MATHIAS BEIGLB ¨OCK AND NICOLAS JUILLET
Abstract. A classical result of Strassen asserts that given probabilitiesµ, νon the real line which are in convex order, there exists amartingale couplingwith these marginals, i.e. a random vector (X1,X2) such thatX1∼µ,X2∼νandE[X2|X1]=X1. Remarkably, it is a non trivial problem to construct particular solutions to this problem. Based on the concept ofshadowfor measures in convex order, we introduce a family of such martingale cou- plings, each of which admits several characterizations in terms of optimality properties/ geometry of the support set/representation through a Skorokhod embedding. As a par- ticular element of this family we recover the (left-)curtain martingale transport, which has recently been studied [11, 22, 15, 9] and which can be viewed as a martingale analogue of the classical monotone rearrangement. As another canonical element of this family we identify a martingale coupling that resembles the usualproduct couplingand appears as an optimizer in the general transport problem recently introduced by Gozlan et al. In addi- tion, this coupling provides an explicit example of a Lipschitz kernel, shedding new light on Kellerer’s proof of the existence of Markov martingales with specified marginals.
Keywords:Strassen’s theorem, Kellerer’s theorem, peacocks, (martingale) optimal trans- port, general transport costs, Skorokhod embedding
1. Introduction
1.1. Outline. Given Polish spacesX,Y, a measureπonX×Y with marginalsµandνis calledtransport plan1 fromµtoνor acoupling ofµandν. LetΠ(µ, ν) be the space of transport plans of marginalsµandν. We will usually consider probability measuresµ,ν on the real line having first moments. Our primary interest lies in the set ofmartingale transport planswhich is defined as
ΠM(µ, ν) ={π=Law(X,Y)∈Π(µ, ν),E(Y|X)=X}
={π∈Π(µ, ν) :R
ydπx,·=xforµ-a.e.x}.
Here the constraintE(Y|X)=Xmeans thatE(Y|X=x)=xforµ-almost everyx∈R, while (πx,·)x∈R denotes the disintegration ofπwith respect to µ, i.e. the family of conditional laws. By Jensen’s inequality, the existence of a martingale transport planπ ∈ ΠM(µ, ν) implies thatµ, νare in the convex orderµ C ν, i.e.R
φdµ ≤ R
φdνfor every convex φ : R → R. Conversely Strassen [47], established thatΠM(µ, ν) is nonempty whenever µ, νare in convex order. While Strassen derived the existence of martingale transport plans as a rather direct consequence of the Hahn-Banach theorem, it seems harder to provide elementary/natural constructions of such martingale transport plans.
2010Mathematics Subject Classification. 60G42.
Key words and phrases.couplings, martingales, peacocks, convex order, optimal transport.
The first author acknowledges support through FWF grant Y782. The second author was partially supported by the “Programme ANR JCJC GMT” (ANR 2011 JS01 011 01).
1Throughout, we will introduce new notations in the place they are needed. For the convenience of the reader, we also collect them in the Appendix.
1
With the aim of defining a systematic martingale coupling we [11] introduced the (left-) curtain couplingπlcwhich can be seen as a martingale analogue of the monotone rearrange- ment coupling (alias thequantile coupling). An explicit description of the curtain coupling is provided whenµis finitely supported in [11, Section 2]. Another construction using differential equations is given by Henry-Labord`ere and Touzi [22] for sufficiently regular distributions. Moreover they establish that the curtain coupling is relevant for the pricing of variance swaps. According to [33] the operation of couplingµandνthrough the cur- tain coupling is continuous so that left-curtain couplings for general measuresµandνcan be approximated using either of the two mentioned constructions, see [33, Remark 2.18].
Hobson and Norgilas [29, 30] deepen the connection to mathematical finance, in particular they use the curtain coupling to obtain robust bounds for the arbitrage free prices of Amer- ican put options. In [9], a link to the field of Skorokhod embedding is established. Nutz and Stebegg provide extensions to a supermartingale setup [40] and, together with Tan, to a multi-period setup [41]. The continuous time setting was explored in [34] and [21] as a limit of the multi-period setting but with a slightly different approach (with respect to [41]).
In this article we reconsider the particular role of the left-curtain coupling as a dis- tinguished coupling in the set ΠM(µ, ν). We will define an infinite family of martingale couplings. Roughly speaking, the (left-)curtain coupling will then be recovered as one extreme element of the this family, while at the other end of the spectrum we will obtain a new and rather different type of systematic coupling that we shall call sunset coupling πsun. Whereas the curtain coupling shares a number of properties with the monotone re- arrangement coupling in (classical optimal transport), the sunset coupling can be seen as the martingale analogue of the product couplingµ×ν. In view of this it is natural that πsundoes not appear as an optimizer of the martingale version of the transport problem.
However, we shall see in Theorem 1.1 and Section 5 below that it enjoys some optimality properties of a different type. A further particular property is thatπsunyields a concrete ex- ample of a martingale transport plan which has theLipschitz(-Markov) property(see§1.3 in this introduction and Section 4).
Before diving into the technical details, we try to convey an intuitive description of the subsequent constructions.
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Figure1. Curtain closing from left to right; sun setting from top to bottom.
We start by describing the curtain coupling. All particles of µare transported to ν, spreading their mass due to the martingale property. Starting with the left most particle
and moving to the right, each particle greedily tries to spread its mass as little as possible, given the portion ofνthat still needs to be filled, see the upper part of Figure 1. Rather than ‘closing the curtain from left to right’ as in Figure 1, we could obtain variants of the curtain coupling by ‘closing the curtain from right to left’, ‘from the middle to boundary’
(see Subsections 3.1.1, 3.1.3, respectively), etc.
Another (and more interesting) variation is to prioritise the mass inµnot from left to right, but rather from top to bottom. This leads to the sunset coupling depicted in the lower part of Figure 1.
Our first aim will be to rigorously describe the class of martingale couplings arising from constructions as hinted here. Subsequently we will describe applications and links to other themes of research.
1.2. Main Theorem. We writeλfor the Lebesgue measure on the unit interval and as- sume thatµ C ν. Throughoutliftorsource2ofµwill refer to a probability ˆµ∈ Π(λ, µ) that will serve as a parameter in the construction of a general version of the left-curtain coupling. The set oflifted martingale transport plansis
ΠˆM( ˆµ, ν) :=n
πˆ ∈Π( ˆµ, ν) :R
yd ˆπu,x,·=xfor ˆµ-a.e. (u,x)o , where ( ˆπu,x,·)(x,u)∈R×[0,1]denotes the disintegration of ˆπwith respect to ˆµ.
“lifted” ˆθ∈ P[0,1]×Rd
disintegration
{{
θˆ[0,u],·(A)=θ([0,ˆ u]×A)
θˆu,·∈ PRd,u∈[0,1]
integration
;;
primitive ..
θˆ[0,u],·∈ MRd,u∈[0,1]
__
derivative
nn
We shall use two further ways to denote the objects ˆµ ∈ Π(λ, µ) ⊆ P([0,1]×R) and πˆ ∈ ΠˆM( ˆµ, ν) ⊆ P([0,1]×R2), respectively: given a measure ˆθ on [0,1]×Rd (where d=1,2 so thatθstands forµorπ), with proj[0,1](ˆθ)=λ, we write
(1) (ˆθu,·)u∈[0,1]for the (λ-a.s. unique) disintegration3of ˆθwith respect toλ.
(2) (ˆθ[0,u],·)u∈[0,1]for the family of measures defined for everyu∈[0,1] by θˆ[0,u],·(A)=θ([0,ˆ u]×A)=Z u
0
θˆs,·(A)ds (1)
whereA⊆Rd.
Our main result is the following.
Theorem 1.1. Letµ, νbe real probability measures in convex order andµˆ∈Π(λ, µ). There exists a uniqueπˆ∈ΠˆM( ˆµ, ν)satisfying any, and then all of the following properties:
2In the recent preprint [14] which considers a continuum time version of the present construction, also the termparametrizationis used.
3We write ˆθu,·instead of the more commonly used ˆθuto emphasize that the disintegration is understood with respect to the first coordinate. In fact, we will also consider disintegrations with respect to to other coordinates subsequently.
(1) πˆminimizes
γˆ 7→
Z (1−u)
q
1+y2d ˆγ(u,x,y) (2)
on the setΠˆM( ˆµ, ν).
(2) πˆ =Law(U,B0,Bτ), where U has uniform law on[0,1],(Bt)is one dimensional Brownian motion,Law(U,B0) = µˆ andτ is the hitting time of the process t 7→
(U,Bt)into aleft barrier(i.e. a Borel set R⊆[0,1]×Rsuch that(u,x)∈R,v≤u implies(v,x)∈R).
(3) π( ˆˆ Γ)=1for a Borel setΓˆ ⊆[0,1]×R×Rwhich ismonotonein the sense that for all s,t,x,x0,y−,y+,y0
s<t,(s,x,y−),(s,x,y+),(t,x0,y0)∈Γ⇒y0<]y−,y+[.
(4) For all u∈[0,1], the projection ofπˆ[0,u],·,·onto the second coordinate is theshadow ofµˆ[0,u],·onto the measureν.
We add some comments to this result:
• In (1) the costc : (u,x,y) 7→ (1−u)p
1+y2 can be replaced by any positive c(u,x,y)=φ(u)ψ(y) whereφ≥0 is strictly decreasing andψ≥0 is strictly convex and the minimum over ˆΠM( ˆµ, ν) is finite. More generally the same conclusions holds for a nonnegativecwith∂(3)uyyc<0 in a weak sense. Alternative assumptions toc≥0 are thatR
|φ|dλ,R
|ψ(y)|dν <∞or thatc(x,y)≥A+Bx+Cy.
• In the setting of (2), ˆπ∈ΠˆM( ˆµ, ν) implies that the martingale (Bt∧τ)t≥0is uniformly integrable, see Proposition 5.3 below.
• To make sense of the last point, note that ifµ0(A)≤µ(A) for every Borel set4and µ C ν, then the set {ν0 : µ0 C ν0andν0 ≤ ν}is nonempty and has a smallest elementSν(µ0) with respect toC, theshadowofµ0onto the measureν(cf. [11, Lemma 4.6] /Definition 2.1 below). Intuitively speaking, among all measures ν0 ≤ νwhich are larger thanµ0in convex order,Sν(µ0) is the most concentrated one.
We call the unique element of ˆΠM( ˆµ, ν) characterized in Theorem 1.1 thelifted shadow couplingwithlift(orsource) ˆµ. Its projection onto the two last coordinates is an elementπ ofΠM(µ, ν) that we callshadow couplingofµandνassociated to the source ˆµ. Note that πis ˆπ[0,1],·,·in the terminology introduced in Equation (1).
Figure 2 illustrates Theorem 1.1. Left part: the measure ˆµhas first marginalλ(as any lifted measure ˆµ), and second marginalµas depicted on the left side of the figure. On the vertical line with abscissaustarts a 1-dimensional Brownian motion (represented on the figure with double arrows for a possible starting position (u,x) in the support of ˆµ) which will hit the setRat a position (u,y) where it is stopped. The measureπis the joint law in R2 of the starting positionxand the end positiony. The measure ˆνis the law of the end position (u,y), where the starting position is (u,x) distributed according to ˆµ(du,dx).
Middle part: the measure ˆνu,·(dy) is the law of the end position for a Brownian motion starting on the vertical line according to ˆµu,·(dx).
Right part: when considering starting positions (u0,x) with u0 ≤ u for a reference u∈[0,1], the measure ˆπrestricted toA=[0,u]×R×Rhas projections (proju0)#π|ˆA=λ|[0,u], (projx)#π|ˆA = µˆ[0,u],· and (projy)#π|ˆA = νˆ[0,u],·. Moreover (projx,y)#π|ˆA = πˆ[0,u],·,·. To (roughly) read ˆπ[0,u],·,· on the picture, we start from the horizontal line of coordinate x,
4This relation is denoted byµ0+µlater.
SHADOW COUPLINGS 5
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consider the mass of ˆµwhen disintegrated until value u (the measure ˆµ·,x|[0,u]), start the usual vertical Brownian motions until they hit the barrier and project them back to the vertical axis (that, here, can properly be calledy-axis). We have described the transition kernel implied by ˆπ[0,u],·,·, fromxdistributed according to ˆµ[0,u],·to the target (probability) measure that we may denote by ˆπ[0,u],x,·.
If the source ˆµis concentrated on the graph of a 1-1 functionT : [0,1] → Rthere is an obvious correspondence between elements ofΠM(µ, ν) and ˆΠM( ˆµ, ν). In particular the optimality property stated in Theorem 1.1 (1) then translates to optimality properties for the martingale version of the transport problem; early papers to investigate such problems include [28, 8, 18, 16, 13, 27, 15, 5, 12]. For general sources, the shadow coupling does not exhibit particular optimality properties for the martingale transport problem. However, it is characterized by a general optimality problem in the sense of Gozlan et al. [20]. We shall discuss this in Section 5 below.
Natural choices of sources lead to natural martingale couplings of shadow type. We shall be particularly interested in the cases where ˆµis either the quantile or the product coupling ofλandµ:
• The quantile coupling (or monotone rearrangement coupling, see§2.1 and§3.1) ofλandµis the unique coupling ˆµwhose support is the graph of an increasing function. Considering the corresponding lifted shadow coupling in ˆΠM( ˆµ, ν), we recover theleft-curtain couplingintroduced in [11]. We shall henceforth denote this coupling byπlc.
Notably most of the results established forπlcin [11] are a particular conse- quence of Theorem 1.1 (cf. Remark 5.6).
• Thesunset couplingπsunis based on the product source ˆµ =λ×µ, i.e. the inde- pendent coupling ofλandµ.
Note that the sources of the above martingale couplings are the two most natural cou- pling methods for elements in the space without constraintΠ(µ, ν). Looking at the measure µas the hypograph of its unit density function, we note that the curves ( ˆµ[0,u],·)u∈[0,1]and ( ˆµu,·)u∈[0,1]reminds the reader of a curtain closed from the left in the first case and from the bottom in the second case. This further motivates the namescurtain couplingandsunset coupling. The lifted versions are naturally denoted by ˆπlc,πˆsun and calledlifted curtain coupling,lifted sunset couplingrespectively.
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Figure3. From quantile to curtain; from product to sunset.
1.3. Other systematic methods to define a martingale transport. A main theme of this paper is to explore systematic methods (µ, ν)7→π∈ΠM(µ, ν) to select a special martingale transport plan for the pairs (µ, ν) withµC ν. By “special” we mean here an elementπ that can be uniquely characterized with respect to one or several mathematical features/ theories. This problem has a rich history and is still being written. Basically, the type of so- lutions may be into two categories. The most classical is the one of Skorokhod embedding, that may be summarized as follows: starting fromµstop a continuous martingale (mostly the Brownian motion distributed likeµat time 0) so that the distribution at the stopping time isν. This problem admits a variety of distinct solutions, e.g. the one obtained by Rost [45] that is particularly natural from the perspective of potential theory, the one of Root [44], which is canonical in the sense that it has minimal variance among all stopping times that solve the Skorokhod problem. Another celebrated solution based on recursion theory was given by Azema-Yor [3]. We note that the original construction of Azema-Yor is re- markably simple and explicit but restricted to the case where Brownian motion is started at a constant. In contrast the extension to non-trivial starting laws (due to Hobson [25]) is no longer explicit. We refer the reader to the comprehensive surveys of Obłl´oj [42] and Hobson [26] on the Skorokhod problem.
The constructions that are the focus of the present paper could be considered more elemen- tary in that their definitions do not rely on an (auxilliary) Brownian motion. Even though computational aspects are not the topic of this paper, we believe that shadow constructions are more adapted to concrete computations (in Section 3.1 we give some explicit formulas whenµandνare uniform measures). In the context of martingale optimal transport we also mention the recent construction of Jourdain and Marghereti [31] that is particularly useful in the context of inequalities for the usual Wasserstein distance.
1.4. Kellerer’s theorem and the sunset coupling. Kellerer’s Theorem [35] states that if a family of measures (µt)t∈R+satisfies s≤t⇒ µs C µt, there exists a martingale (Xt)t∈R+
with Law(Xt) =µt for everyt. The martingale can moreover be supposed to beMarkov and this is, as far as we are concerned, the most spectacular achievement of this theorem.
In contemporary terms (see [23]), (µt)t∈R+ is called a peacockand (Xt)t∈R+ is a Markov martingaleassociatedto this peacock.
To the best of our knowledge, all proofs of Kellerer’s theorem are based on approx- imation arguments using sequences of Markov processes. Here the obstacle is that the Markov-property isnotpreserved when passing to the limit. The key insight of Kellerer was to consider Lipschitz-Markov processes, that is Markov-processes whose transition
kernels have the following Lipschitz property: a kernelP : x7→ P(x) ∈ Pis calledLip- schitz(or more precisely 1-Lipschitz) ifW(P(x),P(x0)) ≤ |x−x0|for allx,x0(see (4) for the definition of the Wasserstein-1 distanceW). It is then not hard to see that the property of being a Lipschitz-Markov processispreserved when passing to the limit in the sense of finite-dimensional distributions.
Martingale transport plans can be seen as one step martingales and it is possible to compose several of them to define a discrete Markov martingale. The main technical step in Kellerer’s proof is therefore to show that given measures µ, ν in convex order there exists a martingale transport planπwhose transition kernelPhas the Lipschitz property, see Theorem 1.3 just bellow. Accordingly, to the concept ofLipschitz kernel Pwe add the one ofLipschitz martingale transport planπ. The following remark provides several reformulations for it.
Remark 1.2. In Kellerer’s terminology [35, 36], martingale transport plans πappear as pairs consisting of an initial measureµand a so-called dilationP, i.e. a transition kernel satisfyingR
ydP(y)= xandµP= ν. Note thatP(x)=πx,·, holdsµ-almost surely. Thus πis a martingale transport plan if there exists a version ( ˜πx,·)x∈Rof (πx,·)x∈R that satisfies W( ˜πx,·,π˜x0,·)≤ |x−x0|for everyx,x0∈R(Recall thatx→πx,·is apriori onlyµ-almost surely defined). As explained in [32] after Definition 5, this is also equivalent to the existence of A⊆Rof full measure (forµ) such thatW(πx,·, πx0,·)≤ |x−x0|is satisfied for everyx,x0∈A.
Slightly abusing notation, we will occasionally identify martingale transport plans with their kernels.
Theorem 1.3 (Kellerer’s key result to Kellerer’s theorem on Markov martingales). Let µ, ν∈ Pbe real probability measures in convex order. Then there exists a Lipschitz mar- tingale transport planπinΠM(µ, ν), i.e.π∈ΠM(µ, ν)such that
W(πx,·, πx0,·)≤ |x−x0| (3)
holds for any x,x0in a set ofµ-full measure.
Let us add that in higher dimensionsd≥2 Lipschitz martingale transport plans may not exist among the elements ofΠM(µ, ν) for some pairsµ ≤ν(see [32, Proposition 1]). As Lipschitz kernels and their variants are the only known methods for proving the Kellerer theorem, still to our knowledge, it is an open problem whether this theorem holds in di- mensions greater than or equal to two.
While the various extremal martingale couplings constructed in [28, 11, 27, 15, 46] do not have the Lipschitz property, we shall see that the sunset coupling has the Lipschitz property. Moreover, as we will see in Section 4 it connects to Kellerer’s original proof (in [35]) of the existence of Lipschitz kernels.
2. Preparations and construction
2.1. Concepts related to the martingale transport problem. We consider the spaceM of positive measures onRwith finite first moments. The subspace of probability measures with finite expectations is denoted byP. In higher dimensions we denote the corresponding spaces byM(Rd) andP(Rd). Forµ, ν∈ M, the Wasserstein-1 distance is defined by
W(µ, ν)= sup
f∈Lip(1)
Z
fdµ− Z
fdν (4)
and endows (P,W) withT1, the usual topology for probability measures with finite first moments5. In (4), the supremum is taken over all 1-Lipschitz functions f : R → R. We also considerW(with the same definition) on the subspacemP={µ∈ M|µ(R)=m} ⊆ M of measures of massm.
According to the Kantorovich duality theorem, an alternative definition in the caseµ, ν∈ Pis
W(µ, ν)= inf
(Ω,X,Y)E(|Y−X|) (5)
whereX,Y : (Ω,F,P)→ Rare random variables of lawsµandν. The infimum is taken over all joint laws (X,Y), the probability space (Ω,F,P) being part of the minimization problem. Without loss of generality (Ω,F,P) can be assumed to be ([0,1],B, λ) whereλis the Lebesgue measure andBtheσ-algebra of Borel sets on [0,1]. On this probability space the quantile functionsGµ andGν in fact realize a minimizing coupling, calledquantile coupling. In particular one hasW(µ, ν)=R1
0 |Gν−Gµ|. Recall that thequantile functionof θis defined byGθ(u) =inf{x∈ [−∞,∞] : θ((−∞,x]) ≥u}as the generalized inverse of thecumulative distribution function Fθ : x∈R 7→ θ((−∞,x]). Finally, an application of Fubini’s theorem yieldsW(µ, ν)=R
R|Fν−Fµ|.
A special choice of a 1-Lipschitz function in (4) is the functionft:x∈R→ |x−t| ∈R. Therefore ifµn→µinM, the sequence of functionsuµn:t7→R
ft(x) dµn(x) converges to uµpointwise. The converse statement also holds if all the measures have the same mass and barycenter (see [24, Proposition 2.3] or [11, Proposition 4.2]). Forµ∈ M, the functionuµ
is usually called thepotential functionofµ. Note thatuµis a convex function with (weak) second derivative is 2µ.
2.2. Bijection between curves, primitive curves, and lifted measures. We elaborate on the equivalent expressions of the lifted measures introduced in Subsection 1.2. In short, we are representing the same mathematical object in three ways: we consider the measure ˆθ that may be ˆµ∈ P([0,1]×R2) or ˆπ∈ P([0,1]×R2), the almost surely defined disintegration (ˆθu,·)u∈[0,1], and the primitive curve (ˆθ[0,u],·)u∈[0,1]. We first recall the integrability conditions.
The lifted measure ˆθis a probability measure on [0,1]×Rd(whered=1 ord=2) such that θ( ˆˆ ρ) <+∞is finite whereρ: x7→ kxkRd and ˆρ(u,x)=ρ(x). This integrability condition corresponds to ˆθ[0,1],·(ρ)<+∞for the primitive curve (ˆθ[0,u],·)u∈[0,1]andR1
0 θˆu,·(ρ) du<+∞ for (ˆθu,·)u∈[0,1]. The marginal condition asserts that ˆθ∈Π(λ, θ) for someθ∈ P(Rd). In terms of the primitive curve this can be expressed by asserting that ˆθ[0,1],·=θand ˆθ[0,u],·(R)=u.
The equivalent condition on ( ˆθu,·)u∈[0,1]is thatλ-almost surely ˆθu,· ∈ Pandθ=R1 0 θˆu,·du.
Note that from a probabilistic point of view, if (U,X) is a random vector of law ˆθwith U ∼λ, the other representations are given by (u×Law(X|U ≤u))u∈[0,1]and (Law(X|U= u))u∈[0,1]. Finally the object that we will ultimately be most interested in is not the source ˆθ butθ:=θˆ[0,1],·=Law(X) (in particular forθ=πwhere the measure in onR2).
In what follows we explain that the derivative of the primitive curve ( ˆθ[0,u],·)u∈[0,1]can be considered with respect toT1, the weak topology with respect to continuous functions which have at most linear growth. Let us start with ˆθ ∈ Π(λ, θ). We disintegrate the measure with respect to the first marginal and obtain an a.s. uniquely determined family (ˆθu,·)u∈[0,1]such that for almost everyu, ˆθu,·∈ P(Rd). (We can assume that the measure is
5A sequence a signed measure (µn)nconverges toµifR
φdµn→R
φdµfor every continuous functionφwith growth at most linear.
zero for the other parameters.) Define ˆθ[0,u],·foru∈[0,1] by θˆ[0,u],·(A)=θ([0,ˆ u]×A)=Ru
0 θˆs,·(A) ds
for A ⊆Rd Borel. Given a function f : x∈ Rd → Rwith f(x)/(1+kxk) bounded, the functions7→θˆs,·(f) is measurable and inL1([0,1]). Hence at almost every timeu∈[0,1]
the function t 7→ θˆ[0,u],·(f) = Ru
0 θˆt,·(f) dt is differentiable with derivative ˆθu,·(f). It is important that the setL⊆[0,1] of times at which the derivative exists for allf is a Borel set of full measure, as we will verify in the next paragraph. Before establishing this claim, note that this permits us to define a canonical disintegration ( ˜ˆθu,·)u∈[0,1]: we define the measure ˜ˆθu,·as the derivative ifu∈L, and zero otherwise.
We turn now to the claim: letXbe a countable set of functions which is dense in the spaceCc(Rd) of continuous functions with compact support and letX+beX ∪ {ρ}where ρ(x) = kxkRd. LetL ⊆ [0,1] be the set such that at any timeu ∈ L, ˆθu,·is a probability measure andu 7→ θˆ[0,u],·(f) has derivative ˆθu,·(f) for any f ∈ X+and note thatLhas full mass. Then, as an incrementh goes to zero the measureh−1(ˆθ[0,u+h],·−θˆ[0,u],·) weakly converges to ˆθu,·and asρis a continuous function with linear growth, convergences holds also inT1, cf. [48, Theorem 7.12]. ThusLis a set of differentiation for any function with finite first moment.
2.3. General description of the construction. In what follows we shortly explain the systematic scheme to define ˆπ∈ΠˆM( ˆµ, ν) when the marginals ˆµ ∈Π(λ, µ) andν∈ Pare given. Recall that the resulting couplingπ = πˆ[0,1],·,· = (projx,y)#πˆ whose marginals are µ=µˆ[0,1],·andνfits more naturally to the theory of optimal transportation than the lifted coupling ˆπ.
Represent ˆµ in the form ( ˆµ[0,u],·)u∈[0,1]. The first canonical operation, calledshadow projectiononν, consists in building the curve (ˆν[0,u],·)u∈[0,1] from it (see Definition 2.1).
Hence the construction is complete if on a setL⊆[0,1] of differentiation (of full measure) of ( ˆµ[0,u],·)uand (ˆν[0,u],·)u we know for everyu ∈ Lhow to canonically choose a joint law πˆu,·,·of the derivatives ˆµu,·and ˆνu,·. In our situation, the martingale constraint on the one side and the fact that we use the convex shadow projection of Definition 2.1 on the other side will make this choice uniquely determined. As we will see ˆπu,·,·is related to Kellerer’s hitting projection (see Definition 2.6). We will thus obtain ( ˆπu,·)u∈[0,1] and equivalently ( ˆπ[0,u],·)u∈[0,1]and ˆπ. This construction will be carried out in detail in the proof of Theorem 2.9.
2.4. Order relations, convex shadow and alternative shadows. OnMwe writeµC,+ νif there existsη ∈ Mwithµ C ηandη + ν. Here+meansη(A) ≤ν(A) for every Borel setA. The orderC,+can also be characterized by assertingµ(f) ≤ν(f) for every convex positive function f. We also introduce the stochastic orderµ sto νthat holds if µ(f) ≤ ν(f) for every integrable increasing function. This is equivalent toGµ ≤ Gν or Fµ ≥ Fν. See [33] for more details on these order relations in the context of martingale optimal transport.
Definition-Proposition 2.1(Definition of the convex shadow). Ifµandνare positive real measures andµC,+νthere exists a unique measureη∈ Msuch that
• µCη
• η+ν
• Ifη0satisfies the two first conditions (i.e.µCη0+ν), one hasηC η0.
This measureηis called theconvex shadowor simplyshadowofµinνand we denote it bySν(µ).
In the two next examples we illustrate the general description of Subsection 2.3 with two alternative types of shadow projections that may at the same time be more intuitive and be useful for future comparisons in the paper.
Example2.2. In this example we illustrate how the quantile coupling ofµandνcan be defined going along the construction lines given in§2.3. Let us first recall that the shortest formula for this quantile coupling is (Gµ,Gν)#λ. Similarly note that, asGλ = Id, the quantile coupling ˆµ = (Id,Gµ)#λis actually the source for the left-curtain coupling. As we will see ˆµis also the source for our alternative definition of (Gµ,Gν)#λas a shadow coupling where a “fake” shadow replaces the (convex) shadow of Definition 2.1).
We first writeµas the superposition of Dirac measures in the following way:
µ=Z 1 0
µˆu,·du
with ˆµu,· = δGµ(u) for everyu ∈ [0,1]. This corresponds to ˆµ[0,u],· = (Gµ)#λ|[0,u] and, as mentioned above, ˆµ=(Id,Gµ)#µ. We perform the same decomposition forν:
ν=Z 1 0
νˆu,·du,
where ˆνu,· =δGν(u) for everyu∈[0,1]. The quantile transport plan is now obtained in the form
π=Z 1 0
πˆu,·,·du
since ˆπu,·,·=δ(Gµ,Gν)(u)is the unique transport plan inΠ( ˆµu,·,νˆu,·).
To see this presentation as an example of shadow coupling we have firstly to see ˆν[0,u],·
as the shadow of ˆµ[0,u],·, with a “fake shadow” projection adapted to the present example.
Secondly the curve (ˆνu,·)uis derived from (ˆν[0,u],·)u. To define the fake shadow in a similar fashion as Definition 2.1 we simply say that ˆν[0,u],·is the measureη + νwith the same mass as ˆµ[0,u],·such that any otherη0+νof massusatisfiesηstoη0. Of course ˆν[0,u],·is simply (Gν)#λ|[0,u]the restriction ofνto the quantiles of level smaller thatuup to the fact that we may have to ‘break’ atoms atGν(u).
Example2.3. Let us go further than Example 2.2: in preparation to§3.1.3 and§3.2, let us notice that ifµsto ν, the “fake shadow” (Gν)#λ|[0,u] is also greater than (Gµ)#λ|[0,u]in stochastic order so that under the conditionµsto ν, the fake shadow is also a “stochastic shadow” in the sense we are going to provide here (compare with Definition 2.1 of the (convex) “true” shadows):
Letµandνbe positive real measures and letEνµbe the set of measuresη∈ Msuch that
• µstoη
• η+ν
AssumeEµν ,∅. Then there existsη ∈ Eµνwithη sto η0for everyη0 ∈ Eνµ. We call this measure thestochastic shadowofµinν.
Based on this definition under the conditionµstoνthe coupling of Example 2.2 is truly astochasticshadow coupling adapted to the stochastic order, exactly as shadow couplings of the present paper are in factconvexshadow couplings adapted to the convex order.
Convex shadows are sometimes difficult to determine. An important fact is that they have the smallest variance among the admissible measuresη0. Indeed, η C η0 implies R xdη=R
xdη0andR
x2dη≤R
x2dη0with equality if and only ifη=η0orR
x2dη= +∞.
Example2.4 (Shadow of an atom, Example 4.7 in [11]). Letδbe an atom of massαat a pointx. Assume thatδ C,+ ν. ThenSν(δ) is the restriction ofνbetween two quantiles, more precisely it isν0=(Gν)#λ]s;s0[wheres0−s=αand the barycenter ofν0isx.
The following result is one of the most important on the structure of shadows (Theorem 4.8 of [11]).
Proposition 2.5(Structure of shadows). Letγ1, γ2 and νbe elements ofMand assume γ1+γ2C,+ν. Then, we haveγ2C,+ν−Sν(γ1)and
Sν(γ1+γ2)=Sν(γ1)+Sν−Sν(γ1)(γ2).
An important consequence is that if ( ˆµ[0,u],·)u∈[0,1]is a primitive curve and ˆµ[0,1],· C ν, then the curve (ˆν[0,u],·)u∈[0,1]satisfies ˆν[0,u],·(R) = uand usingγ1 = µˆ[0,u],·andγ1 +γ2 = µˆ[0,u0],·we obtain ˆν[0,u],·+νˆ[0,u0],·for everyu≤u0. Hence (ˆν[0,u],·)uis a primitive curve. In the next subsection we consider the derivatives of such curves (ˆν[0,u],·)u and introduce for this the proper infinitesimal version of the shadow projection.
2.5. Hitting projection. We denote byF(R) the space of closed subsets ofR, andIthe subspace of those elementsT ∈ F(R) such that supT =−infT = +∞. The spaceF(R) is endowed with a natural topology presented in [36,§2.1], i.e. the coarsest topology such thatF∈ F(R)7→d(x,F) is continuous for everyx∈R.
Definition 2.6(Hitting projection of measure in/to a set). LetT be an element ofI. For everyx∈R, letx−T =sup(T∩(−∞,x]) andx+T =inf(T ∩[x,+∞)). The Kellerer dilation ([36, Definition 16]) is given by
PT(x,·)=
δx ifx∈T;
(xT+−x−T)−1[(x+T−x)δx−T +(x−x−T)δx+T] otherwise.
Hence ifµ ∈ P, the hitting projection ofµinT isν =µPT and the hitting coupling ofµ andνis given byπ(A×B)=R
APT(x,B) dµ(x); we shall abbreviate this byπ=µ(id×PT).
Note that ifT is not an element ofIbut supp(µ) ⊆ [infT,supT], the kernelPT still makes senseµ-almost surely.
Proposition 2.7. Forµˆ ∈Π(λ, µ)andµC ν, let u7→µˆ[0,u],·have right derivativeµˆu0,·at u0 and letνˆ[0,u],·be Sν( ˆµ[0,u],·). Then(ˆν[0,u],·)has a right derivative at u0. This derivative is given byµˆu0,·PT and supp( ˆµu0,·) ⊆ [infT,supT]where T is the support of νˆ]u0,1],· := ν−νˆ[0,u0],·.
Proof. Considerh−1(ˆν[0,u0+h],·−ˆν[0,u0],·)=h−1(Sν( ˆµ[0,u0+h],·)−Sν( ˆµ[0,u0],·))=:σh. According to Proposition 2.5,σhequals
h−1Sνˆ]u0,1],·( ˆµ]u0,u0+h],·),
where we set ˆµ]u,v],· = µˆ[0,v],·−µˆ[0,u],·. However, we know thath−1( ˆµ[0,u0+h],·−µˆ[0,u0],·) = h−1µˆ]u0,u0+h],·tends to ˆµu0,·ash↓ 0. An easy scaling analysis shows thatσhcan in fact be written as
σh=Sh−1νˆ]u0,1],·(h−1µˆ]u0,u0+h],·).
Formally, in the limit, we are considering the shadow projection of ˆµu0,·into the infinite measure∞ ·νˆ]u0,1],·. Here it is easy to believe that the support of ˆν]u0,1],·, denoted byT in
our statement plays the leading role. We are indeed left with the proof thatσhconverges to ˆµu0,·PT ash ↓0 (and of supp( ˆµu0,·)⊆[infT,supT]). SinceT1is metric, it is enough to replacehby a sequence (hn)nand apply the postponed Lemma 2.8 to the sequence (Hn)n
given throughHn :=h−1n , the sequence of probability measuresh−1n µˆ]u0,u0+hn],·forηn, and the measure ˆν]u0,1],·forυ. Note that this lemma delivers at the same time the convergence
and the claimed inclusion.
Lemma 2.8. Let(Hn)n be a sequence of positive numbers tending to infinity,(ηn)na se- quence of probability measures (with finite first moment) converging to ηinPand υ a positive measure. Assumeηn C,+ Hnυfor every n≥1. Then, denoting by T the support supp(υ)ofυit holdssupp(η)⊆ [infT,supT](replacing the bounds by±∞if necessary) and SHnυ(ηn)→ηPT inP.
Proof. Note first that due to the convex order relationηn C,+ Hnυwe have supp(ηn) ⊆ supp(υ) ⊆ [infT,supT] so that ηn([infT,supT]) = 1, for every n. Lettingn tend to infinity we find supp(η)⊆[infT,supT] as well.
1. Let us prove the result ifηn = δxfor everyn ∈ N. We prove in fact a somewhat stronger statement: ifγnhas mass less than or equal to one andγn + Hnυ, the sequence SHnυ−γn(δx) converges toηPT. Moreoverx∈T◦,x∈Tcorx∈∂T. In either of these cases the result easily follows from Example 2.4.
2. We assume now that for everyn∈N, we haveηn =η=Pn
k=1akδxk. We proceed by induction. The initial stepn =1 has just been established. We assume the statement for n−1≥1 and prove it fornby using the decompositionη=η0+anδnwhereη0=Pn−1
k=1akδxk. By Proposition 2.5 we have
SHnυ(η0+anδxn)=SHnυ(η0)+Sβn(anδxn)
whereβn=Hnυ−SHnυ(η0). Each of the two terms converges to the Kellerer projection of η0respectivelyanδn ontoT. Note that for the second projection we used the full strength of the statement proved in 1.
3. A general measureη can be approximated using a convex combination of Dirac massesηkwithηk C ηand such thatηk → η[33, Point 3. in the proof of Proposition 2.34]. We have
W(SHnυ(ηk),SHnυ(η))≤W(ηk, η).
This tends to zero uniformly innas kgoes to infinity. ButSHnυ(ηk) → ηkPT asntends to infinity and the composition withPT is continuous (cf. [36, Section 2.2], this can be understood easily from the action ofPT on the potential functions). Hence we obtain the result for any constant sequenceηn=η.
4. Ifηnis a non-constant sequence
W(SHnυ(ηn), ηPT)≤W(SHnυ(ηn),SHnυ(η))+W(SHnυ(η), ηPT),
which tends to zero as required ([33, Proposition 2.34]).
Based on the preparations above we can now rigorously introduce thelifted shadow couplings.
Theorem 2.9 (Existence, construction, and uniqueness of the lifted shadow coupling).
Letµand νbe elements of Pwithµ C ν. Let µˆ be an element ofΠ(λ, µ). Then there exists a unique elementπˆ ∈ΠˆM( ˆµ, ν), the lifted shadow coupling ofµˆ andν, such that for every u ∈ [0,1], the marginals ofπˆ[0,u],·,·areµˆ[0,u],·and its shadow projection Sν( ˆµ[0,u],·).
Denoting this second marginal byνˆ[0,u],·and the derivative of(ˆν[0,u],·)u∈[0,1]byνˆu,·, we have moreoverνˆu,·=µˆu,·PT(u)where T(u)is the support ofνˆ]u,1],,·:=νˆ[0,1],·−νˆ[0,u],·.
Note that Theorem 2.9 implies in particular that there exists a unique ˆπsatisfying The- orem 1.1 (4).
Proof of Theorem 2.9. Let ˆν[0,u],·=Sν( ˆµ[0,u],·) be as in the statement and let ( ˆµu,·)u∈[0,1]and (ˆνu,·)u∈[0,1]be the derivative curves. According to Proposition 2.7, ˆνu,·can for almost every u ∈ [0,1] be identified with the Kellerer projection of ˆµu,·inT(u) := supp(ˆν]u,1],·) where νˆ]u,1],· =ν−νˆ]0,u],·. Hence we can define ˆπu,· = µˆu,·(id×PT(u)) ∈ ΠM( ˆµu,·,νˆu,·) for almost everyu, and then the corresponding ˆπand ˆπ[0,u],·=Ru
0 πˆt,·dt.
Conversely, let ˆπand ( ˆπ[0,u],·)u∈[0,1] be as in the statement. The curve has marginals µˆ[0,u],·and ˆν[0,u],·=Sν( ˆµ[0,u],·) so that one can apply Proposition 2.7. At pointsuwhere the derivatives ˆπu,·, ˆµu,·and ˆνu,·exist, we have ˆπu,·,·∈ΠM( ˆµu,·,νˆu,·) and ˆνu,·=µˆu,·PT(u)as in the last paragraph, proving the uniqueness part of the statement.
3. Generating martingale couplings from particular liftsµˆ
3.1. Examples of shadow couplings. We now further discuss three special lifts ˆµ ∈ Π(λ, µ) ofµ(with their associated primitive families ( ˆµ[0,u],·)u∈[0,1]) to give rise to particular shadow couplings. The first one is the monotone coupling ofλandµand the second the independent couplingλ×µ.
• µˆ[0,u],· = (Gµ)#λ|[0,u] (corresponding to the left-curtain coupling ; # is the push- forward operator);
• µˆ[0,u],·=u·µ(corresponding to the sunset coupling);
• µˆ[0,u],·=Sµ(u.δm) wherem=R
xdµ(x) (corresponding to the middle curtain cou- pling). Recall Example 2.4 for the shadow of an atom.
Figure 4 illustrate the corresponding Skorokhod embeddings whenµandνare two uniform laws µ = 1[0,1]dx andν = 131[−1,2]dx. Based on Remark 3.1 below he corresponding equations for the shadows are summerized in Table 1, while Table 2 presents the equations of the sources of the particular shadow couplings.
µˆ[0,u],· µˆ[0,u],· νˆ[0,u],·
πlc µ|(−∞,Gµ(u))+(u−Fµ◦Gµ(u))δGµ(u) 1[0,u]dx 131[−u,2u]dy
πsun u·µ u·1[0,1]dx u·1[0,1]dy or 131[1−2u,1+2u]dy
πmid µ|(f(u),g(u))+c(u)δf(u)+d(u)δg(u) 1[1−u
2 ,1+u2 ]dx 1
31[1−2u,1+2u]dy
Table 1. General expression of ˆµ[0,u],· and expressions of ˆµ[0,u],· and ˆν[0,u],· for uniform measures on [0,1] and [−1,2].
µˆ µˆu,· µˆ[0,u],·
πlc (Id×Gµ)#λ δGµ(u) µ|(−∞,Gµ(u))+(u−Fµ◦Gµ(u))δGµ(u)
πsun λ×µ µ u·µ
πmid a·(Id×f)#λ+b·(Id×g)#λ a(u)δf(u)+b(u)δg(u) µ|(f(u),g(u))+c(u)δf(u)+d(u)δg(u)
Table2. General expressions of ˆµ, ˆµu,·and ˆµ[0,u],·for the the main lift methods.
Remark 3.1 (Shadow of a uniform measure in a uniform measure). In the following we illustrate our three examples with the fixed pairµ=1[0,1]dxandν= 131[−1,2]dx. Therefore it is useful to clearly state that two measuresη=m1[a,b]andη0=m01[a0,b0]satisfiyηC,+η0 if and only ifa+2b±mm0
b−a
2 ∈[a0,b0]. In this case the shadow ism01[a+b2 −mm0b−a
2,a+b2 +mm0b−a 2 ], that is the measure with the same mass and barycenter asη, i.e. the restriction ofη0to an interval.
µ ν
λ u x,y
µ ν
λ
µ ν
λ
Figure 4. i. Left-curtain coupling of uniform measures. The mapGµ : [0,1] → Ris identity and ˆµis uniform on this graph, i.e. on the segment of ends (0,0) and (1,1). ii.
Sunset coupling of uniform measures. ˆµ is uniform on the square [0,1]2. iii. Middle- curtain coupling of uniform measures. ˆµis uniform on the segment with ends (0,1/2) and (1,1) and (1,0). These segments are the graphs of f andg. The measuresµandνsatisfy µDC ν(an order relation defined in§3.1.3).
3.1.1. The left- and right-curtain couplings. This case corresponds to the construction given in [11], even though the construction described there appears slightly different. In fact foru =Fµ(x) the three marginals of ˆπ|[0,1]×]−∞,x]×Rareλ|[0,u],µ|]−∞,x] andSν(µ]−∞,x]) so that for everyx∈ R,π|]−∞,x]×Rhas marginalsµ|]−∞,x] andSν(µ]−∞,x]). In [11] this was used to define the left-curtain couplingπlc.
In an entirely symmetric fashion we can define the right-curtain coupling through ˆµ[0,u],·= (Gµ)#λ|[1−u,1]foru∈[0,1].
3.1.2. The sunset coupling. In this case we have ˆµu,·=µfor almost everyu ∈[0,1] and νˆu,·=µPT(u)whereT(u)=supp(ν−Sν(u·µ)). Hence
ν=R1
0 µPT(u)du
Of courseT(u) can be replaced byT∗(u)=T(u)∪(]− ∞,infν]∪[supν,+∞[)∈ I. We refer the reader to Section 4 for further explanations.
3.1.3. The middle-curtain coupling. For diatomic probability measuresµ=aδf+bδgand ν=a0δf0+b0δg0the relationµC νholds if and only ifa f+bg=a0f0+b0g0(same mean) and [a,b]⊆[a0,b0]. In this case, as one can easily check, there exists a unique martingale transport plan inΠM(µ, ν). It is
πmid(µ, ν)= 1 g0−f0
[(g0−f)δf,f0+(f−f0)δf,g0]+[(g0−g)δg,f0+(g−f0)δg,g0] . (6)
This basic fact permits us to construct simple martingale couplings for a special class of ordered pairsµ C ν in a parallel way to the way the quantile coupling is defined in Example 2.3, underµstoν. This martingale coupling is defined in [32] without assigning a particular name. Here we shall call itmiddle-curtain coupling. In our case, diatomic probability measures with meanmreplace the atoms of Example 2.2. There is a almost
surely unique family of diatomic measures ( ˆµu,·)u∈[0,1]such thatR1
0( ˆµu,·)u∈[0,1]du = µand µˆu,· C µˆu0,· for everyu < u0. One (maybe too) elaborate way to see this is to define µˆ[0,u],· = Sµ(u·δm). We recall (see Example 2.4) that this measure is the restriction of µto an interval (with more or less mass at the ends of the interval) of massuand mean m. Consequently the derivative ˆµu,·is the diatomic probability measure announced above.
Note that this corresponds to ˆµu,·=a(u)δf(u)+b(u)δg(u)where
• (a+b)(u)=1;
• a(u)f(u)+b(u)g(u)=m;
• f is decreasing andg is increasing (we can moreover assume that they are right continuous, as quantile functions are).
In the family ( ˆµ[0,u],·)u∈[0,1], the mass appears from the middle which explains the name
‘middle-curtain coupling’.
It may happen that
µˆu,·C νˆu,· for everyu∈[0,1]
(7)
where we denote by (ˆνu,·)u∈[0,1]and ˆν[0,u],·the measures defined in the same way as ˆµu,·and µˆ[0,u],·. Integrating the martingales couplings of (6) gives a martingale coupling
πmid(µ, ν)=Z 1 0
πmid( ˆµu,·,νˆu,·) du (8)
ofµandν, so that, in particular,µC ν. However, contrary to what happens in Example 2.3 for the stochastic order and the quantile coupling,µC νis not equivalent to (7). We call the latter necessary (but not sufficient) condition thediatomic convex orderand denote it byµ DC ν. Due to the non-uniqueness of the derivative curve, a condition that better define this order is probablySµ(u·δm) C Sν(u·δm) for everyu ≤ 1. It can be easily proved to be equivalent.
As the notationπmid suggests, the coupling in (6) and (8) are the middle-curtain cou- plings. The second formula generalises the first one. Let us see that the middle-curtain coupling is a shadow coupling. We already expressed ˆµ[0,u],·=Sµ(u·δm). As we assumed µDC ν, we have
u·δmCSµ(u·δm)
| {z }
µˆ[0,u],·
CSν(u·δm)
| {z }
νˆ[0,u],·
≤ν
so thatSν( ˆµ[0,u],·) =νˆ[0,u],·is not difficult to derive from Definition 2.1. Therefore, under µDC νthe coupling (8) is the shadow coupling of source ˆµ. IfµC νbut notµDC ν, equation (8) has no meaning because ˆµu,·Cνˆu,·is not satisfied for everyu∈[0,1], so that πmid( ˆµu,·,νˆu,·) does not exist. However, the shadow coupling is still defined, which permits to naturally extend the idea of a middle-curtain coupling originally intoduced in [32] to any ordered pairµ C ν. Note that in this case,Sν( ˆµ[0,u],·) ,Sν(u·δm) at least for one u∈[0,1].
Remark 3.2. Finally, note that for every family of probability measures (µt)t∈T indexed by any partial orderT so that s ≤t impliesµs DC µt, there exists a martingale (Xt)t∈T
such that Law(Xs,Xt) is the middle-curtain coupling ofµsandµtfor everys<t∈T, [32, Theorem 4].
3.2. Comparison with the stochastic order; quantile and independent couplings. We have seen in Examples 2.2 and 2.3 that with the quantile coupling as source ˆµ, and modi- fying the definition of shadow (fake shadow or stochastic shadow respectively), the related shadow couplings are in both examples the quantile coupling ofµandν. In this sense the
