Pierre Henry-Labordère

Xiaolu Tan
email: tan@ceremade.dauphine.fr

Nizar Touzi
email: nizar.touzi@polytechnique.edu

An Explicit Martingale Version of the One-dimensional Brenier's Theorem with Full Marginals Constraint *

.

, we also provide a duality result together with the corresponding dual optimizer in explicit form.

As an application to financial mathematics, our results give the model-independent optimal lower and upper bounds for variance swaps.

Introduction

The classical optimal transport (OT) problem was initially formulated by Monge in his treatise "Théorie des déblais et des remblais" as follows. Let µ 0 , µ 1 be two probability measures on R d , c : R d × R d → R be a cost function, then the optimal transport problem consists in minimizing the cost R d c(x, T (x))µ 0 (dx) among all transference plans, i.e. all measurable functions T : R d → R d such that µ 1 = µ 0 • T -1 . The relaxed formulation of the problem, as introduced by Kantorovich, consists in minimizing the value E P [c(X 0 , X 1 )] among all probability measures P such that P • X -1 0 = µ 0 and P • X -1 1 = µ 1 . Under the so-called Spence-Mirrlees or Twist condition, the optimal Monge transference plan is characterized by the Brenier Theorem, and explicitly given by the Fréchet-Hoefding in the one-dimensioanl setting. We refer to Rachev & Ruschendorf [START_REF] Rachev | Mass Transportation Problems[END_REF] and Villani [START_REF] Villani | Optimal Transport: Old and New[END_REF] for a detailed presentation.

The theory has been extended to the multiple marginals case by Gangbo & Świȩch [START_REF] Gangbo | Optimal maps for the multidimensional Monge-Kantorovich problem[END_REF], Carlier [START_REF] Carlier | On a class of multidimensional optimal transportation problems[END_REF], Olkin & Rachev [START_REF] Olkin | Maximum submatrix traces for positive definite matrices[END_REF], Knott & Smith [START_REF] Knott | On a generalization of cyclic monotonicity and distances among random vectors[END_REF], Rüschendorf & Uckelmann [START_REF] Ruschendorf | On the n-coupling problem[END_REF], Heinich [START_REF] Heinich | Problème de Monge pour n probabilités[END_REF], and Pass [START_REF] Pass | Uniqueness and the Monge solutions in the multi marginal optimal transport problem[END_REF][START_REF] Pass | On the local structure of optimal measures in the multi-marginal optimal transportation problem[END_REF][START_REF] Pass | Multi-marginal optimal transport and multi-agent matching problems: uniqueness and structure of solutions[END_REF], etc. We also refer to the full-marginals case addressed by Pass [START_REF] Pass | On a class of optimal transportation problems with infinitely many marginals[END_REF].

Recently, a martingale transportation (MT) problem was introduced in Beiglböck, Henry-Labordère & Penkner [START_REF] Beiglböck | Model-independent Bounds for Option Prices: A Mass-Transport Approach[END_REF] and in Galichon, Henry-Labordère & Touzi [START_REF] Galichon | A stochastic control approach to no-arbitrage bounds given marginals, with an application to Lookback options[END_REF]. Given two probability measures µ 0 and µ 1 , the problem consists in minimizing some expected cost among all a class of cost/reward functions. Two natural questions can then be addressed. The first is whether the discrete binomial process converges when the time step converges to zero, and the second is whether such a limiting continuous-time process is optimal for a corresponding MT problem with full marginals.

Given a continuous family of marginal distributions which is non-decreasing in convex ordering, a stochastic process calibrated to all the marginals is called a peacock (or PCOC "Processus Croissant pour l'Ordre Convexe" in French) in Hirsch, Profeta, Roynette & Yor [START_REF] Hirsch | Peacocks and Associated Martingales, with Explicit Constructions[END_REF]. It follows by Kellerer's theorem that a process is a peacock if and only if there is a martingale with the same marginal distributions at each time, it is then interesting to construct such martingales associated with a given peacock (or equivalently with a given family of marginal distributions). In particular, when the marginal distributions are given by those of a Brownian motion, such a martingale is called a fake Brownian motion. Some examples of martingale peacock (or fake Brownian motion) have been provided by Albin [START_REF] Albin | A continuous non-Brownian motion martingale with Brownian motion martingale distributions[END_REF], Fan, Hamza & Klebaner [START_REF] Fan | Mimicking self-similar processes[END_REF], Hamza & Klebaner [START_REF] Hamza | A family of non-Gaussian martingales with Gaussian marginals[END_REF], Hirsch et al. [START_REF] Hirsch | Constructing self-similar martingales via two Skorokhod embeddings[END_REF], Hobson [START_REF] Hobson | Fake Exponential Brownian Motion[END_REF], Oleszkiewicz [START_REF] Oleszkiewicz | On fake Brownian motions[END_REF], Pagès [START_REF] Pagès | A functional co-monotony principle with an application to peacocks and barrier options[END_REF].

Our procedure gives a new construction of martingales associated with some peacock processes, under technical conditions (see Assumption 3.1 below), and in particular a discontinuous fake Brownian motion. Moreover, assuming that the difference of c.d.f. has exactly one local maximizer and some additional conditions (see Assumption 3.4 below), our constructed martingale is optimal among all martingales with given marginal distributions for a class of cost/reward functions, i.e. it solves a martingale transportation problem. We would like to observe that our procedure depends on the explicit construction of the left-monotone martingale transference plan in [START_REF] Henry-Labordère | An Explicit Martingale Version of Brenier's Theorem[END_REF] under restrictive conditions. This is in contrast with the recent parallel work of Juillet [START_REF] Juillet | Martingales associated to peacocks using the curtain coupling[END_REF], where the convergence of the left-monotone martingale transference plan is investigated under general conditions and/or in various specific situations.

This paper is organized as follows. In Section 2, we recall the discrete-time martingale transport problem under finitely many marginal constraints. Taking the limit as the time step goes to 0, this leads naturally to a continuous-time MT problem under full marginals constraints. which is solved in Section 3 under technical conditions. Namely, by taking the limit of the optimal martingale measure for the multi-marginals MT problem, we obtain a continuous-time martingale calibrated to the given marginals, or equivalently, a martingale associated with the corresponding peacock. Under additional conditions, including the unique local maximizer of the c.d.f. difference, we prove that this limit martingale is a local Lévy process and solves the infinitely-many marginals MT problem for a class of cost/reward functions. In particular, we provide an explicit characterization of this optimal solution as well as the dual optimizer. In Section 4, we discuss some examples of extremal peacock following our construction, including a discontinuous fake Brownian motion and a self-similar martingale. As an application in finance, we provide an optimal robust hedging strategy for the variance swap option in Section 5. Finally, we complete the proofs of our main results in Section 6, where the main idea is to approximate the infinitely-many marginals case by the multi marginals case.

A martingale optimal transport problem

We first consider the martingale optimal transport problem in finite discrete time, and recall some results from [START_REF] Beiglböck | On a problem of optimal transport under marginal martingale constraints[END_REF][START_REF] Henry-Labordère | An Explicit Martingale Version of Brenier's Theorem[END_REF][START_REF] Beiglböck | Monotone Martingale Transport Plans and Skorohod Embedding[END_REF]. By taking the limit, it leads to a continuous-time martingale transport problem.

Discrete-time martingale transport

Let µ 0 , µ 1 be two atomless Borel probability measures on R with extreme left-point and rightpoint of the support -∞ ≤ l µi < r µi ≤ +∞. We assume that µ 0 , µ 1 have finite first moments and µ 0 µ 1 in the convex order, i.e. µ 0 (φ) ≤ µ 1 (φ) for every convex function φ, where µ i (φ) := R φ(x)µ i (dx) for i = 0, 1.1 Let P R 2 be the collection of of all Borel probability measures on R 2 . The corresponding canonical process is denoted by (X 0 , X 1 ), i.e. X i (x 0 , x 1 ) = x i for i = 0, 1 and (x 0 , x 1 ) ∈ R 2 . We consider the set M 2 (µ 0 , µ 1 ) :=

P ∈ P R 2 : X 0 ∼ P µ 0 , X 1 ∼ P µ 1 and E P [X 1 |X 0 ] = X 0
of all martingale measures with marginals µ 0 and µ 1 . Let c : R 2 → R be a reward function such that c(x, y) ≤ a(x) + b(y) for some a ∈ L 1 (µ 0 ) and b ∈ L 1 (µ 1 ), a two-marginals MT problem is defined by

P 2 (µ 0 , µ 1 ) := sup P∈M2(µ0,µ1) E P c(X 0 , X 1 ) . (2.1)
The dual formulation of the MT problem (2.1) turns out to be

D 2 (µ 0 , µ 1 ) := inf (ϕ,ψ,h)∈D2 {µ 0 (ϕ) + µ 1 (ψ)}, (2.2)
where the dual variables set is:

D 2 := (ϕ, ψ, h) : ϕ + ∈ L 1 (µ 0 ), ψ + ∈ L 1 (µ 1 ), h ∈ L 0 and ϕ ⊕ ψ + h ⊗ ≥ c , (2.3)
with notations ϕ + := ϕ∨0, ψ + := ψ∨0, (ϕ⊕ψ)(x, y) := ϕ(x)+ψ(y) and h ⊗ (x, y) := h(x)(y-x).

Under mild conditions, the existence of the optimal martingale as well as the strong duality (i.e. P 2 (µ 0 , µ 1 ) = D 2 (µ 0 , µ 1 )) are proved in Beiglböck, Henry-Labordère & Penkner [START_REF] Beiglböck | Model-independent Bounds for Option Prices: A Mass-Transport Approach[END_REF]. More recently, under additional conditions on the reward function c, the optimal martingale has been characterized as left-monotone transference map in [START_REF] Beiglböck | On a problem of optimal transport under marginal martingale constraints[END_REF], see more extensions in [START_REF] Henry-Labordère | An Explicit Martingale Version of Brenier's Theorem[END_REF][START_REF] Beiglböck | Monotone Martingale Transport Plans and Skorohod Embedding[END_REF]. Let us recall it in a simpler context. Theorem 2.1. Suppose that the partial derivative c xyy exists and c xyy > 0 on (l µ0 , r µ0 ) × (l µ1 , r µ1 ). Then the optimal martingale measure P * (dx, dy) of problem (2.1) is unique, and can be characterized by P * (dx, dy) = µ 0 (dx)T * (x, dy), where the probability kernel T * is of the form

T * (x, dy) := q(x)δ Tu(x) (dy) + (1 -q(x))δ T d (x) (dy), for some functions T u , T d : R → R satisfying T d (x) ≤ x ≤ T u (x) and q(x) ∈ [0, 1].
More precise properties of T d and T u have been obtained in [START_REF] Beiglböck | On a problem of optimal transport under marginal martingale constraints[END_REF][START_REF] Henry-Labordère | An Explicit Martingale Version of Brenier's Theorem[END_REF], which will be recalled and used in Section 6.

The above problem and results can be easily extended to the finitely-many marginals case, when the cost function is given by

n i=1 c(x i-1 , x i ). More precisely, with n + 1 given probability measures (µ 0 , • • • , µ n ) ∈ (P R ) n+1 such that µ 0 • • • µ n in the convex ordering, the problem consists in maximizing E P c(X 0 , • • • , X n ) = E P n i=1 c i (X i-1 , X i ) (2.4)
among all martingales probability measures on R n+1 satisfying the marginal distribution constraints (X i ∼ P µ i , i = 0, • • • , n). Under the same technical condition on c, the optimal martingale measure is obtained by a Markovian iteration of the above left-monotone transference map, thus inducing a binomial tree.

Continuous-time martingale transport under full marginals constraints

We now introduce a continuous-time martingale transportation (MT) problem under full marginals constraints, as the limit of the multi-marginals MT recalled in Section 2.1 above. Namely, given a family of probability measures µ = (µ t ) t∈[0,1] , we consider all continuous-time martingales satisfying the marginal constraints, and optimize w.r.t. a class of reward functions.

To avoid the problem of integration, we define, for every random variable ξ, the expectation

E[ξ] := E[ξ + ] -E[ξ -] with the convention ∞ -∞ = -∞.
Let Ω := D([0, 1], R) denote the canonical space of all càdlàg paths on [0, 1], X the canonical process and F = (F t ) 0≤t≤1 the canonical filtration generated by X, i.e. F t := σ{X s , 0 ≤ s ≤ t}. We denote by M ∞ the collection of all martingale measures on Ω, i.e. the collection of all probability measures on Ω under which the canonical process X is a martingale. The set M ∞ is equipped with the weak convergence topology throughout the paper. By Karandikar [START_REF] Karandikar | On pathwise stochastic integration[END_REF], there is a non-decreasing process ([X] t ) t∈[0,1] defined on Ω which coincides with the Pquadratic variation of X, P-a.s. for every martingale measure P ∈ M ∞ . Denote also by [X] c

• the continuous part of the non-decreasing process [X].

Given a family of probability measures µ = (µ t ) 0≤t≤1 , denote by M ∞ (µ) ⊂ M ∞ the collection of all martingale measures on Ω such that X t ∼ P µ t for all t ∈ [0, 1]. In particular, following Kellerer [START_REF] Kellerer | Markov-Komposition und eine anwendung auf martingale[END_REF] (see also Hirsch and Roynette [START_REF] Hirsch | A new proof of Kellerer's theorem[END_REF]), M ∞ (µ) is nonempty if and only if the family (µ t ) 0≤t≤1 admits a finite first order moment, is non-decreasing in convex ordering, and t → µ t is right-continuous.

Finally, for all t ∈ [0, 1], we denote by -∞ ≤ l t ≤ r t ≤ ∞ the left and right extreme boundaries of the support 2 of µ t . Similar to Hobson & Klimmek [START_REF] Hobson | Model independent hedging strategies for variance swaps[END_REF], our continuous-time MT problem is obtained as a continuous-time limit of the multi-marginals MT problem, by considering the limit of the reward function n i=1 c(x ti-1 , x ti ) as in (2.4), where (t i ) 1≤i≤n is a partition of the interval [0, 1] with mesh size vanishing to zero. For this purpose, we formulate the following assumption on the reward function.

Assumption 2.2. The reward function c : R 2 → R is in C 3 ((l 1 , r 1 ) × (l 1 , r 1 )) and satisfies c(x, x) = c y (x, x) = 0 and c xyy (x, y) > 0, ∀(x, y) ∈ (l 1 , r 1 ) × (l 1 , r 1 ).

In order to obtain the convergence, we need to use the pathwise Itô calculus introduced in Föllmer [START_REF] Föllmer | calcul d'Itô sans probabilités[END_REF], which is also used in Hobson & Klimmek [START_REF] Hobson | Model independent hedging strategies for variance swaps[END_REF] and Davis, Oblój & Raval [START_REF] Davis | Arbitrage Bounds for Weighted Variance Swap Prices[END_REF] (see in particular their Appendix B).

Definition 2.3 (Föllmer [30]). Let π n = (0 = t n 0 < • • • < t n n = 1), n ≥ 1 be partitions of [0, 1] with |π n | := max 1≤k≤n |t n k -t n k-1 | → 0 as n → ∞. A càdlàg path x : [0, 1] → R has a finite quadratic variation along (π n ) n≥1 if the sequence of measures on [0, 1], 1≤k≤n x t n k -x t n k-1 2 δ {t k-1 } (dt), converges weakly to a measure [x] F on [0, 1]. Denote [x] F t := [x] F ([0, t]), and let [x] F,c
• be the continuous part of this non-decreasing path.

The next result follows the same line of proof as in Lemma 7.4 of [START_REF] Hobson | Model independent hedging strategies for variance swaps[END_REF].

2 (i) The functions l and r are non-increasing and non-decreasing, respectively. We only report the justification for the right boundary of the support r, a similar argument applies to l. For 0 ≤ s ≤ t ≤ 1, it follows from the increase in convex order that for all constant c ∈ R, we have ∞ c (x -c) + µt(dx) ≥ ∞ c (x -c) + µs(dx), so that µs((c, ∞)) > 0 implies that µt((c, ∞)) > 0, and therefore r is non-decreasing. (ii) Assume that µt has a density function for all t ∈ [0, 1] and t → µt is continuous w.r.t the weak convergence topology, then the functions 1 (-∞,c) and 1 (c,∞) (x) are µt-a.s. continuous for all t ∈ [0, 1], and it follows that l and r are continuous.

Lemma 2.4. Let Assumption 2.2 hold true. Then for every path x ∈ Ω with finite quadratic variation [x] F along a sequence of partitions (π n ) n≥1 , we have

n-1 k=0 c(x t n k , x t n k+1 ) → 1 2 1 0 c yy (x t , x t )d[x] F,c t + 0≤t≤1 c(x t-, x t ).
We next introduce the non-decreasing process [x] as in Karandikar [START_REF] Karandikar | On pathwise stochastic integration[END_REF], which is defined for every x ∈ Ω, and coincides P-a.s. with the quadratic variation, and also with [x] F , for all

P ∈ M ∞ .
Motivated by the last convergence result, we introduce a reward function

C(x) := 1 2 1 0 c yy (x t , x t )d[x] c t + 0≤t≤1 c(x t-, x t ), for all x ∈ Ω,
where the integral and the sum are defined as the difference of the positive and negative parts, under the convention ∞ -∞ = -∞. We then formulate a continuous-time MT problem under full marginals constraints by [START_REF] Acciaio | A Model-free Version of the Fundamental Theorem of Asset Pricing and the Super-Replication Theorem[END_REF] with some locally bounded positive function K 1 . Since we have, P-a.s. for all P ∈ M ∞ , that s≤t (∆X s ) 2 < ∞, then we also have 0≤t≤1 c(X t-, X t ) < ∞.

P ∞ (µ) := sup P∈M∞(µ) E P C(X) . (2.5) Remark 2.5. Under the condition c(x, x) = c y (x, x) = 0 in Assumption 2.2, we have |c(x, x + h)| ≤ K 1 (x)h 2 for all h ∈ [-1,
Now, let us introduce the dual formulation of the above MT problem (2.5). We first introduce the class of admissible dynamic and static strategies. Denote by H 0 the class of all F-predictable and locally bounded processes H : [0, 1]×Ω → R, i.e. there is an increasing family of F-stopping times (τ n ) n≥1 taking value in [0, 1] ∪ {∞} such that the process H •∧τn is bounded for all n ≥ 1 and τ n → ∞ as n → ∞. Then for every H ∈ H 0 and under every martingale measure P ∈ M ∞ , one can define the integral, denoted by H • X, of H w.r.t. the martingale X (see e.g. Jacod & Shiryaev [START_REF] Jacod | Limit theorems for stochastic processes[END_REF] Chapter I.4). Define

H := H ∈ H 0 : H • X is a P-supermartingale for every P ∈ M ∞ .
For the static strategy, we denote by M ([0, 1]) the space of all finite signed measures on [0, 1] which is a Polish space under the weak convergence topology, and by Λ the class of all measurable maps λ : R → M ([0, 1]) which admit a representation λ(x, dt) = λ 0 (t, x)γ(dt) for some finite non-negative measure γ on [0, 1] and measurable function λ

0 : [0, 1] × R → R which is bounded on [0, 1] × K for all compact K of R. We then denote Λ(µ) := λ ∈ Λ : µ(|λ|) < ∞ , where µ(|λ|) := 1 0 R λ 0 (t, x) µ t (dx)γ(dt).
We also introduce a family of random measures δ X = (δ X t ) 0≤t≤1 on R, induced by the canonical process X, by δ X t (dx) := δ Xt (dx). In particular, we have

δ X (λ) = 1 0 λ(X t , dt) = 1 0 λ 0 (t, X t )γ(dt).
Then the collection of all superhedging strategies is given by

D ∞ (µ) := (H, λ) ∈ H × Λ(µ) : δ X (λ) + (H • X) 1 ≥ C(X • ), P -a.s., ∀P ∈ M ∞ ,
and our dual problem is defined by

D ∞ (µ) := inf (H,λ)∈D∞(µ)
µ(λ).

(2.6)

Main results

We first construct in Subsection 3.1 a martingale transport peacock corresponding to the fullmarginals (µ t ) t∈[0,1] under technical conditions. This is obtained in Proposition 3.2 as an accumulation point of a sequence (P n ) n of solutions of n-periods discrete martingale transport problems. In order to further characterize such a martingale peacock, we next restrict our analysis to the one-maximizer context of Assumption 3.4 below. Then, our second main result in Subsection 3.2 shows that the sequence (P n ) n converges to the distribution of a local Lévy process. Finally our third main result is reported in Subsection 3.3. We show that this limit indeed solves the continuous-time MT problem (2.5), and that the duality result holds with explicit characterization of the solution of the dual problem under Assumption 3.4 below.

A martingale transport plan under full marginals constraint

For every t ∈ [0, 1], we denote by F (t, •) the cumulative distribution function of the probability measure µ t on R, and F -1 (t, •) the corresponding right-continuous inverse with respect to the x-variable. We also denote for t ∈ [0, 1), ε ∈ (0, 1 -t]:

δ ε F (t, x) := F (t + ε, x) -F (t, x), g ε t (x, y) := F -1 t + ε, F (t, x) + δ ε F (t, y) , (3.1)
and f (t, x) > 0 for all K > 0.

E := {(t, x) : t ∈ [0, 1], x ∈ (l t ,
(iii) For every t ∈ [0, 1) and ε ∈ (0, 1 -t], the set of local maximizers M δ ε F (t, x) of function x → δ ε F (t, x) is finite; and moreover, the following irreducibility condition3 holds true,

∞ x (ξ -x)δ ε F (dξ) = - ∞ x δ ε F (t, y)dy > 0, for all x ∈ (l t , r t ). (3.2)
Under the convex ordering condition, by considering the two marginals (µ t , µ t+ε ), the corresponding unique left-monotone martingale measure can be characterized by two functions T ε u (t, x) and T ε d (t, x) as in Theorem 2.1. We recall that Ω := D([0, 1], R) is the canonical space of càdlàg paths, which is a Polish space (separable, complete metric space) equipped with the Skorokhod topology; and X is the canonical process. Let (π n ) n≥1 be a sequence, where every

π n = (t n k ) 0≤k≤n is a partition of the interval [0, 1], i.e. 0 = t n 0 < • • • < t n n = 1. Suppose in addition that |π n | := max 1≤k≤n (t n k - t n k-1 ) → 0.
Then for every partition π n , by considering the marginal distributions (µ t n k ) 0≤k≤n , one obtains an (n + 1)-marginals MT problem, which consists in maximizing

E 0≤k≤n-1 c( Xn k , Xn k+1 )
among all discrete-time martingales Xn = ( Xn k ) 0≤k≤n satisfying the marginal distribution constraints. Under Assumptions 2.2 and 3.1, the left-monotone transference plan denoted by P * ,n is a solution of the last martingale transport problem. Let Ω * ,n := R n+1 be the canonical space of discrete-time process, X n = (X n k ) 0≤k≤n be the canonical process. Then under the optimal martingale measure P * ,n , X n is a discrete-time martingale and at the same time a Markov chain, characterized by T

∆t n k+1 u (t n k , •) and T ∆t n k+1 d (t n k , •) with ∆t n k+1 := t n k+1 -t n k , induced by the two marginals (µ t n k , µ t n k+1
) as in Theorem 2.1. We then extend the Markov chain X n to a continuous-time càdlàg process X * ,n = (X * ,n t ) 0≤t≤1 defined by

X * ,n t := n k=1 X n k-1 1 [t n k-1 ,t n k ) (t), t ∈ [0, 1],
and define the probability measure P n := P * ,n • (X * ,n ) -1 on Ω.

Proposition 3.2. Under Assumption 3.1, the sequence P n n≥1 is tight w.r.t. the Skorokhod topology on Ω. Moreover, every limit point P 0 satisfies P 0 ∈ M ∞ (µ).

Remark 3.3. In a recent work, Juillet [START_REF] Juillet | Martingales associated to peacocks using the curtain coupling[END_REF] analyzes the convergence of the left-monotone martingale transport plan under more general conditions and/or in some other specific context.

A Local Lévy process characterization

We next seek for a further characterization of the limiting peacocks obtained in Proposition 3.2. The remaining part of our results is established under the following unique local maximum condition. Let t ∈ [0, 1) and ε ∈ (0, 1 -t], we denote by

M δ ε F (t, •) (resp. M ∂ t F (t, •) ) the set of all local maximizer of function δ ε F (t, •) (resp. ∂ t F (t, •)). Assumption 3.4. (i) There is a constant ε 0 > 0 such that, for all t ∈ [0, 1] and 0 < ε ≤ ε 0 ∧ (1 -t), we have M δ ε F (t, •) = {m ε (t)} and M ∂ t F (t, •) = {m t }. (ii) Denote m 0 (t) := m t , then (t, ε) → m ε (t) is continuous (hence uniformly continuous with continuity modulus ρ 0 ) on {(t, ε) : 0 ≤ ε ≤ ε 0 , 0 ≤ t ≤ 1 -ε}. (iii) For every t ∈ [0, 1], we have ∂ tx f (t, m t ) < 0.
Example 3.13 below provides some concrete peacocks satisfying the last conditions. Remark 3.5. Under the one local maximizer condition in Assumption 3.4 (i), the irreducibility condition (3.2) holds true. Indeed, assume that (3.2) is not true, the for some t ∈ [0, t), ε ∈ (0, 1 -t], and some K ∈ (l t , r t ), one has

∞ K (ξ -x)δ ε F (dξ) = 0 = K -∞ (x -ξ)δ ε F (dξ).
It follows by Theorem 8.4 of Beiglböck and Juillet [START_REF] Beiglböck | On a problem of optimal transport under marginal martingale constraints[END_REF], we have the decomposition

µ t = ρµ 1 t + (1 -ρ)µ 2 t and µ t+ε = ρµ 1 t+ε + (1 -ρ)µ 2
t+ε for some ρ ∈ (0, 1), where µ 1 t and µ 1 t+ε (resp. µ 2 t and µ 2 t+ε ) are probability measures supported in (-∞, K) (resp. [K, ∞)), and µ i t µ i t+ε in convex ordering for i = 1, 2. Notice that the corresponding differences of c.d.f. δ ε F i satisfies

δ ε F = ρ1 (-∞,K) δ ε F 1 + (1 -ρ)1 (K,∞) δ ε F 2 . Since µ i t µ i t+ε in convex ordering for i = 1, 2, δ ε F 1 (x) (resp. δ ε F 2 (x)) has at least one local maximizer in (-∞, K) ∩ (l t , r t ) (resp. (K, ∞) ∩ (l t , r t )). It follows that δ ε F has at least two local maximizer on (l t , r t ), which is a contradiction to Assumption 3.4 (i).
Recall that the left-monotone transference plan can be characterized by (T ε u , T ε d ), whose explicit construction are provided by [START_REF] Henry-Labordère | An Explicit Martingale Version of Brenier's Theorem[END_REF]. Direct calculation suggests that the sequence (T ε d ) ε is expected to converge to the function T d obtained via the limiting equation

x T d (t,x) (x -ξ)∂ t f (t, ξ)dξ = 0. (3.3)
We also define j d (t, x) and j u (t, x) by

j d (t, x) := x -T d (t, x) and j u (t, x) := ∂ t F t, T d (t, x) -∂ t F (t, x) f (t, x) , t ∈ [0, 1], x > m t . (3.4)
We notice that j u (t, •) and j d (t, •) are both positive and continuous on D c (t) := (m t , r t ). Let us also introduce D(t) := l t , m t and

D c := {(t, x) : t ∈ [0, 1], x ∈ (m t , r t )} and D := (t, x) : t ∈ [0, 1], x ∈ D(t) . (3.5)
Lemma 3.6. Let Assumptions 3.1 and 3.4 hold true. Then for all x ∈ (m t , r t ), the equation

(3.3) has a unique solution T d (t, x) ∈ (l t , m t ). Moreover (i) T d is strictly decreasing in x (ii) j d (t, x)1 x>mt , j u (t,
x)1 x>mt and ju j d (t, x)1 x>mt are all locally Lipschitz in (t, x). Our second main result is the following convergence result for the sequence (P n ) n with an explicit characterization of the limit as the law of a local Lévy process.

Theorem 3.7. Suppose that Assumptions 3.1 and 3.4 hold true, then P n → P 0 , where P 0 is the unique weak solution of the SDE

X t = X 0 - t 0 1 {Xs->m(s)} j d (s, X s-)(dN s -ν s ds), ν s := j u j d (s, X s-)1 Xs->m(s) , (3.6)
and (N s ) 0≤s≤1 is a unit size jump process with predictable compensated process (ν s ) 0≤s≤1 .

The pure jump process (3.6) is in the spirit of the local Lévy models introduced in Carr, Geman, Madan & Yor [START_REF] Carr | From local volatility to local Lévy models[END_REF]. Notice however that the intensity process (ν t ) 0≤t≤1 in our context is state-dependent. We conclude this subsection by providing a point of view from the perspective of the forward Kolmogorov-Fokker-Planck (KFP) equation. Lemma 3.8. Under Assumptions 3.1 and 3.4, the density function f (t, x) satisfies

∂ t f (t, x) = -1 {x<mt} j u f j d (1 -∂ x j d ) t, T -1 d (t, x) -1 {x>mt} j u f j d -∂ x (j u f ) (t, x), (3.7)
for all t ∈ [0, 1) and x ∈ (l t , r t ) \ {m t }.

The first order PDE (3.7) can be viewed as a KFP forward equation of SDE (3.6).

Proposition 3.9. Let Assumptions 3.1 and 3.4 hold true. Suppose that the SDE (3.6) has a weak solution X which is a martingale whose marginal distribution admits a density function

f X (t, x) ∈ C 1 ([0, 1] × R)
. Suppose in addition that E X 1 p < ∞ for some p > 1, and for every t ∈ [0, 1), there is some ε 0 ∈ (0, 1 -t] such that

E t+ε0 t j u (s, X s )1 Xs∈D c (s) ds < ∞. (3.8)
Then, the density function f X of X defined in (3.6) satisfies the KFP equation (3.7)

Optimality of the local Lévy process

The optimality results of this subsection are also obtained in the one-maximizer context of Assumption 3.4. We first introduce the candidates of the optimal dual components for the dual problem (2.6). Following Section 5 of [START_REF] Henry-Labordère | An Explicit Martingale Version of Brenier's Theorem[END_REF], (see also the recalling in Section 6.1.1 below), the optimal superhedging strategy (ϕ ε , ψ ε , h ε ) for the two marginals MT problem associated with initial distribution µ t , terminal distribution µ t+ε , and reward function c : (l 1 , r 1 ) × (l 1 , r 1 ) → R, is explicitly given by:

∂ x h ε (t, x) := c x (x, T ε u (t, x)) -c x (x, T ε d (t, x)) T ε u (t, x) -T ε d (t, x) , x ≥ m ε (t), h ε (t, x) := h ε t, (T ε d ) -1 (t, x) -c y (T ε d ) -1 (t, x), x , x < m ε (t); denoting (T ε ) -1 (t, x) := (T ε u ) -1 (t, x)1 x≥m ε (t) + (T ε d ) -1 (t, x)1 x<m ε (t) , the function ψ ε is given by ∂ x ψ ε (t, x) = c y (T ε ) -1 (t, x), x -h ε t, (T ε ) -1 (t, x) ,
and

ϕ ε (t, x) := x -T ε d (t, x) T ε u (t, x) -T ε d (t, x) c x, T ε u (t, x) -ψ ε t, T ε u (t, x) + T ε u (t, x) -x T ε u (t, x) -T ε d (t, x) c x, T ε d (t, x) -ψ ε t, T ε d (t, x) .
Clearly, h ε and ψ ε are unique up to a constant. More importantly, h ε and ψ ε can be chosen continuous on D so that

c(•, T ε u (t, •)) -ψ ε (T ε u (•)) -c(•, T ε d (•)) + ψ ε (T ε d (•)) -T ε u (•) -T ε d (•) h ε (•)
is continuous, since T ε u and T ε d are both continuous under Assumption 3.4. We shall see later that Assumption 2.2 on the reward function c implies that the continuoustime limit of the optimal dual components is given as follows. The function h * : D → R is defined, up to a constant, by

∂ x h * (t, x) := c x (x, x) -c x (x, T d (t, x)) j d (t, x) , when x ≥ m t , (3.9)
h * (t, x) := h * t, T -1 d (t, x) -c y T -1 d (t, x), x , when x < m t , (3.10)
and ψ * : D → R is defined, up to a constant, by

∂ x ψ * (t, x) := -h * (t, x), for all (t, x) ∈ D. (3.11)
As an immediate consequence of Lemma 3.6 (ii), we obtain the following regularity results

Corollary 3.10. Under Assumptions 2.2, 3.1, and 3.4, we have ψ * ∈ C 1,1 (D).

Proof. Clearly, h * is continuous in (t, x) for x = m t , since the function T d (t, x)1 x≥mt is continuous. We also easily check the continuity of h * at the point (t, m t ) by (3.9), since T -1 d (t, x) → m t as x → m t and c y (x, x) = 0. By its definition in (3.11), ∂ x ψ * is also continuous. We next compute, by (3.9), that

∂ t,x h * (t, x) = ∂ t j d (t, x) c xy x, T d (t, x) j d (t, x) -c x (x, x) -c x (x, T d (t, x)) j d (t, x) 2 ,
which is also locally bounded on D c since T d (t, x) is locally bounded by Lemma 3.6. Then the function

∂ t ψ * (t, x) = C - x 0 ∂ t h * (t, ξ)dξ, for some fixed constant C, is also continuous in (t, x).
In order to introduce a dual static strategy in Λ, we let γ * (dt) := δ {0} (dt) + δ {1} (dt) + Leb(dt) be a finite measure on [0, 1], where Leb(dt) denotes the Lebesgue measure on [0, 1]; we define λ * 0 and λ *

0 by λ * 0 (0, x) := ψ * (0, x), λ * 0 (1, x) := ψ * (1, x), λ * 0 (0, x) := |ψ * (0, x)|, λ * 0 (1, x) := sup t∈[0,1] |ψ * (t, x)|; and for all (t, x) ∈ (0, 1) × R, λ * 0 := ∂ t ψ * + 1 D c ∂ x ψ * j u + ν[ψ * -ψ * (., T d ) + c(., T d )] , λ * 0 := ∂ t ψ * + 1 D c ∂ x ψ * j u + ν[ψ * -ψ * (., T d )] + 1 D c ν[|c(., T d )|],
where we recall that D c = {(t, x) : x > m t }. Finally, we denote λ * (x, dt) := λ * 0 (t, x)γ * (dt) and λ * (x, dt) := λ * 0 (t, x)γ * (dt).

We are now ready for our third main result which states the optimality of the local Lévy process (3.6), as well as that of the dual component introduced above. Similar to [START_REF] Henry-Labordère | An Explicit Martingale Version of Brenier's Theorem[END_REF] and [START_REF] Hobson | Model independent hedging strategies for variance swaps[END_REF], we obtain in addition a strong duality for the MT problem (2.5) and (2.6). Let H * be the F-predictable process on Ω defined by

H * t := h * (t, X t-), t ∈ [0, 1].
Theorem 3.11. Let Assumptions 2.2, 3.1 and 3.4 hold true, suppose in addition that µ(λ

* ) = 1 0 R λ * 0 (t, x)µ t (dx)γ * (dt) < ∞.
Then the martingale transport problem (2.5) is solved by the local Lévy process (3.6). Moreover, (H * , λ * ) ∈ D ∞ (µ) and we have the duality

E P 0 C(X • ) = P ∞ (µ) = D ∞ (µ) = µ(λ * ),
where the optimal value is given by

µ(λ * ) = 1 0 rt mt j u (t, x) j d (t, x) c x, x -j d (t, x) f (t, x) dx dt.
The proofs of Theorems 3.7 and 3.11 are reported later in Section 6, the main idea is to use the approximation technique, where we need in particular the continuity property of the characteristic functions in Lemma 3.10. Remark 3.12. By symmetry, we may consider the right monotone martingale transference plan as in Remark 3.14 of [START_REF] Henry-Labordère | An Explicit Martingale Version of Brenier's Theorem[END_REF]. This leads to an upward pure jump process with explicit characteristics, assuming that x → ∂ t F (t, x) has only one local minimizer mt . Indeed, let ju (t, x) := Tu (t, x) -x and jd (t, x)

:= ∂ t F (t, x) -∂ t F (t, Tu (t, x)) f (t, x) ,
where Tu (t, x) : (l t , mt ] → [ mt , r t ) is defined as the unique solution to

Tu(t,x) x (ξ -x)∂ t f (t, ξ)dξ = 0.
The limit process solves SDE:

dX t = 1 Xt-< mt ju (t, X t-)(d Ñt -νt dt) νt := jd ju (t, X t-)1 Xt-< mt , (3.12)
where ( Ñt ) 0≤t≤1 is an upward jump process with unit jump size and predictable compensated process (ν t ) 0≤t≤1 . Moreover, under Assumption 2.2 together with further technical conditions on µ, this martingale solves a corresponding minimization MT problem with optimal value

1 0 mt lt jd (t, x) ju (t, x) c x, x + ju (t, x) f (t, x) dx dt.
The corresponding dual optimizer is also explicitly given by Theorem 3.11 once ( jd , ju ) is substituted to (j d , j u ).

We conclude this section by some examples of peacocks satisfying Assumption 3.4.

Example 3.13. (i) Let F 0 : R → [0, 1] be a distribution function of random variable such that its density function f 0 (x) := F 0 (x) is strictly positive on R. Define F (t, x) := F 0 (x/t) and f (t, x) := 1 t f 0 (x/t), then the associated marginal distribution is clearly a peacock. Denote further f0 (x) := log f 0 (x). Assume that f 0 (x) > 0 for x ∈ (-∞, 0) and f 0 (x) < 0 for x ∈ (0, ∞), then for every t ∈ (0, 1) and ε ∈ (0, 1 -t), the map y → f0 (y) -f0 (y + εy) is strictly increasing on (-∞, 0) and strictly decreasing on (0, ∞). In this case, the set

{x : f (t, x) = f (t + ε, x)} = x : log(1 + ε/t) = f0 x t + ε -f0 x t + ε (1 + ε/t)
has exactly two points, and one can check that the smaller one is the maximizer m ε (t) of x → δ ε F (t, x), and the second one is the minimizer of x → δ ε F (t, x). Denote by m t the maximizer of x → ∂ t F (t, x), which is also is the smaller solution of

-f 0 x t x t = t ⇐⇒ ∂ t f (t, x) = 0, for a fixed t > 0. (3.13)
Assume in addition that that f 0 (m t ) = 0, then by the fact that

f0 x t + ε -f0 x t + ε (1 + ε/t) = -f 0 x t + ε x t + ε ε t + o(ε/t),
we can also prove the convergence of m ε (t) → m t . In this case, Assumption 3.4 holds true.

(ii) In particular, when the marginals (µ t ) t∈[δ,1+δ] are those of the Brownian motion for some δ > 0, then both Assumptions 3.1 and 3.4 hold true with

f (t, x) = 1 √ 2πt e -x 2 2t , m ε (t) = -t(t+ε) ε log(1 + ε/t) and m t = - √ t.
See also Section 4 for more discussions.

(iii) For general function f0 , it is clear that (3.13) may have more than two solutions, then Assumption 3.4 is no more true.

Examples of extremal peacock processes 4.1 A remarkable fake Brownian motion

In this subsection, we specialize the discussion to the special case µ t :=:= N (0, t), t ∈ [0, 1]. This is the first appearance of peacocks in the literature under the name of fake Brownian motion, see Albin [START_REF] Albin | A continuous non-Brownian motion martingale with Brownian motion martingale distributions[END_REF], Fan, Hamza & Klebaner [START_REF] Fan | Mimicking self-similar processes[END_REF], Hamza & Klebaner [START_REF] Hamza | A family of non-Gaussian martingales with Gaussian marginals[END_REF], Hirsch et al. [START_REF] Hirsch | Constructing self-similar martingales via two Skorokhod embeddings[END_REF],

Hobson [START_REF] Hobson | Fake Exponential Brownian Motion[END_REF], Oleszkiewicz [START_REF] Oleszkiewicz | On fake Brownian motions[END_REF], Pagès [START_REF] Pagès | A functional co-monotony principle with an application to peacocks and barrier options[END_REF].

Our construction in Theorem 3.7 provides a new example of fake Brownian motion which is remarkable by the corresponding optimality property.

It is easily checked that our Assumptions 3.1 and 3.4 are satisfied. By direct computation, we

have m ε (t) = -t(t+ε) ε log(1 + ε/t), m t = - √ t for all t ∈ [δ, 1]
, and it follows that T d (t, x) is defined by the equation:

x T d (t,x) (x -ξ)(ξ 2 -t)e -ξ 2 /2t dξ = 0 for all x ≥ m t .
Direct change of variables provides the scaled solution:

t 1/2 T d t -1/2 x := T d (t, x), x ≥ -1, (4.1)
where

T d (x) is the unique solution in (-∞, -1] of x T d (x) (x -ξ)(ξ 2 -1)
e -ξ 2 /2 dξ = 0, for all x ≥ -1. Equivalently, T d is characterized by:

T d (x) ≤ -1 and e -T d (x) 2 /2 1 + T d (x) 2 -x T d (x) = e -x 2 /2 ; x ≥ -1.
Similarly, we see that , where where the scaled function j is given by:

j u (t, x) := t -1/2 j u t -1/2 x (4.2) $ # ' # # '' $ + / (( Out[105]= T C d +y/ y T C u 2 1 1 2 3
j u (x) := 1 2 x -T d (x)e -( T d (x) 2 -x 2 )/2 = 1 2 x - T d (x) 1 + T d (x) 2 -x T d (x) ; x ≥ -1.
The maps T d (x) and T u (x) := x + j u (x) are plotted in Figure 1.

A new construction of self-similar martingales

In Hirsch, Profeta, Roynette & Yor [START_REF] Hirsch | Constructing self-similar martingales via two Skorokhod embeddings[END_REF], the authors construct martingales M t which enjoy the (inhomogeneous) Markov property and the Brownian scaling property M c 2 t , t ≥ 0 ∼ cM t , t ≥ 0 for all c > 0. When the marginals of M admit a density f (t, x), this property translates to:

f (t, x) = t -1/2 f 1, t -1/2 x , t > 0, x ∈ R.
In order to apply our construction result of Theorem 3.7, we assume that ∂ t F (t, x) has a unique maximizer which is given by m t = √ t m, where m is the smallest solution of

f (1, m) + mf x (1, m) = 0.
The scaling properties (4.1)-(4.2) apply again in this case, and we compute that

x T d (x) (x -ζ) f (1, ζ) + ζf x (1, ζ) dζ = 0, and j u (x) := 1 2 x - T d (x)f (1, T d (x)) f (1, x) ; x ≥ m.
5 Application: Robust sub/superhedging of variance swap

As a final application, we consider the reward function c 0 (x, y) := (ln x-ln y) 2 , which clearly satisfies Assumption 2.2. The so-called "variance swap" is defined by the payoff n-1 k=0 c 0 (X t k , X t k+1 ), and is usually approximated by the continuous-time limit of Lemma 2.4:

C 0 (X) := 1 0 d[X] c t X 2 t + 0<t≤1 ln 2 X t X t- .
We recall that, under any pathwise continuous martingale model P c ∈ M ∞ , the variance swap can be perfectly replicated by a semi-static hedging strategy. This follows from a direct application of Itô's formula:

C 0 (X) = 1 0 d[X] t X 2 t = 2 1 0 dX t X t -2 ln X 1 X 0 , P c -a.s.
However, since our set of admissible models M ∞ also allows for models with jumps, it turns out that the upper and lower optimal bounds for the variance swap define an interval containing ln(x/X 0 )µ 1 (dx) in its interior.

In the following statement, (µ t ) 0≤t≤1 is a non-decreasing family of marginals with support in (0, ∞), and satisfying Assumptions 3.1 and 3.4. The corresponding left-monotone and rightmonotone martingales are defined by the characteristics (m, j u , j d ) and ( m, ju , jd ), respectively. In addition, we assume that the constructed optimal static strategy λ * satisfies the integrability conditions in Theorem 3.11, Proposition 5.1. The optimal martingale measures are given by the local Lévy processes (3.6) and (3.12), and the corresponding optimal upper and lower bounds for the variance swap are given by We have compared these bounds to market values denoted VS mkt for the DAX index (2-Feb-2013) for different maturities (see Table 1). The market marginals (µ t ) 0≤t≤1 are induced from market prices C t (K) of call options, with strike K and maturity t, by µ t (dx) = C t (dx). 4 In Figure 2, we have plotted market marginals ∂ t F (t, K) = -∂ t ∂ K C(t, K) for different maturities t and checked that ∂ t F admits only one local maximizer.

The prices in Table 1 are quoted in volatility ×100. Note that for maturities less than 1.5 years, our upper bound is below the market price, highlighting an arbitrage opportunity. In practice, this arbitrage disappears if we include transaction costs for trading vanilla options with low/high strikes. Recall also that we have assumed that vanilla options with all maturities are traded. 4 Let σ BS t (K) be the Black-Scholes implied volatility of strike K and maturity t, and denote by V BS t (K) := tσ BS t (K) 2 the corresponding total variance. In our numerical experiments, we have used the following interpolation from the observed maturities t 1 , t 2 :

V BS t (K) 2 = t-t 1 t 2 -t 1 V BS t 2 (K) -V BS t 1 (K) + V BS t 1 (K).
This linear interpolation guarantees that C(t 1 , K) ≤ C(t, K) ≤ C(t 2 , K) for all K, and thus preserves the consistency of the interpolation with the no-arbitrage condition.

( 1.9

) (2) 3
2.9

3.9

4.9

K/X_0

4.9

5.9

6.9

7.9 8.9 9.9 6 Proofs

Figure 2: Market marginals ∂ t F (t, K) = ∂ t ∂ K C(t,

Tightness of the monotone martingale plan under the finitelymany maxima condition of the difference of cdf

In order to prove Proposition 3.2, we start by recalling in Subsection 6.1.1 the explicit characterization of the left-monotone transference plan, as derived in [START_REF] Henry-Labordère | An Explicit Martingale Version of Brenier's Theorem[END_REF], and we collect the required properties for our convergence results in Subsection 6.1.2.

Reminders on the left-monotone martingale transport plan

Let (µ 0 , µ 1 ) be a pair of scalar atomless probability measures with µ 0 µ 1 in convex order. For the purpose of the present paper, the corresponding cdf F 0 and F 1 are assumed to be continuous. We also assume that (µ 0 , µ 1 ) is irreducible which translates as a condition on the difference of cdf δF := F 1 -F 0 : ∞ x δF (y)dy < 0, for all x ∈ (l µ1 , r µ1 ), and we denote x 0 := inf x ∈ R : δF increasing on a right neighborhood of x . We further assume that δF has a finite number of local maximizers where:

M(δF ) := {m 0 1 , . . . , m 0 n }, l µ1 < m 0 1 < . . . < m 0 n < r µ1 . Denote by F -1
• A k := (x 0 , m k ] \ ∪ i<k T d ([m i , x i )) ∪ [m i , x i ) = (x 0 , m k ] \ ∪ i<k T d (x i ), x i ; • for x > m k , t A k (x, m k ) is the unique point in A k such that x -∞ F -1 1 (F 0 (ξ)) -ξ dF 0 (ξ) + t A k (x,m k ) -∞ 1 A k (ξ) g(x, ξ) -ξ dδF (ξ) = 0; (6.1)
• for k ≥ 2, define

x k := inf x > m k : g x, t A k (x, m k ) ≤ x and m k+1 := inf M(δF ) ∩ [x k , ∞) . Let D 0 := ∪ k≥0 (x k , m k+1 ]
d(δF • T d ) = -(1 -q)dF 0 , d(F 1 • T u ) = qdF 0 , (6.2)
on the continuity domain of T d , where

q(x) := 1 {Tu(x)-T d (x)=0} + x -T d (x) T u (x) -T d (x) 1 {Tu(x)-T d (x) =0} . (6.3)
Given the two functions T d and T u , we introduce a probability kernel T * from R to R:

T * (x, dy) := 1 D0 (x)δ x (dy) + 1 D c 0 (x) q(x)δ Tu(x) (dy) + (1 -q(x))δ T d (x) (dy) . (6.4)
We next introduce the optimal dual semi-static hedging strategy. The dynamic hedging function h * : R → R is defined up to a constant by:

h * (x) := c x (x, T u (x)) -c x (x, T d (x)) T u (x) -T d (x) , for x ∈ D c 0 , (6.5)
h * (x) := h * T -1 d (x) + c y (x, x) -c y T -1 d (x), x , for x ∈ D 0 .
The stating hedging functions ψ * and ϕ * are defined, also up to a constant, by

ψ * := c y T -1 u , • -h * • T -1 u on D c 0 , ψ * := c y T -1 d , • -h * • T -1 d on D 0 , ϕ * (x) := q(x) c(x, T u (x)) -ψ * (T u (x)) + (1 -q(x)) c(x, T d (x)) -ψ * (T d (x)) ; x ∈ (l µ0 , r µ0 ).
Finally, the free constants are fixed by requiring that: (ii) Assume in addition that the positive parts

c(•, T u (•)) -ψ * (T u (•)) -c(•, T d (•)) + ψ * (T d (•)) -T u (•) -T d (•) h * (•) is continuous.
ϕ + * ∈ L 1 (µ 0 ) and ψ + * ∈ L 1 (µ 1 ). Then (ϕ * , ψ * , h * ) ∈ D 2 (see (2.
3)) and it solves the dual problem (2.2); moreover, we have the duality

E P * c(X 0 , X 1 ) = P 2 (µ 0 , µ 1 ) = D 2 (µ 0 , µ 1 ) = µ 0 (ϕ * ) + µ 1 (ψ * ).
Remark 6.3. By symmetry, one can also consider the c.d.f. Fi (x) := 1-F i (-x), x ∈ R, i = 0, 1, and construct a right monotone martingale transference plan which solves the minimization transportation problem (see more discussions in Remark 5.2 of [START_REF] Henry-Labordère | An Explicit Martingale Version of Brenier's Theorem[END_REF]).

Asymptotic estimates of the left-monotone transference plan

Let t ∈ [0, 1) and ε ∈ (0, 1 -t], we consider the pair (µ t , µ t+ε ) of the two marginal measures. Under Assumption 3.1 (iii), an explicit construction of the left-monotone martingale measure corresponding to (µ t , µ t+ε ) can be obtained by following the procedure recalled in Section 6.1.1. We denote the corresponding characteristics by

x ε k (t), m ε k+1 (t) k≥0 , T ε u (t, •) and T ε d (t, •). Similarly, denote D ε (t) := ∪ k≥1 x ε k-1 (t), m ε k (t) , (D ε ) c (t) := (l t+ε , m t+ε ) \ D ε (t) and A ε k (t) := x ε 0 (t), m ε k (t) \ ∪ i<k T ε d (t, x ε i (t)), x ε i (t) .
Recall that δ ε F and g ε t are introduced in (3.1), and let us define

δ ε f (t, x) := f (t + ε, x) -f (t, x), t ∈ [0, 1), ε ∈ (0, 1 -t]. Then for every x ∈ m ε k (t), x ε k (t) , • the function T ε d (t, x) ∈ A ε k (t) is uniquely determined by x -∞ F -1 t + ε, F (t, ξ) -ξ f (t, ξ)dξ + T ε d (t,x) -∞ 1 A ε k (t) (ξ) g ε t (x, ξ) -ξ δ ε f (t, ξ)dξ = 0, (6.6)
• the function T ε u is given by T ε u (t, x) := g ε t x, T ε d (t, x) . Since the functions T ε d and T ε u are (piecewise) monotone, we may introduce (T ε d ) -1 and (T ε u ) -1 as their right-continuous inverse function in x denotes the of x → T ε d (t, x). We also introduce the jump size function by

J ε u (t, x) := T ε u (t, x) -x, J ε d (t, x) := x -T ε d (t, x).
and the probability of a downward jump by

q ε (t, x) := J ε u (t, x) J ε u (t, x) + J ε d (t, x) 1 {J ε u (t,x)>0} = T ε u (t, x) -x T ε u (t, x) -T ε d (t, x) 1 {T ε u (t,x)>x} .
Lemma 6.4. Let Assumption 3.1 hold true. Then for every K > 0, there is a constant C independent of (t, x, ε) such that

J ε u (t, x) + q ε (t, x) ≤ Cε, ∀x ∈ [-K, K] ∩ (l t , r t ).
Proof. Differentiating g ε t (defined below (6.6)), we have

∂ y g ε t (x, y) = δ ε f (t, y) f t + ε, g ε t (x, y)
.

Notice that |δ ε F (t, x)| + |δ ε f (t, x)| ≤ C 1 ε for some constant C 1 independent of (t, x, ε).
Then for ε > 0 small enough, the value of g ε t (x, y) is uniformly bounded for all t ∈ [0, 1] and all x ∈ [-K, K] ∩ (l t , r t ) and y ∈ R. Further, the density function satisfies inf x∈[-K, K]∩(lt,rt) f (t, x) > 0 for every K > 0 large enough, by Assumption 3.1, then it follows by the definition of T ε u below (6.6) that

q ε (t, x) ≤ T ε u (t, x) -x x -T ε d (t, x) = g ε t (x, T ε d (t, x)) -g ε t (x, x) x -T ε d (t, x) ≤ Cε.
Finally, by the definition of T ε u below (6.6), we have

J ε u (t, x) = F -1 t + ε, F (t, x) + δ ε F t, T ε d (t, x) -F -1 t + ε, F (t, x) + δ ε F (t, x) ≤ δ ε F t, T ε d (t, x) + |δ ε F (t, x)| f t + ε, F -1 t + ε, F (t, x) + ξ ,
for some ξ between δ ε F t, T ε d (t, x) and δ ε F (t, x). We can then conclude the proof by the fact that |δ ε F | ≤ C 1 ε for some constant C 1 .

Proof of Proposition 3.2

We recall that P n is a martingale measure on the canonical space Ω, induced by the continuoustime martingale X * ,n under the probability P * ,n . The martingale X * ,n jumps only on discrete time grid π n = (t n k ) 1≤k≤n . Moreover, at time t n k+1 , the upward jump size is J ε u (t n k , X t n k ) and downward jump size is J ε d (t n k , X t n k ) with ε := t n k+1 -t n k (see Section 3.1). Let C, θ be some positive constants, we define

E n (C, θ) := inf Π k-1 i=j 1 -C(t n i+1 -t n i ) : for some s ∈ [0, 1) and 0 ≤ j ≤ k ≤ n such that s ≤ t n j ≤ t n k+1 ≤ s + θ . Since |π n | := max 1≤k≤n (t n k -t n k-1 ) -→ 0, it follows that E n (C, θ) -→ e -Cθ as n -→ ∞. (i)
To prove the tightness of (P n ) n≥1 , we shall use Theorem VI.4.5 of Jacod & Shiryaev [54, P. 356].

First, Doob's martingale inequality implies that

P n sup 0≤t≤1 |X t | ≥ K ≤ E P n [|X 1 |] K = 1 K R |x|µ 1 (dx) =: L 1 K , ∀K > 0. (6.7)
Let η > 0 be an arbitrary small real number, then there is some K > 0 such that

P n sup 0≤t≤1 |X t | ≥ K ≤ η, for all n ≥ 1.
We can assume K large enough so that -K < m t < K for all t ∈ [0, 1]. Denote then r K (t) := r t ∧ K and l K (t) := l t ∨ (-K). Let δ > 0, it follows by Lemma 6.4 that the upward jump size J ε u (t, x) is uniformly bounded by Cε for some constant C on D K δ := {(t, x) : m t ≤ x ≤ r K (t) -δ/2}. We then consider θ > 0 small enough such that θ ≤ δ 2C and |l

K (t + θ) -l K (t)| + |r K (t + θ) -r K (t)| ≤ δ/2 for all t ∈ [0, 1 -θ].
Let S, T be two stopping times w.r.t to the filtration generated by X * ,n such that 0 ≤ S ≤ T ≤ S + θ ≤ 1. When sup 0≤t≤1 |X * ,n t | ≤ K and X * ,n only increases between S and S + θ, then clearly |X * ,n T -X * ,n S | < δ. Therefore

P * ,n sup 0≤t≤1 |X * ,n t | ≤ K, X * ,n T -X * ,n S ≥ δ ≤ P * ,n sup 0≤t≤1 |X * ,n t | ≤ K,
and there is a down jump of X * ,n on [S, S + θ]

≤ 1 -E n (C, θ),
where the last inequality follows by the estimate of q ε in Lemma 6.4. Then it follows that lim sup

θ→0 lim sup n→∞ P * ,n X * ,n T -X * ,n S ≥ δ ≤ lim sup θ→0 lim sup n→∞ P * ,n sup 0≤t≤1 |X * ,n t | ≤ K, X * ,n T -X * ,n S ≥ δ + P * ,n sup 0≤t≤1 |X * ,n t | ≥ K ≤ lim sup θ→0 lim sup n→∞ 1 -E n (C, θ) + η = η.
Since η > 0 is an arbitrary small real number, we then obtain that lim

θ→0 lim sup n→∞ P * ,n X * ,n T -X * ,n S ≥ δ = 0.
Then it follows by Theorem VI.4.5 of [START_REF] Jacod | Limit theorems for stochastic processes[END_REF] that the sequence (X * ,n , P * ,n ) n≥1 is tight, and hence P n n≥1 is tight. (ii) Let P 0 be a limit of (P n ) n≥1 , let us now check that P 0 • X -1 t = µ t for every t ∈ [0, 1]. By extracting the sub-sequence, we suppose that P n → P 0 , then P * ,n • (X * ,n t )

-1 = P n • X -1 t → P 0 • X -1 t .
By the construction of X * ,n , there is a sequence (s n ) n≥1 in [0, 1] such that s n → t and X * ,n t = X * ,n sn ∼ µ sn under P * ,n . It follows by the continuity of the distribution function F (t, x) that µ sn → µ t , and hence P 0 • X -1 = µ t . (iii) Finally, let us show that X is still a martingale under P 0 . For every K > 0, denote X K t := (-K) ∨ X t ∧ K. Let s < t and ϕ(s, X • ) be a bounded continuous, F s -measurable function, by weak convergence, we have

E P n ϕ(s, X • )(X K t -X K s ) -→ E P 0 ϕ(s, X • )(X K t -X K s ) .
Moreover, since the marginals (µ t ) t∈[0,1] form a peacock, and hence are uniformly integrable, it follows that

E P n ϕ(s, X • )(X K t -X K s ) ≤ 2|ϕ| ∞ sup r≤1 |x|1 {|x|≥K} µ r (dx) -→ 0, as K → ∞,
uniformly in n. Then, by the fact that X is a P n -martingale, we have

E P 0 ϕ(s, X • )(X t -X s ) = 0.
By the arbitrariness of ϕ, this proves that X is a P 0 -martingale.

Convergence to the peacock under the one-maximizer condition on the cdf difference

We recall that under Assumption 3.4, the function x → ∂ t F (t, x) has one unique local maximizer on (l t , r t ). Recall also in this context, one has D(t) := (l t , m t ], D c (t) := (m t , r t ) and

D c := {(t, x) : t ∈ [0, 1], x ∈ D c (t)}, D := (t, x) : t ∈ [0, 1],
x ∈ D(t) .

Full-marginals left-monotone martingale transport

This section is dedicated to the Proof of Lemma 3.6. We first rewrite (3.3) into:

G t, x, T d (t, x) = 0, where G(t, x, y) := x y (x -ξ)∂ t f (t, ξ)dξ. (6.8)
Step 1: We first prove the existence and uniqueness of T d . By the non-decrease of (µ t ) t∈[0,1] in convex order in Assumption 3.1, it follows that

G(t, x, -∞) = x -∞ (x -ξ)∂ t f (t, ξ)dξ ≥ 0. Since M(∂ t F (t, .)) = {m t }, the function x -→ ∂ t F (t, x
) is strictly increasing on the interval (l t , m t ], implying that the last inequality is strict, i.e.

G(t, x, -∞) > 0.
The same argument also implies that y -→ G(t, x, y) is strictly decreasing on (l t , m t ).

Next, notice that G(t, m t , m t ) = 0 and x -→ G(t, x, m t ) is decreasing on interval (m t , r t ):

∂ x G(t, x, m t ) = x mt ∂ t f (t, ξ)dξ = ∂ t F (t, x) -∂ t F (t, m t ) < 0.
In summary, for every x ∈ (m t , r t ), we have G(t, x, m t ) < 0, G(t, x, -∞) > 0 and y → G(t, x, y) is continuous, strictly decreasing on (l t , m t ). It follows that the equation (6.8) has a unique solution T d (t, x) and it takes values in (l t , m t ), which implies that the equation (3.3) has a unique solution in (l t , m t ).

Step 2: We next prove (i). Differentiating both sides of equation (3.3) w.r.t. x ∈ (m t , r t ), it follows that

-x -T d (t, x) ∂ t f t, T d (t, x) ∂ x T d (t, x) + x T d (t,x) ∂ t f (t, ξ)dξ = 0.
Therefore, for every x ∈ (m t , r t ),

∂ x T d (t, x) = ∂ t F (t, x) -∂ t F (t, T d (t, x)) x -T d (t, x) ∂ t f t, T d (t, x) < 0, (6.9)
and hence x → T d (t, x) is strictly decreasing in x on interval (m t , r t ).

Step 3: It remains to prove (ii). We define, for 0 < δ < K < ∞,

E δ := {(t, x) ∈ D c : m t < x < m t + δ}, (6.10)
E δ,K := {(t, x) ∈ D c : m t + δ ≤ x ≤ (m t + K) ∧ r t }. (6.11)
Let us first prove that j d 1 D c is locally Lipschitz in x on [m t , r t ), it is enough to verify that ∂ x T d 1 D c is locally bounded. From (6.9), we have

∂ x T d (t, x) = ∂ t F (t, x) -∂ t F (t, T d (t, x)) x -T d (t, x) ∂ t f t, T d (t, x) , (t, x) ∈ D c . It is clear that ∂ x T d (t, x
) is continuous on D c and hence bounded on E δ,K for every 0 < δ < K.

We then focus on the case (t, x) ∈ E δ . Since ∂ t f (t, m t ) = 0 and ∂ tx f (t, m t ) < 0 by Assumption 3.4, we have

∂ t f (t, ξ) = ∂ tx f t, m t ξ -m t + C 1 (t, ξ) ξ -m t 2 ,
where C 1 (t, ξ) is uniformly bounded for |ξ -m t | ≤ δ. Inserting the above expression into (6.8), it follows that

x T d (t,x) x -ξ ξ -m t dξ = C 2 (t, x) x -T d (t, x) 4 ,
where C 2 is also uniformly bounded on E δ since min 0≤t≤1 ∂ t,x f (t, m t ) < 0 by Assumption 3.4.

By direct computation, it follows that

x -T d (t, x) 2 x -m t + 2 m t -T d (t, x) = C 2 (t, x) x -T d (t, x) 4 ,
which implies that

T d (t, x) = m t - 1 2 x -m t + C 2 (t, x) x -T d (t, x) 2 , (6.12)
Using again the expression (6.9), we have

∂ x T d t, x = - 1 2 + C 3 (t, x) x -T d (t, x) ,
where C 3 is also uniformly bounded on E δ . Finally, by the uniqueness of solution T d of (6.8), we get

∂ x T d t, x = - 1 2 + C 4 (t, x) x -m t , (6.13)
for some C 4 uniformly bounded on E δ , implying that T d 1 D c is locally Lipschitz in x. Moreover, by the expression of j u in (3.4), i.e.

j u (t, x) := ∂ t F t, T d (t, x) -∂ t F (t, x) f (t, x) ,
together with (6.13), it is easy to check hat j u 1 D c and (j u /j d )1 D c are also locally Lipschitz in x.

To prove that these functions are also locally Lipschitz in t, we consider ∂ t T d (t, x). By direct computation, we obtain

∂ t T d (t, x) = x T d (t,x) (x -ξ)∂ 2 tt f (t, ξ)dξ (x -T d (t, x))∂ t f (t, T d (t, x)) ,
which is clearly continuous in (t, x) on D c , and hence uniformly bounded on E δ,K , for K > δ > 0. Using again (6.12), it is easy to verify that ∂ t T d is also uniformly bounded on E 0,δ , and hence ∂ t T d (t, x) is also locally bounded on D c . Therefore, j d (t, x)1 x>mt , j u (t, x)1 x>mt and ju j d (t, x)1 x>mt are all locally Lipschitz. Remark 6.5. From the previous proof, we have in particular that

∂ t T d (t, m t + δ) δ 0 -→ - 3 2 ∂ 2 tt f (t, m t ) ∂ 2 tx f (t, m t )
uniformly in t ∈ [0, 1].

Additional asymptotic estimates of the left-monotone transference plan

We shall use the notations E δ and E δ,K as defined in (6.10)-(6.11).

Lemma 6.6. Let Assumptions 3.1 and 3.4 hold true. Then J ε u and J ε d admit the expansion

J ε u (t, x) = εj ε u (t, x) + ε 2 e ε u (t, x), and J ε d (t, x) = j d (t, x) + ε ∨ ρ(ε) e ε d (t, x), (6.14)
where j d is defined in (3.4), and

j ε u (t, x) := ∂ t F t, x -J ε d (t, x) -∂ t F (t, x) f (t + ε, x) .
Moreover, for all 0 < δ < K < ∞, e ε u (t, x), e ε d (t, x) are uniformly bounded; and consequently, there is constant C δ,K such that q ε admits the asymptotic expansion:

q ε (t, x) = ε j u (t, x) j d (t, x) + C δ,K ε ε ∨ ρ 0 (ε) , for (t, x) ∈ E δ,K . (6.15)
Proof. Let δ < K be fixed. Notice that by its definition, the function T d (t, x) is continuous on compact set E δ,K and hence it is uniformly bounded.

(i) By the definition of T ε u below (6.6), we have

T ε u (t, x) = F -1 t + ε, F (t + ε, x) + δ ε F (t, T ε d (t, x)) -δ ε F (t, x) .
By direct expansion, we see that the first equality in (6.14) holds true with

|e ε u (t, x)| ≤ sup t≤s≤t+ε, T ε d (s,x)≤ξ≤x 2∂ tt F (s, ξ)∂ x f (s, ξ) f 3 (s, ξ) .
(ii) Let us now consider the second equality in (6.14). First,

x -∞ [F -1 (t + ε, F (t, ξ)) -ξ]f (t, ξ)dξ = x -∞ ξ δ ε f (t, ξ)dξ + F -1 (t+ε,F (t,x)) x ξf (t + ε, ξ)dξ = x -∞ ξ δ ε f (t, ξ)dξ -δ ε F (t, x) x + C 1 (t, x)ε = x -∞ (ξ -x) δ ε f (t, ξ)dξ + C 2 (t, x)ε 2 ,
where

|C 1 (t, x)| ≤ |F -1 (t + ε, F (t, x)) -x| 2 |f | ∞ and |C 2 (t, x)| ≤ |C 2 (t, x)||∂ t F | ∞ .
We next note that g ε t (x, ξ) = x + C 3 (t, x, ξ)ε, where |C 3 (t, x, ξ)| ≤ 2 |∂tF |∞ m f . Then it follows by direct computation that Further, for every k ≥ 1,

T ε d (t,x) -∞ g ε t (x, ξ) -ξ δf (t, ξ)dξ = T ε d (t,x) -∞ (x -ξ) δ ε f (t, ξ)dξ + C 4 (t, x)ε 2 ,
where

|C 4 (t, x)| ≤ 2 |∂tF |∞ m f |∂ t F | ∞ .
Combining the above estimates with (6.6), it follows that

x T ε d (t,x) x -ξ 1 ε δ ε f (t, ξ) dξ = C 2 (t, x) ∨ C 4 (t, x) ε. It follows then x T ε d (t,x) (x -ξ)∂ t f (t, ξ) dξ = C 5 (t, x) ε ∨ ρ(ε) ,
where

|C 5 (t, x)| ≤ (x + K)(|∂ t f | ∞ + |∂ 2 tt f | ∞ )
. This implies the first estimation in (6.14) since ∂ t f (t, x) > 0 for x ∈ (l t , m t ). Lemma 6.7. Under Assumptions 2.2, 3.1 and 3.4, we have

T ε d 1 {x>m ε (t)} → T d 1 {x>m(t)} , h ε → h * , ∂ t ψ ε → ∂ t ψ * , and ψ ε → ψ * ,
locally uniformly on {(t, x) : t ∈ [0, 1), x ∈ (l t , r t )}.

Proof. (i) In the one local maximizer case under Assumption 3.4, the definition of T ε d (t, x) in (6.6) is reduced to be

x -∞ F -1 t + ε, F (t, ξ) -ξ f (t, ξ)dξ + T ε d (t,x) -∞ g ε t (x, ξ) -ξ δ ε f (t, ξ)dξ = 0, or equivalently x T ε d (t,x) ξ δ ε f (t, ξ)dξ + T ε u (t,x)
x ξf (t + ε, ξ)dξ = 0, (

with T ε u (t, x) := g ε t x, T ε d (t, x) . Differentiating (6.16), it follows that

∂ t T ε d (t, x) := - A ε (t, x) T ε u -T ε d δ ε f t, T ε d (•) (t, x), (6.17)
with

A ε (t, x) := x T ε d (t,x) ξ∂ t δ ε f (t, ξ)dξ + T ε u (t,x) x ξ∂ t f (t + ε, ξ)dξ + T ε u (t, x) ∂ t F (t, x) -∂ t F t + ε, T ε u (t, x) + ∂ t δ ε F t, T ε d (t, x) , = -T ε u (t, x) -x ∂ t δ ε F (t, x) -∂ t δ ε F (t, T ε d (t, x) - T ε u (t,x) x T ε u (t, x) -ξ ∂ t f (t + ε, ξ)dξ - x T ε d (t,x) (x -ξ)∂ t δ ε f (t, ξ)dξ
and

∂ x T ε d (t, x) := - T ε u (t, x) -x T ε u (t, x) -T ε d (t, x) δ ε f t, T ε d (t, x) f (t, x), (6.18)
where the last term is exactly the same as that induced by ODE (6.2).

(ii) Taking the limit ε → 0, it follows by direct computation and the convergence

T ε d (t, x) → T d (t, x) in Lemma 6.6 that ∂ x T ε d (t, x) → ∂ x T d (t,
E := (t, x, ε) : t ∈ [0, 1], ε ∈ [0, 1 -t], m ε (t) < x < r ε (t) ,
where m 0 (t) := m t and r 0 (t) := r t .

(iii) By exactly the same computation as in Proposition 3.12 of [START_REF] Henry-Labordère | An Explicit Martingale Version of Brenier's Theorem[END_REF], we have

∂ x T ε d (t, x) = 1 + O(ε) + O x -T ε d x -m ε (t) -1 2 x -T ε d + O x -T ε d 2 x -m ε (t) -x -T ε d + O x -T ε d 2 (t, x),
and it follows by similar arguments as in [START_REF] Henry-Labordère | An Explicit Martingale Version of Brenier's Theorem[END_REF] that

T ε d (t, x) -m ε (t) = - 1 2 x -m ε (t) + O (x -m ε ) 2 ,
and hence Next, using the estimation (6.19) and the definition of T ε u , we have

∂ x T ε d (t, m ε (t) + δ) → -
T ε u (t, x) -x = C 1 (ε, t, x) x -T ε d (t, x) 2 and T ε u (t, x) -x ε = C 2 (ε, t, x) x -T ε d (t, x) .
Therefore, by direct computation,

1 ε A ε (t, x) = - 1 2 x -T ε d (t, x) 2 ∂ t 1 ε δ ε f (t, m ε (t)) + C 3 (ε, t, x) x -m ε (t) 3 .
It follows by the uniform convergence in (6. [START_REF] Cox | Robust pricing and hedging of double no-touch options[END_REF]) that

∂ t T ε d (t, x) = - 3 2 ∂ t δ ε f (t, m ε (t)) ∂ x δ ε f (t, m ε (t)) + C 4 (ε, t, x) x -m ε (t) , (6.20)
where we notice that C 4 is uniformly bounded for ε > 0 and x -T ε d (m ε (t)) small enough. Finally, the two uniform convergence results in (6. [START_REF] Cox | Robust pricing and hedging of double no-touch options[END_REF]) and (6.20) together with the continuity of (t, x, ε)

→ ∂ t T ε d (t, x), ∂ x T ε d (t, x) implies that ∂ t T ε d (t, x) and ∂ x T ε d (t,
x) are uniformly bounded on E ∩ {(t, x, ε) : |x| ≤ m ε (t) + K} for every K > 0.

(iv) Therefore, it follows by Arzelà-Ascoli's theorem that T ε d converges to T d locally uniformly. Finally, by the local uniform convergence of T ε d → T d , together with the estimations in (6.13) and (6.19), it is easy to deduce the local uniform convergence of h ε → h, ∂ t ψ ε → ∂ t ψ * and ψ ε → ψ * as ε → 0.

Convergence to the peacock

Recall that the sequence of martingale measures (P n ) n≥1 induced by the left-monotone transference plan is tight by Proposition 3.2, we now show that any limit of (P n ) n≥1 provides a weak solution of (3.6). Let

M t (ϕ, x) := ϕ(x t ) - t 0 j u (s, x s-)Dϕ(x s-)1 xs->m(s) ds + t 0 ϕ(x s--j d (s, x s-)) -ϕ(x s-) j u j d (s, x s-) 1 xs->m(s) ds, (6.21)
for all x ∈ Ω := D([0, 1], R) and ϕ ∈ C 1 (R). Then the process M (ϕ, X) is clearly progressively measurable w.r.t. the canonical filtration F. For the martingale problem, we also need to use the standard localization technique in Jacod & Shiryaev [START_REF] Jacod | Limit theorems for stochastic processes[END_REF]. In preparation, let us introduce, for every constant p > 0, an F-stopping time and the corresponding stopped canonical process

τ p := inf t ≥ 0 : |X t | ≥ p or |X t-| ≥ p , X p t := X t∧τp . (6.22)
Following [START_REF] Jacod | Limit theorems for stochastic processes[END_REF], denote also J(x) := {t > 0 : ∆x(t) = 0}, V (x) := {a > 0 : τ a (x) < τ a + (x)} and V (x) := {a > 0 : τ a (x) ∈ J(x) and |x(τ a (x))| = a}.

Proof of Theorem 3.7. By extracting subsequences, we can suppose without loss of generality that P n → P 0 weakly. To prove that P 0 is a weak solution of SDE (3.6), it is sufficient to show that M t (ϕ, X) t∈[0,1] is a local martingale under P 0 for every ϕ ∈ C 1 b (R). Since the functions j u and j d are only locally Lipschitz (not uniformly bounded) by Lemma 3.10, we need to adapt the localization technique in Jacod & Shiryaev [START_REF] Jacod | Limit theorems for stochastic processes[END_REF], by using the stopping time τ p defined by (6.22). Our proof will be very similar to that of Theorem IX.3.39 in [START_REF] Jacod | Limit theorems for stochastic processes[END_REF].

First, since P n is induced by the Markov chain (X n , P * ,n ) for all n ≥ 1, we have

E P n t n k ϕ(X t n k+1 ) -ϕ(X t n k ) = α u + α d where, denoting ε n k := t n k+1 -t n k , α i := E P n t n k ϕ X t n k + J ε n k i (t n k , X t n k ) -ϕ(X t n k ) J i J d + J u 1 X t n k ≥m ε n k (t n k ) , i = d, u.
By (6.15) in Lemma 6.6 and the uniform continuity of m ε n k (t), we have

α u = E P n t n k t n k+1 t n k Dϕ(X s )j u (s, X s )1 Xs≥m(s) ds + O ε n k (ε n k ∨ ρ 0 (ε n k )) ,
where ρ 0 is the continuity modulus of (t, ε n k ) → m ε n k (t) in Assumption 3.4. Similarly,

α d = E P n t n k t n k+1 t n k ϕ(X s -j d (s, X s )) -ϕ(X s ) j u j d (s, X s )1 Xs≥m(s) ds + O ε n k (ε n k ∨ ρ 0 (ε n k )) .
Therefore, let 0 ≤ s < t ≤ 1, p ∈ N, φ s (X • ) be a F s -measurable bounded random variable on Ω such that φ : Ω → R is continuous under the Skorokhod topology, we have

E P n φ s (X • ) M t∧τp (ϕ, X) -M s∧τp (ϕ, X) ≤ C p |π n | ∨ ρ 0 (|π n |) , (6.23)
for some constant C p > 0.

To proceed, we follow the same localization arguments as in the proof of Theorem IX.3.39 of Jacod & Shiryaev [START_REF] Jacod | Limit theorems for stochastic processes[END_REF]. Since P n → P 0 as n → ∞, then for every p ∈ N, the distribution of the stopped process X p

• under P n also converges, i.e. there is P 0,p such that L P n (X p • ) → P 0,p as n → ∞. Due to the proof of Proposition IX.1.17 of [START_REF] Jacod | Limit theorems for stochastic processes[END_REF], there are at most countably-many a > 0 such that P 0,p ω : a ∈ V (ω) ∪ V (ω) > 0.

So we can choose a p ∈ [p -1, p] such that P 0,p ω : a p ∈ V (ω) ∪ V (ω) = 0. It follows by Theorem 2.11 of [START_REF] Jacod | Limit theorems for stochastic processes[END_REF] that ω → τ ap (ω) is P 0,p -a.s. continuous and the law

L P n X p • , X ap •
converges to L P 0,p X, X τa p . Denote by P0,p the law of X τa p on (Ω, F, P 0,p ), we then have ω → τ ap (ω) is P0,p -a.s. continuous and L P n X ap → P0,p . In particular, since there is a countable set

T * ⊂ [0, 1] such that x → M t∧τa p (ϕ, x) -M s∧τa p (ϕ, x) (6.24)
is P0,p -almost surely continuous for all s < t such that s, t / ∈ T * . Therefore, by taking the limit of (6.23), we obtain

E P0,p φ s (X • ) M t (ϕ, X) -M s (ϕ, X)
= 0, whenever s ≤ t and t / ∈ T * . Combining with the right-continuity of M t (ϕ, x), we know P0,p is a solution of the martingale problem (6.21) between 0 and τ ap , i.e. M t∧τa p (ϕ, X) 0≤t≤1 is a martingale under P0,p . Moreover, since P0,p = P 0 in restriction to (Ω, F τa p ) and τ ap → ∞ as p → ∞, it follows by taking the limit p → ∞ that M t (ϕ, X) 0≤t≤1 is a local martingale under P 0 , i.e. P 0 is a solution to the martingale problem (6.21) and hence a weak solution to SDE (3.6).

Finally, for uniqueness of solutions to SDE (3.6), it is enough to use Theorem III-4 of Lepeltier & Marchal [START_REF] Lepeltier | Problème des martingales et équations différentielles stochastiques associées a un opérateur intégro-différentiel[END_REF] (see also Theorem 14.18 of Jacod [53, P. 453]) together with localization technique to conclude the proof. Remark 6.8. In the multiple local maximizers case under Assumption 3.1, the functions j u and j d are no more continuous, then the mapping (6.24) may not be a.s. continuous and the limiting argument thereafter does not hold true. This is the main reason for which we restrict to the one maximizer case under Assumption 3.4 in Theorem 3.7.

Proof of Lemma 3.8. We recall that by Theorem 3.8 in [START_REF] Henry-Labordère | An Explicit Martingale Version of Brenier's Theorem[END_REF] (ii), the corresponding maps T ε u (t, .) and T ε d (t, .) solve the following ODEs:

d dx δ ε F t + ε, T ε d (t, x) = (1 -q)(t, x)f (t, x), (6.25)
d dx F t + ε, T ε u (t, x) = q(t, x)f (t, x) for all x ∈ (D ε ) c (t), (6.26)
where δ ε F (t + ε, .) := F (t + ε, .) -F (t, .). With the asymptotic estimates

T ε d (t, x) -x = -j d (t, x) + •(ε) and T ε u (t, x) -x = εj u (t, x) + O(ε),
which is locally uniform by Lemma 6.6. By direct substitution of this expression in the system of ODEs (6.25-6.26), we see that the limiting maps (j d , j u ) of (T ε u , T ε d ), as ε 0, satisfy the following system of first order partial differential equations (PDEs):

∂ x j d (t, x) = 1 + j u (t, x) j d (t, x) f (t, x) ∂ t f (t, x -j d (t, x))
, ∂ x {j u f }(t, x) = -∂ t f (t, x) -j u (t, x) j d (t, x) f (t, x).

Since x ∈ D c (t) and x -j d (t, x) ∈ D(t), it follows directly that (3.7) holds true.

Proof of Proposition 3.9. By Lemma 3.8, item (ii) of Proposition 3.9 is a direct consequence of item (i), then we only need to prove (i). Let x ∈ R, the function y → (y -x) + is continuous and smooth on both (-∞, x] and [x, ∞), then it follows by Itô's lemma that d X t -x + = dM t + L t (6.27)

:= 1 { Xt->x} d X t + X t -x + -X t--x + -1 { Xt->x} ∆ X t ,
where (M t ) 0≤t≤1 is a local martingale. Notice that 1 { Xt->x} is bounded and X 1 ∈ L p for some p > 1. Using BDG inequality and then Doob's inequality, it is a standard result that (M t ) 0≤t≤1 is a real martingale. Further, the local Lévy process X is clearly quasi left continuous. Moreover, since for every ε ≤ ε 0 , where ε 0 ∈ (0, 1 -t]. Then, integrating (6.27) between t and t + ε, and taking expectations, it follows that

L s = x -T d s, X s-1 T d s,
E X t+ε -x + -E X t -x + = t+ε t R
x -T d (s, y) j u j d s, y 1 T d (s,y)≤x<y, y∈D c (s) f X (s, y) dy ds. (6.28)

Let us now differentiate both sides of (6.28). For the left hand side, since the density function f X (t, .) of X t is continuous, the function x → E ( X t -x) + = ∞ x (y -x)f X (t, y)dy is differentiable and We now consider the rhs of (6. Finally, differentiating both sides of (6.28) (with (6.29), (6.30) and (6.31)), then dividing them by ε and sending ε 0, it follows that

∂ x E X t -x + = ∞ x -f X (t,
∂ t f X (t, x) = 1 {x>mt} ∂ x f X j u - j u f X j d (t, x) -1 {x<mt} j u f X j d (1 -∂ x j d ) t, T -1 d (t, x) ,
for every t ∈ [0, 1) and x ∈ (l t , r t ) \ {m t }.

Convergence of the robust superhedging strategy

To prove Theorem 3.11, we will consider a special sequence of partitions of [0, 1], π n = (t n k ) 0≤k≤n , where t n k := kε with ε = 1 n . To avoid heavy notation, we will omit the superscript and simplify t n k to t k . We also recall that under every P n , we have P n -a.s. that Next, let us consider the limit of the second term on the left hand side of (6.32).

n-1 k=0 ϕ ε (t k , X t k ) + ψ ε (t k , X t k+1 ) + n-1 k=0 h ε (t k , X t k ) X t k+1 -X t k ≥ n-1 k=0 c X t
Lemma 6.11. Let Assumptions 3.1 and 3.4 hold true. Then we have the following convergence in probability under every martingale measure P ∈ M ∞ :

n-1 k=1 h ε (t k , X t k ) X t k+1 -X t k → 1 0 h * (t, X t-)dX t .
Proof. The functions h ε are all locally Lipschitz uniformly in ε and h ε → h * locally uniformly, as ε → 0, by Lemma 6.7. By the right continuity of martingale X, the above lemma is then a direct application of Theorem I.4.31 of Jacod & Shiryaev [START_REF] Jacod | Limit theorems for stochastic processes[END_REF].

Proof of Theorem 3.11. Using (6.32), together with Lemmas 2.4, 6.9 and 6.11, it follows that under every P ∈ M ∞ (i.e. the canonical process X is a martingale under P), we have the superhedging property

Ψ * (X • ) + 1 0 h * (t, X t-)dX t ≥ 1 0 1 2 c yy (X t , X t )d[X] c t + 0<t≤1
c(X t-, X t ), P-a.s.

Further, by weak duality, we have

E P 0 [C(X • )] ≤ P ∞ (µ) ≤ D ∞ (µ) ≤ µ(λ * ).
Since E P 0 [C(X • )] = µ(λ * ) by Lemma 6.10, this implies the strong duality as well as the optimality of the local Lévy process (3.6) and the semi-static superhedging strategy described by (h * , ψ * ).

Figure 1 :

 1 Figure 1: Fake Brownian motion: Maps Td and Tu

1 1 F 0

 10 the right-continuous inverse of F 1 , and g(x, y) := F -1 (x) + δF (y) . and we set m 1 := m 0 1 , and we introduce the right-continuous functions T u , T d : R → R by T d (x) := t A k (x, m k ), T u (x) := g(x, T d (x)); x ∈ [m k , x k ) and T d (x) = T u (x) = x; x ∈ (x k-1 , m k ],

  and D c 0 := (l µ1 , r µ1 ) \ D 0 , by the continuity of F 0 and F 1 , it follows from the last construction that T d and T u take values in D 0 and D c 0 , respectively. Both functions are continuous except on points (x k ) k≥1 and T -1 d (x k -) k≥1 , where T -1 d denotes the right-continuous version of the inverse function of T d . Moreover, one has T d (x) ≤ x ≤ T u (x) for all x ∈ R. Remark 6.1. (i) In the case δF has only one local maximizer m 0 1 , we have D 0 = (l µ1 , m 0 1 ] and D c 0 = (m 0 1 , r µ1 ), T d maps D c 0 to D 0 and T u maps D c 0 to D c 0 . (ii) In [40], the functions T u and T d are obtained by solving the ODE

Theorem 6 . 2 .

 62 [START_REF] Henry-Labordère | An Explicit Martingale Version of Brenier's Theorem[END_REF] Theorems 4.5 and 5.1] Suppose that the partial derivative c xyy exists and c xyy > 0 on (l µ0 , r µ0 ) × (l µ1 , r µ1 ). Then, (i) the probability P * (dx, dy) := µ 0 (dx)T * (x, dy) ∈ M 2 (µ 0 , µ 1 ) is the unique left-monotone martingale transport plan, which solves the primal problem (2.1).

  t ∈ [0, 1) and ε ∈ [0, ε 0 ∧ (1 -t)], as δ 0. (6.[START_REF] Cox | Robust pricing and hedging of double no-touch options[END_REF]

1 T

 1 Xs-≤ x < Xs-, Xs-∈D c (s) ≤ j d s, X s-1 Xs-∈D c (s)by direct computation, it follows by (3.8) together with dominated convergence theorem thatE t≤s≤t+ε L s = E t+ε t x -T d s, X s- j u j d s, X s-d s, Xs-≤x< Xs-, Xs-∈D c (s)ds ,

u j d s, y 1 Tu j d s, y 1 Tj u j d s, y 1 T

 111 28) and denote l(s, x) := R x -T d (s, y) j d (s,y)≤x<y, y∈D c (s) f X (s, y) dy. Let us fix s ∈ [0, 1) and x ∈ D c (s) := (m s , r s ), then it is clear thatl(s, x) = ∞ x x -T d (s, y) j d (s,y)≤x, y∈D c (s) f X (s, y)dy,where the integrand is smooth in x for every y ∈ R. Hence for every x ∈ D c (s),∂ x l(s, x) = j u (s, x)f X (s, x) + ∞ x d (s,y)≤x, y∈D c (s) f X (s, y)dy,and∂ 2 xx l(s, x) = ∂ x j u f X (s, x) -j u j d f X (s, x).(6.30)We now consider the case x ∈ (l t , m t ). Notice that T d (s, •) : D c (s) → (l s , m s ) is a bijection and X s admits a density function. It follows that the random variable T d (s, X s ) also admits a density function on (m s , r s ), given byf T (s, y) = f X ∂ x T d (s, y), ∀y ∈ (l s , m s ).Then by the expression thatl(s, x) = x -∞ (x -z) j u j d s, T -1 d (s, z) 1 {x≤T -1 d (z)} f T (s, z)dz, we get ∂ 2 xx l(x) = ju j d s, T -1 d (s, x) f T (s, z) = ju j d s, T -1 d (s, x) f X ∂xT d (s, x), ∀x ∈ (l s , m s ).(6.31)

  r t )}.

	Assumption 3.1. (i) The marginal distributions µ = (µ t ) t∈[0,1] are non-decreasing in convex
	ordering and have finite first order moment.
	(ii) For all t ∈ [0, 1], the measure µ t has a density function f (t, •), and t → µ t is continuous
	w.r.t. the weak convergence topology. Moreover,
	F ∈ C 4 b (E) and	inf x∈[-K,K]∩(lt,rt)

  x) and ∂ t T ε d (t, x) → ∂ t T d (t, x) for every (t, x) ∈ D c . Moreover, by the local uniform convergence result in Lemma 6.6, we deduce that ∂ x T ε

	d and
	∂ t T ε d also converge locally uniformly. Denote T 0 d := T d , it follows that the mapping (t, x, ε) →
	∂ t T ε d (t, x), ∂ x T ε d (t, x) is continuous on

  y)dy, ∂ 2 xx E X t -x

+

= f X (t, x).

(6.29)

  k , X t k+1 .(6.32) Moreover, since µ(λ * ) < ∞, it follows by dominated convergence theorem thatE P 0 Ψ * (X • ) = E P 0 X s )1 Xs>m(s) c(X s , X s -j d (s, X s ))ds -j d (t, x) f (t, x)dxdt,since the marginals of X under P 0 are (µ t ) 0≤t≤1 .To computer E P 0 [C(X • )], we notice that [X] c t = 0, P 0 -a.s., and the processY t := s≤t |c(X s-, X s )| -t 0 |c(X s-, X s--j d (s, X s-))| j u (s, X s-) j d (s, X s-) 1 Xs-≥mt ds, is a local martingale. Since µ(λ * ) < ∞, we have 1 0 |c(X s-, X s--j d (s, X s-))| j u (s, X s-) j d (s, X s-) 1 Xs-≥mt ds -j d (t, x) f (t, x)dxdt < ∞,which implies that Y is a martingale and hence E[Y 1 ] = 0. Finally, using similar arguments together with dominated convergence theorem, we get that E

	1 (s, = 0 j u j d 1 0 rt mt j u (t, x) j d (t, x) c x, x = 1 0 rt mt j u (t, x) j d (t, x) c x, x

s≤t c(X s-, X s ) = 1 0 rt mt j u (t, x) j d (t, x) c x, x -j d (t, x) f (t, x)dxdt,

which concludes the proof.

The last integrals are well-defined and valued in R ∪ {∞}, as convex functions are bounded from below by an affine function, which is integrable w.r.t. µ by the finite first moment condition.

We refer to Section 8.1 of[START_REF] Beiglböck | On a problem of optimal transport under marginal martingale constraints[END_REF] for the decomposition of any pair (µ 0 , µ 1 ) in irreducible components.

The authors gratefully acknowledge the financial support of the ERC 321111 Rofirm. Xiaolu Tan and Nizar Touzi are grateful for the support of 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).

By taking the limit of every term, we obtain a superhedging strategy for the continuous-time reward function, and we can then check that this superhedging strategy induces a duality of the transportation problem as well as the optimality of the local Lévy process (3.6).

Let us first introduce Ψ * : Ω → R by

Lemma 6.9. Let Assumptions 3.1 and 3.4 hold true. Then for every càdlàg path x ∈ D([0, 1], R) taking value in (l 1 , r 1 ), we have

Proof. By direct computation, we have for every n ≥ 1,

First, we have ψ ε (t n-1 , x 1 ) → ψ * (1, x 1 ) and by Lemma 6.7,

Further, when x > m t ,

which concludes the proof.

Lemma 6.10. Let Assumptions 3.1 and 3.4 hold true, and µ(λ * ) < ∞. Then for the limit probability measure P 0 given in Theorem 3.7, we have

Proof. We notice that under the limit probability measure P 0 , X is a pure jump martingale with intensity ju j d (s, X s-). Then by Itô's formula, the following process is a local martingale ψ * (t, X t ) -ψ * (0, X 0 ) -t 0 ∂ t ψ * (t, X s )dt t 0 j u (s, X s )∂ x ψ * (s, X s ) + j u j d (s, X s ) ψ * (s, X s -j d (s, X s )) -ψ * (s, X s ) 1 Xs>m(s) ds.