Instability of Martingale optimal transport in dimension d \ge 2
aa r X i v : . [ m a t h . O C ] J a n INSTABILITY OF MARTINGALE OPTIMAL TRANSPORTIN DIMENSION d ≥ MARTIN BRÜCKERHOFF NICOLAS JUILLET
Abstract.
Stability of the value function and the set of minimizers w.r.t. the givendata is a desirable feature of optimal transport problems. For the classical Kan-torovich transport problem, stability is satisfied under mild assumptions and ingeneral frameworks such as the one of Polish spaces. However, for the martingaletransport problem several works based on different strategies established stabilityresults for R only. We show that the restriction to dimension d = 1 is not accidentalby presenting a sequence of marginal distributions on R for which the martingaleoptimal transport problem is neither stable w.r.t. the value nor the set of minimiz-ers. Our construction adapts to any dimension d ≥ . For d ≥ it also provides acontradiction to the martingale Wasserstein inequality established by Jourdain andMargheriti in d = 1 . Keywords: Mathematics Subject Classification (2010): Introduction
For two probability measures µ and ν on R d let Π( µ, ν ) denote the set of all couplingsbetween µ and ν , i.e. the set of all probability measures on R d × R d which have marginaldistributions µ and ν . Let c : R d × R d → R be measurable and integrable with respectto the elements of Π( µ, ν ) . The classical optimal transport problem is given by(OT) V c ( µ, ν ) = inf π ∈ Π( µ,ν ) Z R d × R d c ( x, y ) d π ( x, y ) . For the cost function c ( x, y ) := k y − x k (where k · k is the Euclidean norm) the -Wasserstein distance W := V c is a metric on P ( R d ) , the space of probability measures µ that satisfy R R d k x k dµ ( x ) < ∞ .Two probability measures µ, ν ∈ P ( R d ) are said to be in convex order, denoted by µ ≤ c ν , if R R d ϕ d µ ≤ R R d ϕ d ν for all convex functions ϕ ∈ L ( ν ) . If µ ≤ c ν , Strassen’stheorem yields that there exists at least one martingale coupling between µ and ν . Amartingale coupling is a coupling π ∈ Π( µ, ν ) for which there exists a disintegration ( π x ) x ∈ R d such that(1.1) Z R y d π x ( y ) = x for µ -a.e. x ∈ R d . If µ ≤ c ν , the martingale optimal transport problem is given by(MOT) V Mc ( µ, ν ) = inf π ∈ Π M ( µ,ν ) Z R d × R d c ( x, y ) d π ( x, y ) where Π M ( µ, ν ) denotes the set of all martingale couplings between µ and ν . Date : January 19, 2021.MB is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foun-dation) under Germany’s Excellence Strategy EXC 2044 –390685587, Mathematics Münster:Dynamics–Geometry–Structure.
Stability in d = 1 . Let us recall two reasons why stability results are crucial froman applied perspective. Firstly, they enable the strategy of approximating the problemby a discretized problem or by any other that can rapidly be solved computationally(cf. [1, 12]). Secondly, any application to noisy data would require stability for theresults to be meaningful. In relation with (MOT), this discussion is motivated by itsconnection to (robust) mathematical finance (cf. [3, 10]).Let µ, ν ∈ P ( R ) with µ ≤ c ν , and ( µ n ) n ∈ N and ( ν n ) n ∈ N be sequences of prob-ability measures on R with finite first moment such that lim n →∞ W ( µ n , µ ) = 0 , lim n →∞ W ( ν n , ν ) = 0 and µ n ≤ c ν n for all n ∈ N . The following stability resultsare available:(S1) Accumulation Points of Minimizers:
Let c be a continuous cost function whichis sufficiently integrable (e.g. | c ( x, y ) | ≤ A (1 + | x | + | y | ) ) and let π n be a min-imizer of the (MOT) problem between µ n and ν n for all n ∈ N . Any weakaccumulation point of ( π n ) n ∈ N is a minimizer of (MOT) between µ and ν .(S2) Continuity of the Value:
Let c be a continuous cost function which is sufficientlyintegrable (e.g. | c ( x, y ) | ≤ A (1 + | x | + | y | ) ). There holds lim n →∞ V Mc ( µ n , ν n ) = V Mc ( µ, ν ) . (S3) Approximation:
For all π ∈ Π M ( µ, ν ) there exists a sequence ( π n ) n ∈ N of mar-tingale couplings between µ n and ν n that converges weakly to π .This constitutes the heart of the theory of stability recently consolidated for themartingale transport problem on the real line. Before we go more into the details ofthe literature let us stress that with (S3) any minimizer can be approximated by asequence of martingale transport with prescribed marginals. Therefore, under mildassumptions (S3) implies (S2). Moreover, due to the tightness of S n ∈ N Π M ( µ n , ν n ) ,(S2) implies (S1).Early versions of (S1) and (S2) for special classes of cost-functions were obtainedby Juillet [14] and later by Guo and Obloj [9]. The general version of (S1) and (S2)was first shown by Backhoff-Veraguas and Pammer [2, Theorem 1.1, Corollary 1.2]and Wiesel [19, Theorem 2.9]. Only very recently, Beiglböck, Jourdain, Margheritiand Pammer [4] have proven (S3). We want to stress that (S1), (S2) and (S3) aregiven in a minimal formulation and that in the articles some aspects of the results arenotably stronger. For instance, the cost function c in (S1) and (S2) can be replaced bya uniformly converging sequence ( c n ) n ∈ N [2]. Moreover, it is an important achievementthat on top of weak convergence we have convergence w.r.t. (an extension of) theadapted Wasserstein metric for the approximation in (S3) [4] and for the convergencein (S1) [5], see also [19]. Finally, these stability results also hold for weak martingaleoptimal transport which is an extension of (MOT) w.r.t. the structure of the costfunction (cf. [5, Theorem 2.6]). For further details we invite the interested reader todirectly consult the articles.The martingale Wasserstein inequality introduced by Jourdain and Margheriti in[11, Theorem 2.12] belongs also to the context of stability and approximation and itappears for example as the important last step in the proof of (S3) in [4]. In dimension d = 1 there exists a constant C > independent of µ and ν such that(MWI) M ( µ, ν ) ≤ C W ( µ, ν ) . where M ( µ, ν ) is the value of the (MOT) problem between µ and ν w.r.t. the costfunction k x − y k . Moreover, they proved that C = 2 is sharp. For their proof Jourdain NSTABILITY OF MARTINGALE OPTIMAL TRANSPORT 3 y xm = n = 2 y xm = n = 3 Figure 1.
The construction for m = n = 2 and m = n = 3 . The redcircles indicate the Dirac measures of µ m each with mass m and theblue circles indicate the Dirac measures of ν m,n each with mass m .and Margheriti introduce a family of martingale couplings π ∈ Π M ( µ, ν ) that satisfy R R × R | x − y | d π ( x, y ) ≤ W ( µ, ν ) (including the particularly notable inverse transformmartingale coupling). Instability in d ≥ . The stability of (OT) (and its extension to weak optimal trans-port [5, Theorem 2.5]) is independent of the dimension. However, Beiglböck et al. hadto restrict their stability theorem for (weak) (MOT) in the critical step to dimension d = 1 (cf. [5, Theorem 2.6 (b’)]). Similarly, in dimension d ≥ , Jourdain and Margher-iti could only extend the martingale Wasserstein inequality for product measures andfor measures in relation through a homothetic transformation, see [11, Section 3]. Thedifficulties in expanding these stability results to higher dimensions are not of technicalnature but a consequence of instability of (MOT) in higher dimensions without furtherassumptions.In the following we construct a sequence of probability measures on R for which(S1), (S2) and (S3) do not hold. Moreover, we provide an example that shows thatthe inequality (MWI) does not hold in dimension d = 2 for any fixed constant C > without further assumptions. Since one can embed this example into R d for any d ≥ by the map ι : ( x, y ) ( x, y, , ..., , these results also fail in any higher dimension.We denote by P θ the one step probability kernel of the simple random walk alongthe line l θ that makes an angle θ ∈ (cid:2) , π (cid:3) with the x -axis. More precisely: P θ : R ∋ ( x , x ) (cid:0) δ ( x ,x )+(cos( θ ) , sin( θ )) + δ ( x ,x ) − (cos( θ ) , sin( θ )) (cid:1) ∈ P ( R ) . For m, n ∈ N ≥ we define two probability measures on R by µ m := m X i =1 m δ ( i, and ν m,n := µ m P π n where µ m P π n denotes the application of the kernel P π n to µ m . Figure 1 illustrates ( µ , ν , ) and ( µ , ν , ) .Since for any convex function ϕ : R → R Jensen’s inequality yields Z R ϕ d ν m,n = Z R (cid:18)Z R ϕ d P π n ( x, · ) (cid:19) d µ m ( x ) ≥ Z R ϕ d µ m , we have µ m ≤ c ν m,n for all m, n ∈ N ≥ . Lemma 1.1.
The martingale coupling π m,n := µ m ( Id , P π n ) is the only martingalecoupling between µ m and ν m,n for all m, n ∈ N ≥ . NSTABILITY OF MARTINGALE OPTIMAL TRANSPORT 4
The sequence ( µ , ν ,n ) n ∈ N serves as a counterexample to analogue versions of (S1),(S2) and (S3) in dimension d = 2 . The crucial observation is that whereas Π M ( µ , ν ,n ) consists of exactly one element for all n ∈ N by Lemma 1.1, there are infinitely manydifferent martingale couplings between µ and the limit of ( ν ,n ) n ∈ N . Proposition 1.2.
There holds lim n →∞ W ( ν ,n , µ P ) = 0 .Moreover, we have the following:(i) Let c ( x, y ) := k y − x k for all x, y ∈ R and π n := µ ( Id , P π n ) for all n ∈ N ≥ .The martingale couplings π n are minimizers of the (MOT) problem between µ and ν ,n w.r.t. c . Moreover, ( π n ) n ∈ N is weakly convergent but the limit is notan optimizer of (MOT) w.r.t. c between (its marginals) µ and µ P .(ii) Let c be defined as in (i). There holds lim n →∞ V Mc ( µ , ν ,n ) = 1 > V Mc ( µ , µ P ) . (iii) The set Π M ( µ , µ P ) \ { µ ( Id , P ) } is non empty and no element in this setcan be approximated by a weakly convergent sequence ( π n ) n ∈ N of martingalecouplings π n ∈ Π M ( µ , ν ,n ) . The sequence ( µ n , ν n,n ) n ∈ N shows that there cannot exist a constant C > for whichthe inequality (MWI) holds in dimension d = 2 . Proposition 1.3.
There holds lim n →∞ M ( µ n , ν n,n ) W ( µ n , ν n,n ) = + ∞ . Remark 1.4.
The theory of MOT in dimension two and further is also challenging inother aspects. For instance, the concept of irreducible components and convex pavingcan be directly reduced to the study of potential functions in dimension d = 1 , whereasthere are at least three different advanced approaches in dimension d ≥ (cf. [8, 6, 17] ).On the level of processes we would like to remind the reader that a higher dimensionalversion of Kellerer’s theorem is still not proved or disproved. One major obstacle isthat the one-dimensional proof via Lipschitz-Markov kernels cannot be extrapolated, see [13, Section 2.2] where a method similar to ours is used. Proofs
We denote by f µ the push-forward of the measure µ under the function f . Proof of Lemma 1.1.
Let m, n ≥ be integers and θ n := π n . We denote by L θ n theprojection parallel to the line l θ n = { ( x , tan( θ n ) x ) : x ∈ R } onto the x -axis, i.e. L θ n : R ∋ ( x , x ) x − tan( θ n ) − x ∈ R . Moreover, by setting ˜ ν := ( L θ n ) ν m,n and ˜ µ := ( L θ n ) µ m one has(2.1) ˜ ν = 1 m m X i =1 δ i = ˜ µ. Let π ∈ Π M ( µ m , ν m,n ) . As L θ n is a linear map, ˜ π := ( L θ n ⊗ L θ n ) π is a martingalecoupling of ˜ µ and ˜ ν . Indeed, for all ϕ ∈ C b ( R ) there holds Z R ϕ ( x )( y − x ) d˜ π ( x ) = L θ n (cid:18)Z R ϕ ( L θ n ( x ))( y − x ) d π ( x ) (cid:19) = 0 NSTABILITY OF MARTINGALE OPTIMAL TRANSPORT 5 and this property is equivalent to ˜ π being a martingale coupling. Jensen’s inequalityin conjunction with (2.1) yields Z R x d˜ µ ( x ) ≤ Z R (cid:18)Z R y d˜ π x ( y ) (cid:19) d˜ µ ( x ) = Z R y d˜ ν ( y ) = Z R x d˜ µ ( x ) where (˜ π x ) x ∈ R is a disintegration of ˜ π that satisfies (1.1). Since the square is a strictlyconvex function, there holds R R y d˜ π x = x if and only if ˜ π x = δ x . Thus, ˜ π = ˜ µ ( Id , Id ) and we obtain x = L θ n ( x , x ) = L θ n ( y , y ) for π -a.e. (( x , x ) , ( y , y )) ∈ R × R because supp( µ m ) ⊂ R × { } . Hence, the martingale transport plan π is only trans-porting along the lines parallel to l θ n . Since there are exactly two points in the supportof ν m,n that lie on the same line, and we are looking for a martingale coupling, we have π = µ m ( Id , P θ n ) . (cid:3) Lemma 2.1.
For all m ∈ N \ { } and θ ∈ (cid:2) , π (cid:3) one has W ( µ m P , µ m P θ ) ≤ θ. Proof.
The inequality consists merely of a comparison of angle and chord. Alternatively,for all m ∈ N and θ ∈ (cid:2) , π (cid:3) we directly compute W ( µ m P , µ m P θ ) ≤ Z R W ( δ x P , δ x P θ ) d µ m ( x )= | (sin( θ ) , cos( θ ) − | = p − cos( θ )) = 2 sin( θ/ ≤ θ. (cid:3) Proof of Proposition 1.2.
By Lemma 2.1, we know lim n →∞ W ( ν ,n , µ P ) = lim n →∞ W ( µ P π n , µ P ) = 0 . Moreover, for all n ∈ N Lemma 1.1 yields that π n := µ ( Id , P π n ) is the only martingalecoupling between µ and ν ,n and therefore automatically the unique minimizer of the(MOT) problem between these two marginals w.r.t. to any cost function. The sequence ( π n ) n ∈ N converges weakly to π := µ ( Id , P ) ∈ Π M ( µ , µ P ) . Note that π ′ := 16 (cid:0) δ ((1 , , (1 , + 2 δ ((2 , , (2 , + δ ((3 , , (3 , (cid:1) + 124 (cid:0) δ ((1 , , (0 , + δ ((1 , , (4 , + δ ((3 , , (0 , + 3 δ ((3 , , (4 , (cid:1) is a martingale coupling between µ and µ P = µ − (cid:16) δ + δ − δ + δ (cid:17) different from π where only the mass not shared by µ and µ P is moved.Item (i): Since π is the weak limit of the sequence ( π n ) n ∈ N , it is the only accumulationpoint. But as we see below in (ii), π is not the minimizer of the (MOT) problem between µ and µ P w.r.t. c .Item (ii): There holds lim n →∞ V Mc ( µ , ν ,n ) = lim n →∞ Z R × R k x − y k d π n = 1 >
12 = Z R × R k x − y k d π ′ ≥ V Mc ( µ , µ P ) . In fact, according to Lim’s result [16, Theorem 2.4], under an optimal martingaletransport w.r.t. c the shared mass between the marginal distribution is not moving. NSTABILITY OF MARTINGALE OPTIMAL TRANSPORT 6
Since π ′ is the unique martingale transport between µ and µ P with this property, itis the minimizer of this (MOT) problem and V Mc ( µ , µ P ) = .Item (iii): Since π is the weak limit of the solitary elements of Π M ( µ , ν ,n ) , noelement of Π M ( µ , µ P ) \ { µ ( Id , P ) } can be approximated and π ′ is an element ofthis set. (cid:3) Proof of Proposition 1.3.
Let n ∈ N . By Lemma 1.1, µ n ( Id , P π n ) is the only martingalecoupling between µ n and ν n,n . Thus, M ( µ n , ν n,n ) = 1 . Since W is a metric on P ( R ) , the triangle inequality yields W ( µ n , ν n,n ) ≤ W ( µ n , µ n P ) + W ( µ n P , ν n,n ) . We can easily compute µ n P = 12 n n X i =1 δ ( i − , + n X i =1 δ ( i +1 , ! and therefore W ( µ n , µ n P ) = n . By Lemma 2.1, there holds W ( µ n P , ν n,n ) ≤ π n .Hence, we obtain lim n →∞ M ( µ n , ν n,n ) W ( µ n , ν n,n ) ≥ lim n →∞ n + π n = + ∞ . (cid:3) Remark 2.2 (Variations) . Our construction may appear somewhat degenerate since µ m is discrete and supported on a lower dimensional subspace of R . However, it is notparticularly difficult to adapt the present construction with new measures that appearmore general but yield the same result:(ii) One could replace the rows of Dirac measures by uniform measures on parallel-ograms. More precisely, we could set ˜ µ m,n := Unif F m,n and ˜ ν m,n := 12 (cid:16) Unif F + m,n + Unif F − m,n (cid:17) where F m denotes the parallelogram spanned by the points − v n , − v n + ( m, , v n + ( m, and v n with v n := (cid:0) cos (cid:0) π n (cid:1) , sin (cid:0) π n (cid:1)(cid:1) ∈ R and F ± m,n is the translation of this par-allelogram by ± v n (cf. Figure 2). By the same argument as in Lemma 1.1,any martingale coupling between ˜ µ m,n and ˜ ν m,n can only transport along linesparallel to { ( x, tan (cid:0) π n (cid:1) x ) : x ∈ R } . In contrast to the situation in Lemma 1.1,the martingale transport along one of these parallel lines is no longer unique butevery π ∈ Π M (˜ µ m,n , ˜ ν m,n ) satisfies π (cid:0) | x − y | < (cid:1) = 0 for all m, n ∈ N becausethe supports are disjoint. This restriction carries over to any weak accumu-lation point of those martingale couplings and is sufficient to show analogousversions of Proposition 1.2 and Proposition 1.3.(iii) One could replace µ m and ν m,n by ˜ µ m := (1 − ǫ ) µ m + ǫγ and ˜ ν m,n := (1 − ǫ ) ν m,n + ǫγ where ε ∈ (0 , and γ is a probability measure with full support (e.g. a standardnormal distribution). There holds W (˜ µ m , ˜ ν m,n ) = (1 − ǫ ) W ( µ m , ν m,n ) since W derives of the Kantorovich-Rubinstein norm [15] (alternatively see [18, Bib. NSTABILITY OF MARTINGALE OPTIMAL TRANSPORT 7 y xm = n = 2 y xm = n = 3 Figure 2.
The construction in Remark 2.2 (i) for m = n = 2 and m = n = 3 . The red area is the support of ˜ µ m,n and the blue area thesupports of ˜ ν m,n .Notes of Ch.6 ] or [7, §*11.8] ) and M (˜ µ m , ˜ ν m,n ) = (1 − ǫ ) M ( µ m , ν m,n ) by aresult of Lim [16, Theorem 2.4] . Remark 2.3.
Finally, we would like to point out that Propositions 1.2 and 1.3 andtheir proofs are not depending on the choice of the Euclidean norm while defining c . References [1] A. Alfonsi, J. Corbetta, and B. Jourdain. Sampling of probability measures in the convex orderby Wasserstein projection.
Ann. Inst. Henri Poincaré Probab. Stat. , 56(3):1706–1729, 2020.[2] J. Backhoff-Veraguas and G. Pammer. Stability of martingale optimal transport and weak optimaltransport. preprint, arXiv:1904.04171 , 2019.[3] M. Beiglböck, P. Henry-Labordère, and F. Penkner. Model-independent bounds for option prices—a mass transport approach.
Finance Stoch. , 17(3):477–501, 2013.[4] M. Beiglböck, B. Jourdain, W. Margheriti, and G. Pammer. Approximation of martingale cou-plings on the line in the weakadapted topology. preprint, arXiv:2101.02517 , 2020.[5] M. Beiglböck, B. Jourdain, W. Margheriti, and G. Pammer. Monotonic-ity and stability of the weak martingale optimal transport problem.
Online, https: // cermics. enpc. fr/ ~margherw/ Documents/ stabilityWMOT. pdf , 2020.[6] H. De March and N. Touzi. Irreducible convex paving for decomposition of multidimensionalmartingale transport plans.
Ann. Probab. , 47(3):1726–1774, 2019.[7] R. M. Dudley.
Real analysis and probability , volume 74 of
Cambridge Studies in Advanced Math-ematics . Cambridge University Press, Cambridge, 2002. Revised reprint of the 1989 original.[8] N. Ghoussoub, Y.-H. Kim, and T. Lim. Structure of optimal martingale transport plans in generaldimensions.
Ann. Probab. , 47(1):109–164, 2019.[9] G. Guo and J. Obloj. Computational methods for martingale optimal transport problems. preprint, arXiv:1710.07911 , 2017.[10] P. Henry-Labordère.
Model-free hedging . Chapman & Hall/CRC Financial Mathematics Series.CRC Press, Boca Raton, FL, 2017. A martingale optimal transport viewpoint.[11] B. Jourdain and W. Margheriti. A new family of one dimensional martingale couplings.
Electron.J. Probab. , 25:Paper No. 136, 50, 2020.[12] B. Jourdain and G. Pagès. Quantization and martingale couplings. preprint, arXiv:2012.10370 ,2020.[13] N. Juillet. Peacocks parametrised by a partially ordered set. In
Séminaire de Probabilités XLVIII ,volume 2168 of
Lecture Notes in Math. , pages 13–32. Springer, Cham, 2016.[14] N. Juillet. Stability of the shadow projection and the left-curtain coupling.
Ann. Inst. HenriPoincaré Probab. Stat. , 52(4):1823–1843, 2016.[15] L. V. Kantorovič and G. v. Rubinšte˘ın. On a space of completely additive functions.
VestnikLeningrad. Univ. , 13(7):52–59, 1958.[16] T. Lim. Optimal martingale transport between radially symmetric marginals in general dimen-sions.
Stochastic Process. Appl. , 130(4):1897–1912, 2020.[17] J. Obloj and P. Siorpaes. Structure of martingale transports in finite dimensions. preprint, arXiv:1702.08433 , 2017.
NSTABILITY OF MARTINGALE OPTIMAL TRANSPORT 8 [18] C. Villani.
Optimal Transport: Old and New . Grundlehren der mathematischen Wissenschaften.Springer, 2009.[19] J. Wiesel. Continuity of the martingale optimal transport problem on the real line. preprint, arXiv:1905.04574 , 2020.(MARTIN BRÜCKERHOFF)
UNIVERSITÄT MÜNSTER, GERMANY
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UNIVERSITÉ DE STRASBOURG ET CNRS, FRANCE
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