On the stability of the martingale optimal transport problem: A set-valued map approach
aa r X i v : . [ m a t h . P R ] F e b ON THE STABILITY OF THE MARTINGALE OPTIMAL TRANSPORTPROBLEM: A SET-VALUED MAP APPROACH
ARIEL NEUFELD AND JULIAN SESTER
Abstract.
Continuity of the value of the martingale optimal transport problem on the real linew.r.t. its marginals was recently established in [2] and [19]. We present a new perspective of thisresult using the theory of set-valued maps. In particular, using results from [4], we show that the setof martingale measures with fixed marginals is continuous, i.e., lower- and upper hemicontinuous,w.r.t. its marginals. Moreover, we establish compactness of the set of optimizers as well as upperhemicontinuity of the optimizers w.r.t. the marginals.
Keywords:
Martingale optimal transport, Stability, Set-valued map, Berge’s maximum theorem Introduction
The martingale optimal transport problem (as introduced in [3]) consists, given a measurable func-tion Φ : R → R , in solving(1.1) m ( µ , µ ) := sup Q ∈M ( µ ,µ ) Z R Φ( x , x ) d Q ( x , x ) , where M ( µ , µ ) describes the set of martingale measures on R with fixed marginals µ and µ .Recently, the stability of the martingale optimal transport problem w.r.t. its marginal distribu-tions, i.e., the question whether the solutions m ( µ , µ ) and m ( f µ , f µ ) of martingale optimal trans-port problems are close, whenever the marginals µ i , and e µ i , i = 1 ,
2, are close in the Wasserstein-distance, attracted a lot of attraction. One particular reason for the importance of a positiveanswer to this question is that in this case martingale transport problems involving finitely sup-ported marginal distribution that are close to the original marginals yield solutions that are close tothe original value. Since these approximated solutions can be computed with tractable linear pro-gramming methods (compare e.g. [9] and [10]), the stability result builds an important theoreticalfoundation for the numerics of martingale optimal transport.The stability result was indeed established for a Lipschitz-continuous payoff function and a relaxedformulation of the transport problem in [9], for particular payoffs and special marginals in [14], andeventually in a great generality by [2] and [19]. Recently, [4] established a result which allows toapproximate martingale measures with fixed marginals ( µ , µ ) in the adapted Wasserstein-distance(compare [2]) by a sequence of approximating martingale measures with fixed marginals ( µ ( n )1 , µ ( n )2 )which converge in Wasserstein-distance to ( µ , µ ) for n → ∞ .In this work, we regard the problem from a new perspective by studying properties of set-valuedmaps. More precisely, we show that the set-valued map(1.2) ( µ , µ )
7→ M ( µ , µ ) , is continuous, i.e., is upper hemicontinuous and lower hemicontinuous. This in turn implies throughan application of Berge’s maximum theorem [6] the stability of the martingale optimal transportproblem w.r.t. its marginals. Moreover, we prove compactness of the set of optimizers, i.e., of the setof two-dimensional martingale measures maximizing m ( µ , µ ) as well as the upper-hemicontinuityof the correspondence mapping from marginals to these optimizers.2. Main Result
We first introduce some notation. For every n ∈ N let P ( R n ) denote the set of all probabilitymeasures on R n . Then, we define the 1- Wasserstein space P ( R n ) := (cid:26) P ∈ P ( R n ) (cid:12)(cid:12)(cid:12)(cid:12) Z R n n P i =1 | x i | d P ( x , . . . , x n ) < ∞ (cid:27) describing the probability measures with existing first moment. Moreover, we let C lin ( R n ) := ( f ∈ C ( R n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup ( x ,...,x n ) ∈ R n | f ( x , . . . , x n ) | P ni =1 | x i | < ∞ ) denote the set of all continuous functions on R n with linear growth. Next, we define for every n ∈ N the set Π( µ , µ ) of all probability measures on R n with fixed marginal distributions µ , µ ∈P ( R n ), also called couplings of µ , µ . We further define for P , Q ∈ P ( R n ) the 1-Wassersteindistance W ( P , Q ) := inf π ∈ Π( P , Q ) Z R n × R n n P i =1 | x i − y i | d π ( x , . . . , x n , y , . . . , y n ) , as well as the sum of Wasserstein distances between measures P i , Q i ∈ P ( R n ) for i = 1 , . . . , m W ⊕ (cid:0) ( P , . . . , P m ) , ( Q , . . . , Q m ) (cid:1) := m X i =1 W ( P i , Q i ) . The set of all martingale measures on R with fixed marginal distributions µ , µ ∈ P ( R ) is thengiven by(2.1) M ( µ , µ ) := (cid:26) Q ∈ Π( µ , µ ) (cid:12)(cid:12)(cid:12)(cid:12) Z R H ( x )( x − x ) d Q ( x , x ) = 0 for all H ∈ C b ( R ) (cid:27) . Moreover, we denote by µ (cid:22) µ the convex order of measures µ , µ ∈ P ( R ), which is characterizedby the inequality R f d µ ≤ R f d µ required to hold for all convex functions f : R → R .When speaking of a correspondence from a set X to a set Y , denoted by ϕ : X ։ Y , we refer toa set-valued function, i.e., a map that assigns each x ∈ X a set ϕ ( x ) ⊆ Y , see also [1, Chapter 17].What continuity means for a correspondence is clarified in the following definition. Definition 2.1.
Let ϕ : X ։ Y be a correspondence between two topological spaces. (i) ϕ is called upper hemicontinuous , if { x ∈ X | ϕ ( x ) ⊆ A } is open for all open sets A ⊆ Y . (ii) ϕ is called lower hemicontinuous , if { x ∈ X | ϕ ( x ) ∩ A = ∅} is open for all open sets A ⊆ Y . (iii) We say ϕ is continuous, if ϕ is upper and lower hemicontinuous. Our main result is the following.
Proposition 2.2.
Let X := { ( µ , µ ) ∈ P ( R ) × P ( R ) | µ (cid:22) µ } be equipped with the topology induced by the convergence w.r.t. W ⊕ , and let Y := P ( R ) be equipped with the topology induced by the convergence w.r.t. W . Then, the correspondence ϕ : X ։ Y ( µ , µ )
7→ M ( µ , µ ) is non-empty, convex, compact, and continuous. As a consequence one obtains the following. (i) For any Φ ∈ C lin (cid:0) R (cid:1) the functional m : X → R ( µ , µ ) max Q ∈M ( µ ,µ ) Z R Φ( x , x ) d Q ( x , x ) is continuous. (ii) For any ( µ , µ ) ∈ X and Φ ∈ C lin (cid:0) R (cid:1) the set of optimizers Q ∗ ( µ , µ ) := (cid:26) Q ∗ ∈ M ( µ , µ ) (cid:12)(cid:12)(cid:12)(cid:12) max Q ∈M ( µ ,µ ) Z R Φ( x , x ) d Q ( x , x ) = Z R Φ( x , x ) d Q ∗ ( x , x ) (cid:27) has non-empty, compact values. (iii) For any Φ ∈ C lin (cid:0) R (cid:1) the correspondence µ : X ։ Y ( µ , µ )
7→ Q ∗ ( µ , µ ) is upper hemicontinuous. N THE STABILITY OF THE MARTINGALE OPTIMAL TRANSPORT PROBLEM 3
Remark 2.3. (i)
If a correspondence Θ ∋ θ
7→ S ( θ ) is single-valued, i.e., S ( θ ) = { s θ } for all θ ∈ Θ , then upper hemicontinuity of the set-valued map Θ ∋ θ
7→ S ( θ ) is equivalent to thecontinuity of the (single-valued) map Θ ∋ θ s θ , compare e.g. [1, Lemma 17.6] . Thus, ifthe set of optimizers Q ∗ ( µ , µ ) = { Q ∗ ( µ , µ ) } is a singleton for all ( µ , µ ) ∈ X , we haveby Proposition 2.2 that the map ( µ , µ ) Q ∗ ( µ , µ ) is continuous, if Φ ∈ C lin ( R ) . It is possible to state explicit conditions on the payoff function Φ which imply uniqueness of the associated optimizers. These conditions were among othersstudied in detail in [5] , [11] , [12] , and [13] . (ii) If a correspondence is non-empty, convex, compact, and continuous, then there exist resultsenabling a (numerically tractable) approximation of the correspondence, whenever its imageis contained in a Euclidean space, compare for example [7] and [8] . The correspondence ( µ , µ )
7→ M ( µ , µ ) fulfils, according to Proposition 2.2, all of the above mentioned re-quirements, but its image is not contained in a Euclidean space. If a suitable approximationwould be possible also for this measure-valued correspondence, then such an approximationcould one allow to approximate the martingale optimal transport problem (1.1) under consid-eration. Moreover, the continuity property of the set valued map ( µ , µ )
7→ M ( µ , µ ) couldbe fruitful to analyze (lower hemi-) continuity of the set valued map of optimizers w.r.t. thegiven marginals, i.e. ( µ , µ )
7→ Q ∗ ( µ , µ ) , compare for example with [15] , which in turnleads to the existence of continuous selectors, see, e.g., [16] .We leave these questions open for future research. Proof of Proposition 2.2
Preliminaries and useful Results.
For the proof of the main result we make use of thefollowing results. The first result is Berge’s maximum theorem. Note that for a corespondence ϕ : X ։ Y the graph is defined asGr ϕ := { ( x, y ) ∈ X × Y | y ∈ ϕ ( x ) } . Theorem 3.1 ([1], Theorem 17.31) . Let ϕ : X ։ Y be a continuous correspondence betweentopological spaces with non-empty compact values, and suppose that f : Gr ϕ → R is continuous.Then the following holds. (i) The function m : X → R x max y ∈ ϕ ( x ) f ( x, y ) is continuous. (ii) The correspondence c : X ։ Yx
7→ { y ∈ ϕ ( x ) | f ( x, y ) = m ( x ) } has non-empty, compact values. (iii) If either f has a continuous extension to all X × Y or Y is Hausdorff, then c is upperhemicontinuous. By Definition 2.1 a correspondence is called continuous if it is upper hemicontinuous and lowerhemicontinuous. These notions are characterized through the following two lemmas.
Lemma 3.2 ([1], Lemma 17.20) . Assume that the topological space X is first countable and that Y is metrizable. Then, for a correspondence ϕ : X ։ Y the following statements are equivalent. (i) The correspondence ϕ is upper hemicontinuous and ϕ ( x ) is compact for all x ∈ X . (ii) For any x ∈ X , if a sequence (cid:0) ( x ( n ) , y ( n ) ) (cid:1) n ∈ N ⊆ Gr ϕ satisfies x ( n ) → x for n → ∞ , thenthere exists a subsequence (cid:0) y ( n k ) (cid:1) k ∈ N with y ( n k ) → y ∈ ϕ ( x ) for k → ∞ . Lemma 3.3 ([1], Theorem 17.21) . For a correspondence ϕ : X ։ Y between first countable topo-logical spaces the following statements are equivalent. (i) The correspondence ϕ is lower hemicontinuous. ARIEL NEUFELD AND JULIAN SESTER (ii)
For any x ∈ X , if x ( n ) → x for n → ∞ , then for each y ∈ ϕ ( x ) there exists a subsequence (cid:0) x ( n k ) (cid:1) k ∈ N and elements y ( k ) ∈ ϕ (cid:0) x ( n k ) (cid:1) for each k ∈ N such that y ( k ) → y for k → ∞ . The following result from [4] turns out to be crucial for our arguments. Note that the adapted1-Wasserstein distance between two measures Q , Q ′ (with first marginals µ and µ ′ , respectively)is defined as AW ( Q , Q ′ ) := inf π ∈ Π( µ ,µ ′ ) Z R | x − x ′ | + W ( Q x , Q ′ x ′ ) d π ( x , x ′ ) , where Q x , Q ′ x ′ denote the disintegration of the measures Q and Q ′ respectively w.r.t. its firstmarginals, i.e., Q = µ ⊗ Q x and Q ′ = µ ′ ⊗ Q ′ x ′ . Theorem 3.4 ([4], Theorem 2.5) . Let µ ( k )1 , µ ( k )2 ∈ P ( R ) , µ ( k )1 (cid:22) µ ( k )2 , k ∈ N with W ⊕ (cid:16) ( µ ( k )1 , µ ( k )2 ) , ( µ , µ ) (cid:17) → for k → ∞ . Let Q ∈ M ( µ , µ ) . Then there exists a sequence ( Q ( k ) ) k ∈ N with Q ( k ) ∈ M ( µ ( k )1 , µ ( k )2 ) for all k ∈ N s.t. AW ( Q ( k ) , Q ) → for k → ∞ . Proof of Proposition 2.2.
Proof.
Let Φ ∈ C lin (cid:0) R (cid:1) and define onGr ϕ := (cid:8) ( µ , µ , Q ) (cid:12)(cid:12) µ , µ ∈ P ( R ) , µ (cid:22) µ , Q ∈ M ( µ , µ ) (cid:9) the function f : Gr ϕ → R ( µ , µ , Q ) Z R Φ( x , x ) d Q ( x , x ) . We aim at showing1.) ϕ is non-empty valued.2.) ϕ is convex and compact valued.3.) ϕ is upper hemicontinuous.4.) ϕ is lower hemicontinuous.5.) f is continuous.If the above listed requirements are established, then the result of Proposition 2.2 follows directly byBerge’s maximum theorem stated in Theorem 3.1. Before proving the requirements we note that,according to [18, Definition 6.8 (d) and Theorem 6.9], the convergence lim n →∞ W ( P , P ( n ) ) = 0 of asequence (cid:0) P ( n ) (cid:1) n ∈ N ⊆ P ( R ) to some limit P ∈ P ( R ) is equivalent to(3.1) lim n →∞ Z R f ( x , x ) d P ( n ) ( x , x ) = Z R f ( x , x ) d P ( x , x ) for all f ∈ C lin ( R ) . Moreover, note that with the assigned metrics W ⊕ and W respectively, the spaces X and Y areindeed first countable, thus Lemma 3.2 as well as Lemma 3.3 are applicable. Further, Y is Hausdorff(since W is a metric on P ( R )) as required in Theorem 3.1.1.) Pick some ( µ , µ ) ∈ X . Then we have by definition of X that µ (cid:22) µ , which implies dueto the well-known result from [17, Theorem 8] that ϕ (( µ , µ )) = M ( µ , µ ) = ∅ .2.) Pick some ( µ , µ ) ∈ X . First note that the convexity of M ( µ , µ ) follows by definition.To see that M ( µ , µ ) ⊆ Y equipped with W is compact, let ( Q ( n ) ) n ∈ N ⊆ ϕ (( µ , µ )) = M ( µ , µ ). Then, according to the weak-compactness of M ( µ , µ ) (see [3, Proposition2.4.]), there exists a subsequence ( Q ( n k ) ) k ∈ N ⊆ ( Q ( n ) ) n ∈ N such that ( Q ( n k ) ) k ∈ N convergesweakly to some Q ∈ M ( µ , µ ). According to [3, Lemma 2.2], we then havelim k →∞ Z R f ( x , x ) d Q ( n k ) ( x , x ) = Z R f ( x , x ) d Q ( x , x ) for all f ∈ C lin ( R ) , which implies by (3.1) that W ( Q ( n k ) , Q ) → k → ∞ . Thus, we have shown thecompactness of M ( µ , µ ) ⊆ Y w.r.t. W . N THE STABILITY OF THE MARTINGALE OPTIMAL TRANSPORT PROBLEM 5 ϕ we apply Lemma 3.2. Thus, let ( µ , µ ) ∈ X andconsider a sequence (cid:16) µ ( n )1 , µ ( n )2 , Q ( n ) (cid:17) n ∈ N ⊆ Gr( ϕ ) such that(3.2) lim n →∞ W ⊕ (cid:16) ( µ ( n )1 , µ ( n )2 ) , ( µ , µ ) (cid:17) = 0 . Observe that by Prokhorov’s theorem and by the weak convergence implied through (3.2)the sets { µ ( n )1 , n ∈ N } and { µ ( n )2 , n ∈ N } are tight. Denote by e Π := Π (cid:16) { µ ( n )1 , n ∈ N } , { µ ( n )2 , n ∈ N } (cid:17) ⊆ P ( R )the set of probability measures on R with first marginal in { µ ( n )1 , n ∈ N } and secondmarginal in { µ ( n )2 , n ∈ N } . The set e Π is tight according to [18, Lemma 4.4]. Thus, since( Q ( n ) ) n ∈ N ⊆ e Π, according to Prokhorov’s theorem, there exists a subsequence ( Q ( n k ) ) k ∈ N which converges weakly to Q ∈ P ( R ). From the Wasserstein-convergence of the marginals,stated in (3.2), we also obtain by (3.1) that lim k →∞ W (cid:0) Q ( n k ) , Q (cid:1) = 0 as well as Q ∈ Π( µ , µ ). Next, let H ∈ C b ( R ). We have that ( x , x ) H ( x )( x − x ) ∈ C lin ( R ) andthus the Wasserstein convergence lim k →∞ W (cid:0) Q ( n k ) , Q (cid:1) = 0 implies, according to (3.1), that0 = lim n →∞ Z R H ( x )( x − x ) d Q ( n ) ( x , x ) = Z R H ( x )( x − x ) d Q ( x , x ) . Since the function H was chosen arbitrarily, we obtain by the definition of M ( µ , µ ), statedin (2.1), that Q ∈ M ( µ , µ ) = ϕ (( µ , µ )).4.) To show the lower hemicontinuity of ϕ , we apply Lemma 3.3. Let ( µ , µ ) ∈ X and considera sequence (cid:16) ( µ ( n )1 , µ ( n )2 ) (cid:17) n ∈ N ⊆ X such thatlim n →∞ W ⊕ (cid:16) ( µ ( n )1 , µ ( n )2 ) , ( µ , µ ) (cid:17) = 0 . Note that, by step 1.), M ( µ , µ ) is non-empty. Let Q ∈ M ( µ , µ ). Then, by Theorem 3.4,there exists some sequence ( Q ( n ) ) n ∈ N with Q ( n ) ∈ M ( µ ( n )1 , µ ( n )2 ) for all n ∈ N , convergergingw.r.t. AW towards Q . Moreover, we have by definition the pointwise inequality W ≤ AW .Thus, convergence w.r.t. AW implies convergence w.r.t. W , i.e., we obtainlim n →∞ W (cid:16) Q ( n ) , Q (cid:17) ≤ lim n →∞ AW (cid:16) Q ( n ) , Q (cid:17) = 0 . µ ( n )1 , µ ( n )2 , Q ( n ) ) n ∈ N ⊆ Gr ϕ be a sequence such thatlim n →∞ W ⊕ (cid:16) ( µ ( n )1 , µ ( n )2 ) , ( µ , µ ) (cid:17) + lim n →∞ W (cid:16) Q ( n ) , Q (cid:17) = 0 , for some ( µ , µ , Q ) ∈ Gr ϕ . Then, we obtain by (3.1) and since Φ ∈ C lin ( R ) thatlim n →∞ f (cid:16) ( µ ( n )1 , µ ( n )2 , Q ( n ) ) (cid:17) = lim n →∞ Z R Φ( x , x ) d Q ( n ) ( x , x )= Z R Φ( x , x ) d Q ( x , x ) = f (( µ , µ , Q )) . (cid:3) Acknowledgments
Financial support by the Nanyang Assistant Professorship Grant (NAP Grant)
Machine Learningbased Algorithms in Finance and Insurance is gratefully acknowledged.
ARIEL NEUFELD AND JULIAN SESTER
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