the interpretation of a semi-static super-replicating strategy under the model P. If the aggregation above were possible, then the dual problem D 2 turns to the quasi-sure formulation of the robust superhedging problem, see also Beiglböck, Nutz & Touzi [START_REF] Beiglböck | Robust super-replication for time-invariant derivatives[END_REF], and the duality P = D 2 reduces to the well know pricing-hedging duality.

Third dual problem Following the pioneering work [START_REF] Hobson | Robust hedging of the lookback option[END_REF] of Hobson, the martingale optimal transport approach is applied to study the robust hedging problems in finance. We do not postulate any specific model on the underlying assets and pursue here a robust approach. Assume further that all call/put options are liquid in the market for maturities t ∈ T, thus yielding a family of marginal distributions µ = (µ t ) t∈T that is considered to be exogenous, see e.g. Breeden & Litzenberger [START_REF] Breeden | Prices of state-contingent claims implicit in options prices[END_REF]. Then, the time 0 market price of any derivative λ(X t ) is given by µ t (λ). Hence, the cost of a static strategy λ ∈ Λ T is µ(λ).

The return from a zero-initial cost dynamic trading, defined by a suitable process H = (H t ) 0≤t≤1 , is given by the stochastic integral (H • X) which we define similarly to Dolinsky & Soner [START_REF] Dolinsky | Martingale optimal transport in the Skorokhod space[END_REF]. We restrict H : [0, 1] → R d to be left-continuous with bounded variation. Then, we may define the stochastic integral by integration by parts:

(H • X) t := H t • X t -H 0 • X 0 - t 0 X u • dH u for all t ∈ [0, 1], ( 2
D 3 (ξ) := (λ, H) ∈ Λ T × A : λ(ω) + (H • X) 1 (ω) ≥ ξ(ω) for all ω ∈ Ω + .
For a peacock µ ∈ P T satisfying (2.5), the third dual problem is defined by

D 3 (µ) := inf (λ,H)∈D 3 (ξ)
µ(λ).

(2.8)

Remark 2.6. It is clear by definition that the weak duality P(µ) ≤ D 1 (µ) holds. Moreover, if the peacock µ satisfies (2.5), then

P(µ) ≤ D 1 (µ) ≤ D 2 (µ) ≤ D 3 (µ).

Main results

We aim to study the existence of optimal transport plans and establish the dualities in a systematic way. Before providing these results in Sections 3.2 and 3.3, we first introduce some notions of topology on Ω and the associated space of probability measures in Section 3.1.

Preliminaries

In the classical optimal transport problem, the relevant results (existence of optimizers, duality, etc.) rely essentially on the compactness condition of M(µ, ν). However, when passing to the continuous-time case, as shown by Example 3.1 below, the set M(µ) is in general not tight with respect to the topologies L ∞ (uniform topology) and J 1 (Skorokhod topology). For our purpose, we endow Ω with the S-topology introduced by Jakubowski [START_REF] Jakubowski | A non-Skorohod topology on the Skorohod space[END_REF] such that the Borel σ-field agrees with the projection σ-filed F 1 , and more importantly, the S-topology facilitates the tightness issue and both Skorokhod representation theorem and Prohorov's theorem hold true. Before introducing the S-topology, we give an example which shows that the topologies L ∞ and J 1 are not convenient to handle the tightness of M(µ).

Example 3.1. Let M = (M 0 , M 1 , M 2 ) be a discrete-time martingale on some probability space such that P M 0 = M 1 and

M 1 = M 2 > 0. Define P n := P • (M n ) -1 for n ≥ 3, where M n = (M n t ) 0≤t≤1 is defined by M n t := M 0 1 [0, 1 2 -1 n ) (t) + M 1 1 [ 1 2 -1 n , 1 2 + 1 n ) (t) + M 2 1 [ 1 2 + 1 n ,1] (t).
Clearly, P n ∈ M(µ) for all n ≥ 3 with T = {0, 1} and

µ = P • M -1 0 , P • M -1 2
. However, it follows from Theorem VI.3.21 in Jacod & Shiryaev [START_REF] Jacod | Limit theorems for stochastic processes[END_REF] that, the sequence (P n ) n≥3 is not J 1 -tight and thus not L ∞ -tight.

Definition 3.2 (S-topology).

The S-topology on Ω is the sequential topology induced by the following S-convergence, i.e. a set F ⊆ Ω is closed under S-topology if it contains all limits of its S-convergent subsequences, where the S-convergence (denoted by

S -→) is defined as follows. Let (ω n ) n≥0 ⊆ Ω, we say that ω n S -→ ω 0 as n → ∞ if for each ε > 0, we may find a sequence (v n ε ) n≥0 ⊆ Ω such that v n ε has bounded variation, ω n -v n ε ≤ ε for all n ≥ 0 and lim n→∞ [0,1] f (t) • dv n ε (t) = [0,1] f (t) • dv 0 ε (t) for all f ∈ C([0, 1], R d ).
We denote by

S *
-→ the convergence induced by the S-topology.

Remark 3.3. (i) It is shown in Jakubowski [START_REF] Jakubowski | A non-Skorohod topology on the Skorohod space[END_REF] that the S-topology is not metrizable. However, its associated Borel σ-field coincides with F 1 . Moreover, a function

ξ : Ω → R is S-continuous (semicontinuous) if and only if ξ is S * -continuous (semi- continuous). (ii) The functions ω → ω i,1 , ω → 1 0 ω i,t dt and ω → 1 0 |ω t |dt for i = 1, • • • , d are S-continuous. The functions ω → ω and ω → sup 0≤t≤1 ω i,t for i = 1, • • • , d are S-lower semicontinuous.
Notice that the S-topology is not metrizable, then instead of the usual weak convergence, we use another convergence of probability measures introduced in [START_REF] Jakubowski | A non-Skorohod topology on the Skorohod space[END_REF], which induces easy criteria for S-tightness and preserves the Prohorov's theorem, i.e. tightness yields sequential compactness. Definition 3.4. Let (P n ) n≥1 be a sequence of probability measures on the space (Ω, F 1 ). We say P n * =⇒ D P if for each subsequence (P n k ) k≥1 , one can find a further subsequence (P n k l ) l≥1 and stochastic processes (Y l ) l≥1 and Y defined on the probability space

[0, 1], B [0,1] , ℓ such that L(Y l ) = P n k l for all l ≥ 1, L(Y ) = P, Y l (e) S * -→ Y (e) for all e ∈ [0, 1],
and for each ε > 0, there exists an S * -compact subset

K ε ⊆ Ω such that ℓ Y l ∈ K ε for all l ≥ 1 > 1 -ε.
It follows from Jakubowski [START_REF] Jakubowski | A non-Skorohod topology on the Skorohod space[END_REF] (see Theorem A.1) that the convergence * =⇒ D implies in some sense the convergence of finite dimensional distributions that is specified later, and more importantly, the limit of every convergent sequence of martingale measures is still a martingale measure.

We next introduce the Wasserstein distance for the purpose of deriving the duality P = D 1 . Recall the set P(µ, ν) introduced in (1.1). Definition 3.5. The Wasserstein distance of order 1 is defined by

W 1 (µ, ν) := inf P∈P(µ,ν) E P X -Y for all µ, ν ∈ P. A sequence (µ n ) n≥1 ⊆ P converges to µ ∈ P in W 1 if W 1 (µ n , µ) → 0 as n → ∞ or, equivalently, lim n→∞ µ n (λ) = µ(λ)
for all λ ∈ Λ, see e.g. Theorem 6.9 in Villani [START_REF] Villani | Optimal Transport: Old and New[END_REF].

For µ n = (µ n t ) t∈T n≥1 ⊆ P T and µ = (µ t ) t∈T ∈ P T , we say that µ n converges to µ if µ n t converges to µ t in W 1 for all t ∈ T and this convergence is denoted by

W T 1 -→.
We now provide a crucial tightness result for the present paper which is a consequence from [START_REF] Jakubowski | A non-Skorohod topology on the Skorohod space[END_REF].

Let T 0 ⊆ T be the collection of all condensation points under the lower limit topology, i.e. t = 1 or [t, t + ε) ∩ T is uncountable for any ε > 0.

Lemma 3.6. Let (P n ) n≥1 be a sequence of probability measures such that P n ∈ M(µ n ) for some µ n ∈ P T , satisfying

µ n W T 1 -→ µ ∈ P T . ( 3.1)
(i)Then, (P n ) n≥1 is S-tight, i.e. any subsequence (P n k ) k≥1 admits a further convergent subsequence under * =⇒ D . Moreover, any limit point P of (P n ) n≥1 is again a martingale measure.

(ii) Assume in addition that T 0 = T, then P ∈ M(µ).

Proof. (i) By Theorem A.1, it is clear that (P n ) n≥1 is S-tight and there exist a convergent subsequence (P n k ) k≥1 with limit P ∈ P. Moreover, one has a countable subset T ⊆ [0, 1) such that for any finite set

{u 1 , • • • , u r } ⊂ [0, 1] \ T , P n k • X u 1 , • • • , X ur -1 L -→ P • X u 1 , • • • , X ur -1 as k → ∞. (3.2)
Let s, t ∈ [0, 1] \ T such that s < t, and take a finite subset

{u 1 , • • • , u r } ⊆ [0, s]
\T and a sequence of bounded continuous functions {f i } 1≤i≤r . Notice that for every u ∈ [0, 1], X u is uniformly integrable with respect to (P n ) n≥1 . Indeeed,

lim R→∞ sup n≥1 E Pn X u 1 |Xu|≥R ≤ lim R→∞ sup n≥1 E Pn |X u | -R/2 + ≤ lim R→∞ sup n≥1 E Pn |X 1 | -R/2 + = lim R→∞ sup n≥1 µ n 1 (|x| -R/2) + = 0. (3.3)
Combining (3.2) and (3.3), one has

E P f 1 (X u 1 ) • • • f r (X ur ) X t -X s = lim k→∞ E Pn k f 1 (X u 1 ) • • • f r (X ur ) X t -X s = 0.
Since T is at most countable, it follows that E P [X t |F s ] = X s for any s, t ∈ [0, 1] \ T such that s < t. It follows by the right continuity of X that P ∈ M.

(ii) To prove that P ∈ M(µ), it remains to show that P • X -1 t = µ t for all t ∈ T.

When t ∈ T \ T , by the convergence (3.2) and the fact that

µ n W T 1 -→ µ, it follows that P • X -1 t = µ t .
Further, notice that T 0 = T, then for every t ∈ T, there exists a sequence (t i ) i≥1 ⊆ T\T decreasing to t. Using again the right continuity of X, we conclude P • X -1 t = lim i→∞ P • X -1 t i = µ t . As a consequence, the set M(µ) is S-tight and it is closed if T 0 = T. The following example shows that the closeness may fail when T 0 = T. Example 3.7. Let T = {0, 1} and consider a random variable Y such that P(Y = 1) = P(Y = -1) = 1/2. Define P n := P • (M n ) -1 for n ≥ 1, where M n = (M n t ) 0≤t≤1 is defined by

M n t := Y 1 [ 1 n ,1] (t).
Define a peacock µ = (µ 0 , µ 1 ) by µ 0 := δ {0} and µ 1 = δ {-1} + δ {1} /2. Obviously, P n ∈ M(µ) for all n ≥ 1. However, the limit of (P n ) n≥1 is a martingale measure P such that X t = X 0 , P-a.s. and P • X -1 0 = µ 1 , which does not lie in M(µ).

Finitely-many marginal constraints

We start by studying the finitely-many marginal case and assume throughout this subsection that

T = {0 = t 0 < • • • < t m = 1}. Denote ∆t i := t i -t i-1 for all i = 1, • • • ,
ω n S * -→ ω and ω n t i -→ ω t i for all i = 0, • • • , m -1. For ε = (ε 1 , • • • , ε m ) ∈ R m + such that |ε| < ∆T, let f ε (forward function
) and b ε (backward function) be two non-decreasing functions defined on [0, 1]:

f ε (t) := m i=1 1 (t i-1 ,t i ] (t) t i-1 + ∆t i ∆t i -ε i t -t i-1 -ε i + , (3.4) b ε (t) := m i=1 1 (t i-1 ,t i ] (t) t i -∆t i - ∆t i ∆t i -ε i t -t i-1 + . (3.5)
Assumption 3.9. There is a continuous function α : R + → R + with α(0) = 0 such that the following inequality holds for any

ε ∈ R m + satisfying |ε| < ∆T ξ(ω) -ξ ω fε ≤ α(|ε|) 1 + m i=0 |ω t i | + 1 0 |ω t |dt , (1)
ξ(ω) -ξ ω bε ≤ α(|ε|) 1 + m i=0 |ω t i | + 1 0 |ω t |dt , (2)
where ω fε (resp. ω bε ) denotes the composition of ω and f ε (resp. b ε ).

Theorem 3.10. Le ξ be bounded from above and satisfies Assumptions 3.8 and 3.9 [START_REF] Acciaio | A Trajectorial Interpretation of Doob's Martingale Inequalities[END_REF]. Then for all µ ∈ P T : (i) The duality P(µ) = D 1 (µ) holds.

(ii) Assuming further that ξ is bounded, the duality D 1 (µ) = D 2 (µ) holds for all µ satisfying (2.5).

To establish the duality D 1 (µ) = D 3 (µ), we need more regularity conditions on ξ. Define a distance ρ T on Ω by

ρ T (ω, ω ′ ) := m i=1 ρ [t i-1 ,t i ] (ω, ω ′ ) + 1 0 ω u -ω ′ u du for all ω, ω ′ ∈ Ω, (3.6) where ρ [s,t] : D([s, t], R d ) × D([s, t], R d ) → R + dentoes the Skorokhod metric on the space D([s, t], R d ). Clearly, |ω t i -ω ′ t i | ≤ ρ T (ω, ω ′ ) for all ω, ω ′ ∈ Ω and i = 1, • • • , m.
Assumption 3.11. ξ is locally ρ T -uniformly continuous.

Theorem 3.12. Let ξ be bounded and µ ∈ P T satisfying (2.5). Then under Assumptions 3.8, 3.9 and 3.11, the duality P(µ) = D 3 (µ) holds.

Remark 3.13. Using the pathwise Doob's inequality in Acciaio, Beiglböck, Penkner, Schachermayer & Temme [START_REF] Acciaio | A Trajectorial Interpretation of Doob's Martingale Inequalities[END_REF], the boundeness condition in Theorem 3.12 may be removed when µ 1 (|x| p ) < +∞ for some p > 1, see also Dolinsky & Soner [START_REF] Dolinsky | Martingale optimal transport in the Skorokhod space[END_REF].

Infinitely-many marginal constraints

Using approximation techniques, we then obtain some results for the martingale transport problem under infinitely-many marginal constraints.

Proposition 3.14. Let ξ be S * -upper semicontinuous and bounded from above. For all µ ∈ P T : (i) Assume that there exists an increasing sequence of finite sets {T n } n≥1 such that 1 ∈ T n ⊆ T for all n ≥ 1 and ∪ n≥1 T n is dense in T under the lower limit topology. Then

lim n→∞ P(µ n ) = P(µ) with µ n := (µ t ) t∈Tn .
(ii) Assume T 0 = T, then there exists an optimal transport plan P * ∈ M(µ), i.e.

P(µ) = E P * [ξ(X)]. (3.7)
Proof. (i) It follows by the definition of µ n that P(µ n ) is non-increasing with respect to n. Take a sequence (P n ) n≥1 such that P n ∈ M(µ n ) and

P(µ) ≤ lim n→∞ P(µ n ) = lim n→∞ E Pn [ξ].
By Lemma 3.6 (i) , there is a convergent subsequence (P n k ) k≥1 with some limit P ∈ M. It follows by the same arguments in the proof of Lemma 3.6 that P ∈ M(µ) and, the upper semicontinuity of ξ yields

lim n→∞ P(µ n ) = lim k→∞ E Pn k [ξ] ≤ E P [ξ] ≤ P(µ).
(ii) Take a maximizing sequence (P n ) n≥1 ⊆ M(µ), then we may get a limit point P * and by Lemma 3.6 (ii) , P * is the required optimal transport plan.

Consequently, we obtain immediately the dualities for general T through Proposition 3.14.

Theorem 3.15. Let ξ be S * -upper semicontinuous and bounded from above and µ ∈ P T , consider an increasing sequence of finite sets {T n } n≥1 such that T n ⊆ T, ∪ n≥1 T n is dense in T, and set µ n := (µ t ) t∈Tn .

(i) Assume that P(µ n ) = D 1 (µ n ) for all n ≥ 1. Then P(µ) = D 1 (µ).

(ii) Assume further that µ satisfies (2.5) and

D 1 (µ n ) = D 2 (µ n ) = D 3 (µ n ) for all n ≥ 1. Then D 1 (µ) = D 2 (µ) = D 3 (µ).
Proof. It is enough to show (i). Notice by definition that D 1 (µ n ) ≥ D 1 (µ) for all n ≥ 1, then it follows by Proposition 3.14 (i) that

P(µ) = lim n→∞ P(µ n ) ≥ D 1 (µ).
Then the proof is fulfilled by the weak duality P(µ) ≤ D 1 (µ).

Remark 3.16. In the present setting, the marginal constraint µ = (µ t ) t∈T is given by a family of joint distributions µ t on R d . If we replace the probability distribution µ t by, either

d marginal distributions (µ 1 t , • • • , µ d t ) on R, or a joint distribution µ t on R l×d for some t := (t 1 , • • • , t l ) with 0 ≤ t 1 < • • • < t l ≤ 1,
then all the arguments still hold true and we can obtain similar duality results as in Theorems 3.10, 3.12 and 3.15.

The dualities P

= D 1 = D 2
In the following, we focus on the finite-marginal case, i.e.

T = {0 = t 0 < • • • < t m = 1}
and start by proving the first duality. To prove the equality P = D 1 , we shall apply the following well-known result from convex analysis. Next we show that the map µ → P(µ) is W T 1 -upper semicontinuous and concave and then identify its dual space to be Λ T by µ, λ = µ(λ).

Space of signed measures on R d and its dual space

Let M denote the space of all finite signed Borel measures µ on R d satisfying

R d 1 + |x| |µ|(dx) < +∞.
It is clear that M is a linear vector space. We endow M with a topology (of Wasserstein kind) induced by the following convergence: Let (µ n ) n≥0 ⊆ M be a sequence of bounded signed measures, we say µ n converges to

µ 0 if lim n→∞ R d λ(x)µ n (dx) = R d λ(x)µ 0 (dx) for all λ ∈ Λ.
Notice that the above topology restricted on the subspace P ⊆ M of probability measures is exactly that induced by the Wasserstein distance. As for the space M 0 of all finite signed Borel measures on R d equipped with the weak convergence topology, it is well known that its dual space M * 0 can be identified as the space of all bounded continuous functions Λ 0 , see e.g. Deuschel & Stroock [START_REF] Deuschel | Large Deviations[END_REF]. The following lemma identifies the dual space of M. Lemma 4.2. The space M is a Hausdorff locally convex space, and the duality relation

(λ, µ) ∈ Λ × M -→ µ(λ) determines a representation of M * as Λ.
The proof is almost the same as that of M * 0 = Λ 0 . For completeness, we provide a short proof in Appendix. For the finite set T, let us endow M T with the product topology and obviously, the dual space of M T is given by Λ T .

Proof of the duality P = D 1

In preparation for the first duality, we show first the upper semicontinuity of µ → P(µ) in the context of Theorem 3.10 (i) . For ε

= (ε 1 , • • • , ε m ) ∈ R m + such that |ε| < ∆T, we introduce M ε (µ) := P ∈ M(µ) : X t = X t k on [t k , t k + ε k ) for all k = 0, • • • , m -1, P-a.s.
and

P ε (µ) := sup P∈M ε (µ) E P ξ(X) .
Proposition 4.3. Let ξ be bounded from above and satisfying Assumptions 3.8 and 3.9 (1), then µ → P(µ) is W T 1 -upper semicontinuous on P T .

Proof. (i) First notice P ε (µ) ≤ P(µ) since M ε (µ) ⊆ M(µ). Next, for each P ∈ M(µ), define

P ε := P • X -1 fε , where f ε is defined in (3.4). It is clear that P ε ∈ M ε (µ) and E P ε [ξ(X)] = E P [ξ(X fε )]
. It follows by Assumption 3.9 (1),

E P ξ(X) ≤ E P ξ(X fε ) + α(|ε|) 1 + (m + 2)E P |X 1 | = E P ε ξ(X) + α(|ε|) 1 + (m + 2)µ 1 (|x|) ≤ P ε (µ) + α(|ε|) 1 + (m + 2)µ 1 (|x|) ,
which implies that

P(µ) = inf 0<|ε|<∆T P ε (µ) + α(|ε|) 1 + (m + 2)µ 1 (|x|) .
(ii) In order to prove that µ → P(µ) is upper semicontinuous, it suffices to verify that µ → P ε (µ) is upper semicontinuous µ → P ε (µ) is upper semicontinuous. To see this, let (µ n ) n≥1 ⊆ P T be a sequence such that µ n W T 1 -→ µ ∈ P T . By definition, we have a sequence (P n ) n≥1 such that P n ∈ M ε (µ n ) and lim sup

n→∞ P ε (µ n ) = lim sup n→∞ E Pn ξ .
Then one may find a convergent subsequence (P n k ) k≥1 with limit P ∈ M. It follows by exactly the same arguments as in Lemma 3.6 (ii) that P ∈ M ε (µ). Since ξ is bounded from above, then it follows from Fatou's lemma that lim sup

n→∞ E Pn [ξ] = lim k→∞ E Pn k [ξ] ≤ E P 0 [ξ] ≤ P ε (µ),
which concludes the proof. Now we are ready to provide the first duality P(µ) = D 1 (µ). To apply Fenchel-Moreau theorem, we need to embed P T to a locally convex space. Recall that M is the space of all finite signed measures µ such that

R d 1 + |x| |µ|(dx) < +∞,
and M T is its T-product. We then extend the map P from P T to M T by

P(µ) := P(µ), if µ ∈ P T , -∞, otherwise.
Proof of Theorem 3.10 (i) . The concavity of the map µ → P(µ) is immediate from its definition. Together with the upper semicontinuity of Proposition 4.3, we may directly verify that the extended map P is also W T 1 -upper semicontinuous and concave. Then, combining the Fenchel-Moreau theorem and Lemma 4.2, it follows that for all µ ∈ M T , P(µ) = P * * (µ), where P * * denotes the biconjugae of P. In particular, for µ ∈ P T one has

P(µ) = P(µ) = P * * (µ) = inf λ∈Λ T µ(λ) -P * (λ) = inf λ∈Λ T µ(λ) -inf ν∈M T ν(λ) -P(ν) ≥ inf λ∈Λ T µ(λ) + sup ν∈P T sup P∈M(ν) E P ξ -λ(X) = inf λ∈Λ T µ(λ) + sup P∈M E P ξ -λ(X) = D 1 (µ) ≥ P(µ),
which yields P(µ) = D 1 (µ).

Proof of the duality

D 1 = D 2
For technical reasons, we need to restrict the static strategy λ to a smaller class of functions Λ T lip defined by Λ T lip := λ = (λ t i ) 1≤i≤m ∈ Λ T : each λ t i is boundedly supported and Lipschitz . Proof. Clearly, by the definition of D 1 and the fact that µ 0 = δ 1 (dx) and supp(µ 1 ) ⊆ R d + , one obtains by interchanging inf and sup that

D 1 (µ) ≥ inf λ∈Λ T sup P∈M + µ(λ) + E P ξ -λ(X) ≥ sup P∈M + inf λ∈Λ T µ(λ) + E P ξ -λ(X) = P(µ) = D 1 (µ),
by Theorem 3.10 (i). Hence

D 1 (µ) = inf λ∈Λ T µ(λ) + sup P∈M + E P ξ -λ(X) .
Next for every λ = (λ t i ) 1≤i≤m ∈ Λ T , there exists some constant L > 0 such that for every 1

≤ i ≤ m, λ L t i (x) := λ t i (x) -L(1 + 1 • x) ≤ 0 for all x ∈ R d + .
Denote λ L := (λ L t i ) 1≤i≤m , then for every martingale measure P ∈ M + , we have

µ(λ) + E P ξ -λ(X) = µ(λ L ) + E P ξ -λ L (X) .
Further, for each R > 0, let ψ R : R d → [0, 1] be some continuous function such that

ψ R (x) = 1 whenever |x| ≤ R and ψ R (x) = 0 whenever |x| > R + 1. Let λ L,R := λ L,R t i 1≤i≤m with λ L,R t i (x) := λ L t i (x)ψ R (x) ≥ λ L t i (x), then sup P∈M + E P ξ -λ L,R (X) ≤ sup P∈M + E P ξ -λ L (X) .
On the other hand, for all P ∈ M + we have by monotone convergence theorem lim

R→∞ E P ξ -λ L,R (X) = E P ξ -λ L (X) . Hence lim R→∞ sup P∈M + E P ξ -λ L,R (X) = sup P∈M + E P ξ -λ L (X) . It follows that lim R→∞ µ(λ L,R ) + sup P∈M + E P ξ -λ L,R (X) = µ(λ L ) + sup P∈M + E P [ξ -λ L (X) = µ(λ) + sup P∈M + E P [ξ -λ(X) .
Finally, by a convolution argument each λ L,R t i can be approximated uniformly by some Lipschitz function that is also boundedly supported, which yields the required result. For all (ω, t) ∈ Ω + × [0, 1], denote by B sem ω,t ⊆ P the set of probability measures P such that P X s = ω s for all 0 ≤ s ≤ t = 1 and (X s ) s≥t is a non-negative semimartingale under P. Denote further M loc ω,t := P ∈ B sem ω,t : (X s ) s≥t is a local martingale under P .

Write in particular B sem = B sem ω,0 and M loc = M loc ω,0 . Let ζ : Ω → R be a measurable function and put

V t (ω) := sup P∈M loc ω,t E P ζ(X) . (4.2)
Our objective now is to show that the process (V t ) 0≤t≤1 is F U -adapted and that the dynamic programming principle holds. To achieve this, we use the related results in Neufeld & Nutz [START_REF] Neufeld | Measurability of Semimartingale Characteristics with Respect to the Probability Law[END_REF][START_REF] Neufeld | Nonlinear Lévy Processes and their Characteristics[END_REF]. Let P ∈ B sem be a semimartingale measure with the triplet (B P , C P , ν P ) of predictable semimartingale characteristics, see e.g. Chapter II of Jacod & Shiryaev [START_REF] Jacod | Limit theorems for stochastic processes[END_REF]. Notice that

M loc = P ∈ B sem : B P t = 0 for all t ∈ [0, 1] .
By Theorem 2.5 in [START_REF] Neufeld | Measurability of Semimartingale Characteristics with Respect to the Probability Law[END_REF], the map P → (B P , C P , ν P ) is measurable, then it follows that M loc is Borel. Moreover, by the same arguments we have the following lemma.

Lemma 4.5. The set (ω, t, P) ∈ Ω × [0, 1] × P(Ω) : P ∈ M loc ω,t is Borel.

By Theorem 2.1 in [START_REF] Neufeld | Nonlinear Lévy Processes and their Characteristics[END_REF], we have the following lemma.

Lemma 4.6. Let P ∈ M loc ω,t and τ be an F-stopping time taking values in [t, 1]. (i) There is a family of conditional probability (P ω ) ω∈Ω of P with respect to F τ such that P ω ∈ M loc ω,τ (ω) for P-a.e. ω ∈ Ω. (ii) Assume that there exists a family of probability measures

(Q ω ) ω∈Ω + such that Q ω ∈ M loc ω,τ (
ω) for P-a.e. ω ∈ Ω, and the map ω

→ Q ω is F τ -measurable, then P ⊗ Q ∈ M loc ω,t
, where

P ⊗ Q(•) := Ω Q ω (•)P(dω).
The dynamic programming principle follows by Lemmas 4.5 and 4.6, and as a consequence we have the following proposition. Proposition 4.7. Assume that ζ is bounded, then the process V = (V t ) 0≤t≤1 defined in (4.2) is a M + -supermartingale, i.e. V ∈ S.

E P [ζ(X)] = inf V 0 : (V t ) 0≤t≤1 ∈ S such that V 1 (ω) ≥ ζ(ω) for all ω ∈ Ω + .
Proof. By Proposition 4.8 with the process V defined in (4.2), it remains to show that sup

P∈M + E P [ζ(X)] = sup P∈M loc E P [ζ(X)].
It is clear that sup )] since M + ⊆ M loc , then it suffices to prove the converse inequality. For each P ∈ M loc , there exists an increasing sequence of stopping times (σ n ) n≥1 such that σ n → +∞, P-almost surely and X σn∧• is a P-martingale, where X σn∧• := (X σn∧t ) 0≤t≤1 . Hence

P∈M + E P [ζ(X)] ≤ sup P∈M loc E P [ζ(X
E P [ζ(X σn∧• )] ≤ sup Q∈M + E Q [ζ(X)].
The required result follows from the dominated convergence theorem.

Proof of Theorem 3.10 (ii) . It remains to show D 1 (µ) ≥ D 2 (µ). Indeed, one has by Proposition 4.4,

D 1 (µ) = inf λ∈Λ T lip µ(λ) + sup P∈M + E P ξ(X) -λ(X) .
For each λ ∈ Λ T lip and ε > 0, by Proposition that 4.8 there exists a process

V ε = (V ε t ) 0≤t≤1 ∈ S such that D 1 (µ) + ε ≥ µ(λ) + V ε 0 and V ε 1 (ω) ≥ ξ(ω) -λ(ω).
This implies λ, (V ε t -V ε 0 ) 0≤t≤1 ∈ D 2 (ξ), and therefore D 1 (µ) ≥ D 2 (µ) .

5 Proof of the duality D 1 = D 3

Now let us turn to prove the third duality D 1 = D 3 in Theorem 3.12. We will follow the idea in Dolinsky & Soner [START_REF] Dolinsky | Martingale optimal transport in the Skorokhod space[END_REF] to discretize the underlying paths and then use the classical constrained duality result of Föllmer & Kramkov [START_REF] Föllmer | Optional decompositions under constraints[END_REF]. The proof in [START_REF] Dolinsky | Martingale optimal transport in the Skorokhod space[END_REF] relies on the min-max theorem and the explicit approximation of a martingale measure. We emphasize that the present proof is less technically involved than [START_REF] Dolinsky | Martingale optimal transport in the Skorokhod space[END_REF] as the marginals constraints have already been reduced by the first duality.

Reduction of ξ to be boundedly supported

In this section we denote P(µ, ξ) and D 3 (µ, ξ) in place of P(µ) and D 3 (µ) to emphasize the dependence on ξ, then clearly for any ξ, ξ ′ : Ω → R and c ∈ R, one has

D 3 (µ, ξ + ξ ′ ) ≤ D 3 (µ, ξ) + D 3 (µ, ξ ′ ) and D 3 (µ, ξ + c) = D 3 (µ, ξ) + c.
In particular for c > 0 one has

D 3 (µ, cξ) = cD 3 (µ, ξ).
Hence, under the conditions of Theorem 3.12, we may assume that 0 ≤ ξ ≤ 1. Indeed, we show next that it suffices to establish the duality P(µ, ξ) = D 3 (µ, ξ) for ξ that is boundedly supported. For all R > 0, define the continuous function χ R : R + → [0, 1] by

χ R (x) := 1 [0,R] (x) + (R + 1 -x)1 (R,R+1] (x) for all x ∈ R + . Denote further for R > 0 ξ R (ω) := ξ(ω)χ R ( ω ) for all ω ∈ Ω. Notice that 0 ≤ ξ ≤ 1 yields ξ R (ω) ≤ ξ(ω) ≤ ξ R (ω) + 1 { ω ≥R} , then it follows that D 3 (µ, ξ R ) ≤ D 3 (µ, ξ) ≤ D 3 (µ, ξ R ) + D 3 (µ, 1 { X ≥R} ).
(5.1)

Lemma 5.1. Let ξ be bounded and µ ∈ P T,+ . Then

D 3 (µ, ξ) = lim R→∞ D 3 (µ, ξ R )
Proof. It is enough to prove by (5.1) that lim

R→+∞ D 3 (µ, 1 { X ≥R} ) = 0.
This is indeed a direct consequence of the pathwise inequality, see e.g. Lemma 2.3 of Brown, Hobson and Rogers [START_REF] Brown | Robust hedging of barrier options[END_REF] 1

{ X i ≥R} ≤ (|X i,1 | -K) + R -K + 1 { X i ≥R} R -X i,1 R -K for all i = 1, • • • , d
holds for every 0 < K < R. It follows by taking

K = R/2 that D 3 (µ, 1 { X ≥R} ) ≤ d i=1 D 3 (µ, 1 { X i ≥R/d} ) ≤ 2d R d i=1 µ 1 (x i - R 2d ) + .
The proof is fulfilled by letting R → +∞.

Next we show that ξ R inherits almost the same properties as ξ.

Lemma 5.2. For each R > 0: (i) If ξ satisfies Assumptions 3.8 and 3.9, then so does ξ R .

(ii) If ξ satifies Assumption 3.11, then ξ R is L ∞ -uniformly continuous, and ξ R (ω) -ξ R (ω ′ ) ≤ β ρ T (ω, ω ′ ) for all ω, ω ′ ∈ Ω such that ω ′ ≤ ω (5.2)
for some continuous increasing function β : R + → R + with β(0) = 0.

Proof. (i) follows by the fact that ω → ω is S * -lower semicontinuous and ω fε = ω bε = ω . Let us turn to show (ii) . Notice that ξ is ρ T -uniformly continuous on ω : ω ≤ R , i.e. there exists a continuous increasing function β : R + → R + with β(0) = 0 such that for all ω , ω ′ ≤ R

|ξ(ω) -ξ(ω ′ )| ≤ β ρ T (ω, ω ′ ) .
Hence, for any ω, ω ′ ∈ Ω such that ω ′ ≤ ω , one has

ξ R (ω) -ξ R (ω ′ ) ≤ 1 { ω ≤R} ξ(ω) -ξ(ω ′ ) ≤ β ρ T (ω, ω ′ ) . Moreover, |ξ R (ω ′ ) -ξ R (ω)| ≤ |ξ(ω ′ ) -ξ(ω)|χ R ( ω ) + |χ R ( ω ) -χ R ( ω ′ )| ≤ β ρ T (ω, ω ′ ) + ω -ω ′ ≤ β 2 ω -ω ′ + ω -ω ′ , which yields the L ∞ -uniform continuity of ξ R .
Therefore, in the following it suffices to consider the function ξ that is boundedly supported such that the Assumptions 3.8, 3.9 and Condition (5.2) hold. Similar to the proof of the duality P(µ) = D 2 (µ), it remains to prove a duality without marginal constraints.

Duality without marginal constraints

We consider in this section the optimization problem without marginal constraints. Let ζ : Ω → R be bounded and define

P(ζ) := sup P∈M + E P [ζ(X)] and D(ζ) := inf (z,H)∈D(ζ)
z.

(

where, with the same definition of integral in (2.7),

D(ζ) := (z, H) ∈ R × A : z + (H • X) 1 (ω) ≥ ζ(ω) for all ω ∈ Ω + .
We provide immediately a duality result for the above optimization problems, and leave its proof in Section 5.3.

Theorem 5.3. Suppose that ζ satisfies the Assumptions 3.8, 3.9 and Condition (5.2), then

P(ζ) = D(ζ). (5.4)
By exactly the same arguments as in the proof of Theorem 3.10 (ii) , the duality

P(µ, ξ R ) = D 3 (µ, ξ R ) follows immediately by taking ζ = ξ R -λ in Theorem 5.3.
Proof of Theorem 3.12. Using Lemma 5.1 as well as the first duality P = D 1 for ξ R , one has

P(µ, ξ) ≥ lim R→∞ P(µ, ξ R ) = lim R→∞ D 3 (µ, ξ R ) = D 3 (µ, ξ).
Hence we conclude the proof by the weak duality P(µ, ξ) ≤ D 3 (µ, ξ).

Proof of Theorem 5.3

Recall that T = {0 = t 0 < • • • < t m = 1}, ∆t i = t i -t i-1 for i = 1, • • • , m and ∆T = min 1≤i≤m ∆t i . Let ζ : Ω → R
be measurable and boundedly supported. Then for each 0 ≤ δ < ∆T, denote Ω δ := D([0, 1 + δ], R d ) and all its elements by ω δ . Put

T δ := {0 = t δ 0 < • • • < t δ m = 1 + δ}, where t δ i := k δ t i for all i = 0, • • • , m with k δ := 1 + δ. Define ζ δ : Ω δ → R by ζ δ (ω δ ) := ζ(ω δ ),

2).

(ii) There is a continuous function η : R + → R + with η(0) = 0 such that for all 0 ≤ δ < δ ′ < ∆T the following inequality holds

ζ δ (ω δ ) -ζ δ ′ (ω δ ′ ,δ ) ≤ η δ ′ -δ 1 + δ ′ 1 + m i=0 |ω δ t δ i | + 1+δ 0 |ω δ t |dt for all ω δ ∈ Ω δ ,
where ω δ ′ ,δ ∈ Ω δ ′ is defined by

ω δ ′ ,δ t := ω δ (t-t δ ′ i +t δ i )∧t δ i+1 for all t ∈ [t δ ′ i , t δ ′ i+1 ] and i = 0, • • • , m -1.
Proof. (i) will be proved in Lemmas 5.10, 5.11 and 5.12 in Section 5.5.

(ii) Clearly,

ζ δ ′ (ω δ ′ ,δ ) = ζ(ω δ ′ ,δ ), where ωδ ′ ,δ t := ω δ ′ ,δ k δ ′ t for all t ∈ [0, 1]. Direct computation reveals that ωδ, ′ δ = ωδ • b ε with ε := δ ′ -δ 1 + δ ′ ∆t 1 , • • • , ∆t m .
Hence by Assumption 3.9 one obtains

ζ δ (ω δ ) -ζ δ ′ (ω δ ′ ,δ ) = ζ(ω δ ) -ζ(ω δ ′ ,δ ) ≤ α(|ε|) 1 + m i=0 |ω δ t i | + 1 0 |ω δ t |dt ≤ α |∆T| δ ′ -δ 1 + δ ′ 1 + m i=0 |ω δ t δ i | + 1+δ 0 |ω δ t |dt .
The proof is completed by taking η(•) = α(|∆T|•).

We are now ready to prove the required duality. Define

Ω δ + := ω δ ∈ Ω δ : ω δ 0 = 1 and ω δ t ∈ R d + for all t ∈ [0, 1 + δ]
and the corresponding martingale optimal transport problem

P δ := sup P∈M δ + E P [ζ δ (X δ )],
where similarly, X δ = (X δ t ) 0≤t≤k δ denotes the canonical process and M δ + denotes the set of martingale measures supported on Ω δ + . The dual problem is slightly different. Denote further

Ω c,δ + := ω δ ∈ Ω δ + : ω δ t δ i -= ω δ t δ i for all i = 1, • • • , m
and define the dual problem by

D c δ := inf (z,H)∈D c δ z,
with D c δ given by

D c δ := (z δ , H δ ) ∈ R × A δ : z δ + (H δ • ω δ ) 1 ≥ ζ δ (ω δ ) for all ω δ ∈ Ω c,δ + ,
where, similarly to Definition 2.5, A δ denotes the collection of all left-continuous adapted processes with bounded variation such that the stochastic integral (H δ • X δ ) is a supermartingale under all probability measures in M δ + . The main technical step for our result is the following. Lemma 5.5. Suppose that ζ satisfies Assumptions 3.8, 3.9 and Condition (5.2). Then D c δ ≤ P δ for all δ ≥ 0.

(

The proof of Lemma 5.5 is adapted from Dolinsky & Soner [START_REF] Dolinsky | Martingale optimal transport in the Skorokhod space[END_REF] and is reported in Section 5.4.

Proof. (i) For each (z δ , H δ ) ∈ D c δ with δ > 0 let us construct a robust super-replication on Ω + . For any ω ∈ Ω + define H 0 (ω) = H δ 0 (ω δ,0 ) and

H t (ω) := H δ t-t i +t δ i (ω δ,0 ) for all t ∈ (t i , t i+1 ] and i = 0, • • • , m -1,
where ω δ,0 ∈ Ω c,δ + is defined as before by

ω δ,0 t = ω (t-t δ i +t i )∧t i+1 for all t ∈ [t δ i , t δ i+1 ] and i = 0, • • • , m -1.
It is clear that H is F-adapted, left-continuous, with bounded variation, and (H • X) is a supermartingale under every P ∈ M + , hence H ∈ A. Moreover,

z δ + H δ • ω δ,0 1+δ ≥ ζ δ (ω δ,0
) for all ω ∈ Ω + .

(5.8)

Notice that H δ • ω δ,0 1+δ = (H • ω) 1 , thus we obtain by Assumption 3.9 and Condition (5.8) (ii) Let (δ n ) n≥1 be such that δ n > 0 and δ n ↓ 0. Then there is a sequence (P n ) n≥1 such that lim sup

z δ + (H • ω) 1 ≥ ζ(ω) -η δ 1 + δ 1 + m i=0 |ω t i | +
n→∞ P δn = lim sup n→∞ E Pn ζ δn (X δn ) .
For any fixed δ 0 > 0, we assume without loss of generality that δ n ≤ δ 0 for all n ≥ 1.

Then for each n ≥ 1, let us define Pn := P n • Xδn -1 where Xδn (ω δn ) := X δ 0 (ω δ 0 ,δn ) is the extended process from Ω δn to Ω δ 0 . It follows by Proposition 5.4 (ii) that

E Pn ζ δn (X δn ) ≤ 1 + (m + 2)d η δ 0 -δ n 1 + δ 0 + E Pn ζ δ 0 ( Xδn ) = 1 + (m + 2)d η δ 0 -δ n 1 + δ 0 + E Pn ζ δ 0 (X δ 0 ) .
Again by the same argument in Proposition 4.

Proof of Lemma 5.5

The arguments are mainly adapted from Dolinsky & Soner [START_REF] Dolinsky | Martingale optimal transport in the Skorokhod space[END_REF] and the main idea is to discretize the paths on the Skorokhod space. By Proposition 5.4 (i), the proof of D c δ ≤ P δ is not altered by the value of δ. We therefore consider δ = 0 in this subsection.

A probabilistic hedging problem

For all n ∈ N, put

A (n) := 2 -n q : q ∈ N d and B (n) := i √ d2 -n : i ∈ N ∪ √ d2 -n /j : j ∈ N * .
We then define a subspace Ω := Ω(n) ⊆ Ω + as follows.

Definition 5.7. A path ω ∈ Ω + belongs to Ω if there exist non-negative integers

0 = K 0 < K 1 < • • • < K m + m and a partition 0 = τ0 < τ1 < • • • < τKm+m = 1 such that τK i +i = t i for 1 ≤ i ≤ m and ω t = m-1 i=0 K i+1 +i-1 k=K i +i ω τk 1 [τ k ,τ k+1 ) (t) + ω t i+1 1 [τ K i+1 +i ,t i+1 ) (t) + ω 1 1 {t=1} ,
where

ω t i ∈ A (n) for 1 ≤ i ≤ m and for 0 ≤ i < m ω τk ∈ A (n+k-K i -i) , K i + i < k < K i+1 + i + 1, τk -τk-1 ∈ B (n+k-K i -i) , K i + i < k < K i+1 + i + 1.
Notice that Ω is countable, then there exists a probability measure P := P(n) on Ω + supported on Ω which gives positive weight to every element of Ω. In particular, the canonical process X has finitely many jumps P-almost surely. Denote by F the completed filtration of F under P. Put

Ĥ(n) := Ĥ : [0, 1] × Ω + → R d is F -predictable such that Ĥ ≤ n and Â(n) := Ĥ ∈ Ĥ(n) : ( Ĥ • X) t ≥ K for all t ∈ [0, 1], P-a.s. for some K ∈ R . Let D(n) (ζ) := (z, Ĥ) ∈ R × Â(n) : z + ( Ĥ • X) 1 ≥ ζ(X), P-a.s.
and define the robust superhedging problem under the dominating measure P

D (n) (ζ) := inf (z, Ĥ)∈ D(n) (ζ)
z.

Let P ⊆ P be the subset of probability measures supported on Ω, and Mn ⊆ P be the subset of probability measures Q that have the following properties: Proof. (i) From Example 2.3 and Proposition 4.1 in Föllmer & Kramkov [START_REF] Föllmer | Optional decompositions under constraints[END_REF], it follows that

E Q Km+m k=1 E Q [X τk |F τk -] -X τk-1 ≤ 1 n , where 0 < τ1 (ω) < • • • < τKm+m-1 (ω) < 1
D (n) (ζ) = sup Q∈ P E Q ζ -n Km+m k=1 X τk-1 -E Q [X τk |F τk -] .
Since 0 ≤ ζ ≤ 1, we determine that D (n) (ζ) ≥ 0 and we have for every Q ∈ P\ Mn ,

E Q ζ -n Km+m k=1 X τk-1 -E Q [X τk |F τk -] ≤ 0, which yields
D (n) (ζ) ≤ sup Q∈ Mn E Q [ζ(X)]. (ii) Let us take a sequence (Q n ) n≥1 with Q n ∈ Mn such that lim sup n→∞ sup Q∈ Mn E Q [ζ(X)] = lim sup n→∞ E Qn [ζ(X)].
Since under each Q n the canonical process X is piecewise constant with jump times 0

< τ1 < • • • < τKm+m-1 < 1, X is a Q n -semimartingale.
Then we have the decomposition X = M Qn -A Qn , where A Qn is a predictable process of bounded variation and M Qn is a martingale under Q n . Moreover, A Qn is identified by

A Qn t = Km+m-1 k=1 1 [τ k ,τ k+1 ) (t) k j=1 X τj-1 -E Qn [X τj |F τj -] for all t ∈ [0, 1),
and

A Qn 1 = lim t→1 A Qn t . It follows then E Qn |X 1 -M Qn 1 | ≤ E Qn |A Qn 1 | ≤ 1/n and Q n A Qn ≥ n -1/2 ≤ n 1/2 E Qn Km+m-1 k=1 X τk-1 -E Qn [X τk |F τk -] ≤ n -1/2 . Since ζ is L ∞ -uniformly continuous, one obtains lim sup n→∞ E Qn [ζ(X)] ≤ lim sup n→∞ E Qn [ζ(M Qn )]. Let P n = Q n • (M Qn ) -1 , then sup n≥1 E Pn [|X 1 |] = sup n≥1 E Qn [|M Qn 1 |] ≤ sup n≥1 E Qn [|M Qn 1 -X 1 |] + sup n≥1 E Qn [X 1 ] ≤ sup n≥1 E Qn [|M Qn 1 -X 1 |] + sup n≥1 E Qn [X 1 -M Qn 1 ] + sup n≥1 E Qn [M Qn 1 ] ≤ 1 + 2 n ≤ 3.
By Assumptions 3.8 and 3.9, it follows that for any ε

∈ R m + such that 0 < |ε| < ∆T lim sup n→∞ E Pn [ζ(X)] ≤ lim sup n→∞ E Pn [ζ(X fε )] + 1 + (m + 2)d α(|ε|).
Again with the same reasoning, we may prove

lim sup n→∞ E Pn [ζ(X fε )] ≤ P(ζ).
Since ε is arbitrary we get lim sup

n→∞ sup Q∈ Mn E Q [ζ(X)] ≤ P(ζ),
and hence the required result.

Time-space discretization

Discretization:

For each ω ∈ Ω c + let us define τ k := τ (n) k (ω) and K i := K (n) i (ω) by τ 0 := 0, K 0 := 0, τ 1 := t 1 ∧ √ d2 -n ∧ inf t > 0 : |ω t -ω 0 | ≥ 2 -n , τ k+1 := t 1 ∧ (τ k + ∆τ k ) ∧ inf t > τ k : |ω t -ω τ k | ≥ 2 -n , ∆τ k = τ k -τ k-1 for k ≥ 1.
Set further

K 1 := min {k ∈ N : τ k = t 1 } .
Recursively, we define for 1 ≤ i ≤ m -1 and k ≥ K i ,

τ K i +1 := t i+1 ∧ (t i + √ d2 -n ) ∧ inf t > t i : |ω t -ω t i | ≥ 2 -n , τ k+1 := t i+1 ∧ (τ k + ∆τ k ) ∧ inf t > τ k : |ω t -ω τ k | ≥ 2 -n for k ≥ K i + 1
and

K i+1 := min {k ∈ N : τ k = t i+1 } . It is clear that 0 = τ 0 < τ 1 • • • < τ Km = 1 and τ K i = t i for all i = 1, • • • , m. Moreover, for 0 ≤ i ≤ m -1, K i < k ≤ K i+1 and t ∈ [τ k-1 , τ k ), |ω t -ω τ k-1 | ≤ 2 -n and ∆τ k+1 ≤ ∆τ k ≤ 2 -n .
Also by the continuity of ω at τ

K i = t i for all i = 1, • • • , m |ω t -ω τ K i -1 | ≤ 2 -n for all t ∈ [τ K i -1 , t i ] and i = 1, • • • , m.
Lifting: Set τ0 := 0 and for 0

≤ i ≤ m -1 τK i +1 := τK i + √ d2 -n , τk := τk-1 + 1 - √ d2 -n /∆t i+1 sup ∆t > 0 : ∆t ∈ B (n+k-K i -i) , ∆t < ∆τ k-1 , for all K i + i + 2 ≤ k ≤ K i+1 + i, τK i+1 +i+1 := t i+1 . Denote Π(ω) = Π(ω) 0≤t≤1 by Πt (ω) := m-1 i=0 K i+1 +i-1 k=K i +i π (n+k-K i -i) (ω τ k )1 [τ k ,τ k+1 ) (t) + π (n) (ω t i )1 [τ Kr ,t i+1 ) (t) +π (n) (ω 1 )1 {t=1} , then Π(ω) ∈ Ω.
For each Ĥ ∈ Â we may define

H t (ω) := Km-1 k=0 Ĥτ k+1 (ω) ( Π(ω))1 (τ k (ω),τ k+1 (ω)] (t) for all (ω, t) ∈ Ω + × [0, 1]. (5.9)
Following the argument of Lemmas 3.5 and 3.6 of Dolinsky & Soner [START_REF] Dolinsky | Martingale optimal transport in the Skorokhod space[END_REF], we see that the process H defined by (5.9) belongs to A, and more importantly, there exists some constant C > 0 independent of n such that for all ω ∈ Ω + , Proof. Take an arbitrary (ẑ, Ĥ) ∈ D. Then for any ω ∈ Ω + one has Π(ω) ∈ Ω and thus

ρ T (ω, Π(ω)) ≤ C2 -n 1 + ω and (H • ω) 1 -( Ĥ( Π(ω)) • Π(ω)) 1 ≤ Cn2 -n . ( 5
ẑ + ( Ĥ( Π(ω)) • Π(ω)) 1 ≥ ζ( Π(ω)) for all ω ∈ Ω + .
Take H constructed as (5.9), then by (5.10), we have

H ∈ A and ẑ + (H • ω) 1 ≥ ζ( Π(ω)) -Cn2 -n for all ω ∈ Ω + .
Moreover, by the construction of Π(ω) one has Π(ω) ≤ ω . Notice that ζ is boundedly supported, saying by ω ∈ Ω : ω ≤ R . Then by (5.2) one has a continuous increasing function β : R + → R + with β(0) = 0 such that for all ω ∈ Ω + ,

ζ( Π(ω)) ≥ ζ(ω) -1 { ω ≤R} β ρ T ω, Π(ω) ≥ ζ(ω) -β C(1 + R)2 -n , which implies that ẑ + β C(1 + R)2 -n + Cn2 -n , H ∈ D c (ζ). Hence D c (ζ) ≤ D (n) (ζ) + β C(1 + R)2 -n + Cn2 -n ,
which yields the required result.

Proof of Proposition 5.4 (i)

Recal that ξ satisfies Assumptions 3.8, 3.9 and Condition (5.2). The required statement follows from the three lemmas below.

Lemma 5.10. lim sup n→∞ ξ δ (ω δ,n ) ≤ ξ(ω δ,0 ) holds for any sequence (ω δ,n ) n≥0 ⊆ Ω δ such that ω δ,n S * -→ ω δ,0 and ω δ,n

t δ i -→ ω δ,0 t δ i for all i = 0, • • • , m -1.
Proof. For the sake of simplicity we may assume that lim sup

n→∞ ξ δ (ω δ,n ) = lim n→∞ ξ δ (ω δ,n ).
Since (ω δ,n ) n≥1 is S-tight, then by the S-tightness criteria and the construction in (5.5) we determine that (ω δ,n ) n≥1 is again S-tight, which yields a convergent subsequence (ω δ,n k ) k≥1 and a limit ω 0 ∈ Ω, i.e. ωδ,n k S * -→ ω 0 . Clearly, ω δ,n

t δ i -→ ω δ,0 t δ i implies in particular that ωδ,n k t i -→ ωδ,0 t i for all i = 0, • • • , m -1.
Next, ω δ,n k S * -→ ω δ,0 yields a countable set T ⊆ [0, 1 + δ) such that The proof is completed by taking α δ (•) = α(•/k δ ).

Lemma 5.12. ξ δ is L ∞ -uniformly continuous and satisfies Condition (5.2) for ρ T δ .

Proof. For any ω δ , v δ ∈ Ω δ such that v δ ≤ ω δ , one has ξ δ (ω δ ) -ξ δ (v δ ) = ξ(ω δ ) -ξ(v δ ) ≤ β ρ T (ω δ , vδ ) , ξ δ (ω δ ) -ξ δ (v δ ) = ξ(ω δ ) -ξ(v δ ) .

It is thus enough to show that

ρ [t i

Theorem 4 . 1 (

 41 Fenchel-Moreau). Let (E, Σ) be a Hausdorff locally convex space and F : E → R be a concave and upper semicontinuous function. Then F is equal to its biconjugate F * * which is defined by F * * (e) := inf e * ∈E * e, e * + sup e ′ ∈E F (e ′ ) -e ′ , e * and E * denotes the dual space of E.

Proposition 4 . 4 .D 1 (

 441 Under the conditions of Theorem 3.10 (ii) one has

Proposition 4 . 8 .

 48 Let ζ be a measurable and bounded function, then one has sup P∈M +

5 ) 5 . 4 .

 554 where ωδ ∈ Ω is defined by ωδ t := ω δ k δ t for all t ∈ [0, 1]. (5.Proposition Assume that ζ satisfies the Assumptions 3.8, 3.9 and Condition (5.2). Then: (i) For all 0 ≤ δ < ∆T, the ζ δ defined by (5.5) satisfies the Assumptions 3.8, 3.9 and Condition (5.

Lemma 5 . 6 .

 56 Suppose that ζ satisfies Assumption 3.8, 3.9 and Condition (5.2). Then lim inf δ↓0 D c δ ≥ D(ζ) and lim sup δ↓0 P δ ≤ P(ζ).

1 0≥

 1 |ω t |dt for all ω ∈ Ω + , which yields D c δ + 1 + (m + 2)d η δ 1+δ

  required result since δ 0 > 0 is arbitrary. Proof of Theorem 5.3. Let (z, H) ∈ D(ζ), we know by definition z + (H • ω) 1 ≥ ζ(ω), ∀ω ∈ Ω + . Taking expectation over each sides, it follows that z ≥ E P [ζ(X)] for all P ∈ M + . Then we get the weak duality P(ζ) ≤ D(ζ). The reverse inequality follows by Lemmas 5.5 and 5.6.

9 . 5 . 8 .

 958 are the jumps times of the piecewise constant process X(ω) with τ0 (ω) = 0 and τKm+m (ω) = 1. Then the required result D c (ζ) ≤ P(ζ) follows from the following Propositions 5.8 and 5.Proposition Assume that ζ satisfies Assumptions 3.8, 3.9 and is L ∞ -uniformly continuous, then lim sup n→∞ D (n) (ζ) ≤ P(ζ).

. 10 ) 5 . 9 .

 1059 Proposition Assume that ζ satisfies Condition (5.2), then one has lim inf n→∞ D (n) (ζ) ≥ D c (ζ).

Lemma 5 . 11 .

 511 ω δ,n k t -→ ω δ,0t for all t ∈ [0, 1 + δ]\T , which yields another countable setT ′ ⊆ [0, 1) such that ωδ,n k t -→ ωδ,0 t for all t ∈ [0, 1]\T ′ .Hence one has ωδ,0 = ω 0 and thus ωδ,n k S * -→ ωδ,0 and ωδ,nk t i -→ ωδ,0 t i for all i = 0, • • • , m -1, which implies that lim k→∞ ξ δ (ω δ,n k ) = lim k→∞ ξ(ω δ,n k ) ≤ ξ(ω δ,0 ) = ξ δ (ω δ,0). There exists a continuous functionα δ : R + → R + with α δ (0) = 0 such that for all ε = (ε 1 , • • • , ε m ) ∈ R m+ sufficiently small one hasξ δ (ω δ ) -ξ δ ω δ f δ ε , ξ δ (ω δ ) -ξ δ ω ε , b δ ε : [0, 1 + δ] → [0, 1 + δ]are two non-decreasing functions defined as in (3.4) and (3.5). Proof. We only prove the inequality on f δ ε , while the inequality on b δ ε follows by the same arguments. Define f δ ε : [0, 1 + δ] → [0, 1 + δ] by f δ ε (t) := 1 k δ f δ ε (k δ t) for all t ∈ [0, 1].

ωδ f δε

  (t) = ω δ k δ f δ ε (t) for all t ∈ [0, 1]and thus ωδf δ ε = ωδ • f δ ε . Hence ξ δ (ω δ ) -ξ δ ωδ f δ ε = ξ(ω δ ) -ξ ωδ • f δ ε ≤ α(|ε|/k δ ) 1 +

α U λ 1 α

 1 ω δ γ δ (t) -v δ t 3.2.3 of Deuschel & Stroock[START_REF] Deuschel | Large Deviations[END_REF]. Notice that the topology on M is generated by all the following open ballsU λ 1 ,••• ,λ m ,c (µ) := ν ∈ M : µ(λ i ) -ν(λ i ) < c for all 1 ≤ i ≤ m ,whereλ i ∈ Λ for 1 ≤ i ≤ mand c > 0. Let O be the collection of open sets generated by the open balls above, then clearly, every open set U ∈ O could be expressed asU = ,••• ,λ nα α ,c α (µ α ) with λ ∈ Λ for 1 ≤ 1 ≤ n α , n α ∈ N and c α > 0.Theorem A.2. The space (M, O) is a Hausdorff locally convex space, whose dual space can be identified by M * = Λ.Proof. (i) First, (M, O) is clearly a topological vector space. For every µ ∈ M, letU (µ) := U λ 1 α ,••• ,λ nα α ,c α (µ) : λ i α ∈ Λ 1 for 1 ≤ i ≤ n α , n α ∈ N and c α > 0 .By definition, one can check that U (µ) is a local basis of µ for every µ ∈ M. Moreover, by denoting 0 ∈ M the null measure, U (0) is a local basis of absolutely convex absorbent sets and thus M is a locally convex space.(ii) Now, let us identify the dual space of M. First, for every λ ∈ Λ, the map F λ : M → R defined by F λ (µ) := µ(λ) gives a unique element in M, and hence Λ ⊆ M * . On the other hand, for any F ∈ M * , we define a function λ F by λ F (x) := F (δ {x} ) for all x ∈ R d .Clearly one has the following implicationx n → x 0 =⇒ δ {xn} O -→ δ {x 0 } =⇒ λ F (x n ) → λ F (x 0 ), which implies that λ F is continuous. It follows that the set F -1 (-1, 1) is open and thus there exists someU λ 1 ,••• ,λ m ,c (0) such that U λ 1 ,••• ,λ m ,c (0) ⊆ F -1 (-1, 1),where λ i ∈ Λ for all i = 1, • • • , m and c > 0. Now for any µ ∈ M such that m i=1 µ(λ i )| > 0, we define μ := cµ m i=1 µ(λ i ) . Then μ ∈ U λ 1 ,••• ,λ m ,c (0) and thus |F (μ)| < 1. It follows that |F (µ)| ≤ m i=1 µ(λ i )| for all µ ∈ M,and hence λ F ∈ Λ. When µ is a linear combination of Dirac measures, it is obvious that F (µ) = µ(λ F ). Moreover, since such µ are dense in M, it follows that F (µ) = µ(λ F ) holds for all µ ∈ M.

  An F-adapted process H : [0, 1] × Ω + → R d is called a dynamic strategy if t → H t (ω)is left-continuous and of bounded variation for every ω ∈ Ω + and (H • X) is a supermartingale under every P ∈ M + . Let A be the set of all dynamic strategies and define the set of robust super-replications

	Definition 2.5.
		.7)
	where	t 0 X u • dH u refers to the scalar Lebesgue-Stieltjes integration.

  Let Γ [s,t] denotes the collection of strictly increasing continuous functions γ defined on [s, t] such that γ(s) = s and γ(t) = t. For anyγ δ ∈ Γ [t δ i-1 ,t δ i ] , define γ ∈ Γ [t i-1 ,t i ] by γ(t) := 1 k δ γ δ k δ t for all t ∈ [t i-1 , t i ].

		0	1	ωδ t -vδ t dt ≤	0	1+δ	ω δ t -v δ t dt .
	Hence
	sup t i-1 ≤t≤t i	ωδ γ(t) -vδ

-1 ,t i ] (ω δ , vδ ) ≤ ρ [t δ i-1 ,t δ i ] (ω δ , v δ ) for all i = 1, • • • , m and t = sup t i-1 ≤t≤t i ω δ k δ γ(t) -v δ k δ t = sup t i-1 ≤t≤t i ω δ γ δ ((k δ t) -v δ k δ t = sup t δ i-1 ≤t≤t δ i

We gratefully acknowledge the financial support of the ERC 321111 Rofirm, the ANR Isotace, and the Chairs Financial Risks (Risk Foundation, sponsored by Société Générale) and Finance and Sustainable Development (IEF sponsored by EDF and CA).

and sup

which implies that

). We may thus conclude by

A Appendix

A.1 Tightness under S-topology

is the Skorokhod space of càdlàg paths on [0, 1], with canonical process X = (X t ) 0≤t≤1 and canonical filtration F = (F t ) 0≤t≤1 ; and P denotes the set of all probability measures on (Ω, F 1 ). The following result is recalled from Jakubowski [START_REF] Jakubowski | A non-Skorohod topology on the Skorohod space[END_REF] and shows that under S-topology, both Skorokhod representation theorem and Prohorov's theorem hold true.

Theorem A.1 (Jakubowski). (i)Let (P n ) n≥1 ⊆ P be a sequence of probability measures such that X is a P n -supermartingale for all n ≥ 1, then

(ii) Let (P n ) n≥1 ⊆ P be a S-tight sequence of probability measures. Then there exist a subsequence (P n k ) k≥1 , a probability measure P ∈ P and a countable subset T ⊂ [0, 1) such that for all finite sets

In particular, X n k * =⇒ D X 0 as k → ∞.

A.2 Dual space of M

Recall that M denotes the space of all finite signed measures µ on R d satisfying

and it is equipped with the topology induced by the convergence W 1 -→. We would like identity its dual space as Λ, where the arguments are mainly adapted from Lemma