2 Proof of Theorem 1.1

ELECTRONIC COMMUNICATIONS in PROBABILITY

MEASURABILITY OF OPTIMAL TRANSPORTATION AND STRONG COUPLING OF MARTINGALE MEASURES

JOAQUIN FONTBONA¹

DIM-CMM, UMI(2807) UCHILE-CNRS, Universidad de Chile, Casilla 170-3, Correo 3, Santiago-Chile.

email: [email protected] HÉLÈNE GUÉRIN²

IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes-France.

email: [email protected] SYLVIE MÉLÉARD³

CMAP, Ecole Polytechnique, CNRS, route de Saclay, 91128 Palaiseau Cedex-France.

email: [email protected]

SubmittedAugust 5, 2008, accepted in final formMarch 22, 2010 AMS 2000 Subject classification: 49Q20;60G57

Keywords: Measurability of optimal transport. Coupling between orthogonal martingale mea- sures. Predictable transport process.

Abstract

We consider the optimal mass transportation problem inR^dwith measurably parameterized marginals under conditions ensuring the existence of a unique optimal transport map. We prove a joint mea- surability result for this map, with respect to the space variable and to the parameter. The proof needs to establish the measurability of some set-valued mappings, related to the support of the optimal transference plans, which we use to perform a suitable discrete approximation procedure.

A motivation is the construction of a strong coupling between orthogonal martingale measures.

By this we mean that, given a martingale measure, we construct in the same probability space a second one with a specified covariance measure process. This is done by pushing forward the first martingale measure through a predictable version of the optimal transport map between the covariance measures. This coupling allows us to obtain quantitative estimates in terms of the Wasserstein distance between those covariance measures.

1 Introduction

We consider the optimal mass transportation problem inR^dwith measurably parameterized marginals, for general cost functions and under conditions ensuring the existence of a unique optimal trans- port map. The aim of this note is to prove a joint measurability result for this map, with respect

1SUPPORTED BY FONDECYT PROYECT 1070743, ECOS-CONICYT C05E02 AND BASAL-CONICYT

2SUPPORTED BY ECOS-CONICYT C05E02

3SUPPORTED BY ECOS-CONICYT C05E02

124

to the space variable and the parameter. One of our motivations is the construction of couplings between martingale measures (in the sense of Walsh [15]). This problem classically arises in the literature, especially in cases where the martingale measures are compensated Poisson point measures or space-time white noises, for which the covariance measures are deterministic (cf.

Grigelionis[5], El Karoui-Lepeltier[1], Tanaka[12], El Karoui-Méléard[2], Méléard-Roelly[7], Guérin[6]). A classical approach for constructing such couplings is to use the Skorokhod repre- sentation theorem. However, this does not give any quantitative estimate.

Here, we aim at a method to construct what we call “strong coupling” between martingale mea- sures. By this we mean that, given a martingale measure, we search to construct in the same probability space a second one with specified covariance measure process. We will do this by pushing forward the given martingale measure through the optimal transport map between the two covariance processes. The interest of such a construction is that it will provide quantitative estimates in terms of the Wasserstein distance between the covariance measures. But in order to make this construction rigorous, the existence of apredictableversion of the above described transport map is needed. To our knowledge, available measurability results on the mass trans- portation problem with respect to some parameter require a topological structure on the space of parameters, or concern measurability oftransference plans, but not oftransport maps(see e.g.

[10], or Corollaries 5.22 and 5.23 in[14]). Our main result provides the existence of the jointly- measurable optimal transport map which is required.

Let us establish notations and recall basic facts. We denote the space of Borel probability measures inR^dbyP(R^d)endowed with the usual weak topology, and byPp(R^d)the subspace of probability measures having finitep−order moment. Givenπ∈ P(R^2d), we write

π <^µ_ν

ifµ,ν ∈ P(R^d)are respectively its first and second marginals. Suchπis referred to as a “trans- ference plan” betweenµandν.

Letc:R^2d→R₊be a continuous function. The mapping π→I(π):=

Z

R^2d

c(x,y)π(d x,d y)

is then lower semi continuous. The Monge-Kantorovich or optimal mass transportation problem with costcand marginalsµ,νconsists in finding

π<inf^µν

I(π).

It is well known that the infimum is attained as soon as it is finite, see[13], Ch.1. In this case, we denote byΠ^∗c(µ,ν)the subset ofP(R^2d)of minimizers. If otherwise,I(π) = +∞for allπ <^µ_ν, then by convention we setΠ^∗c(µ,ν) =;.

We shall say thatAssumption H(µ,ν,c)holds if

there exists a unique optimal transference planπ∈Π^∗_c(µ,ν), and it has the form π(d x,d y) =µ(d x)⊗δT(x)(d y)

for aµ(d x)−a.s.unique mapping T:R^d→R^d.

SuchT is called anoptimal transport map betweenµandνfor the cost function c.

We recall that the condition thatµdoes not give mass to sets with Hausdorff dimension smaller than or equal tod−1 is optimal both for existence and uniqueness ofT, see Remark 9.5 in[14]. Moreover, ifΠ^∗_c(µ,ν)6=;, then the latter condition impliesH(µ,ν,c)in the following situations (see Gangbo and McCann[4]):

i) c(x,y) =˜c(|x− y|)with ˜c:R₊→R₊ strictly convex, superlinear and differentiable with locally Lipschitz gradient.

ii) c(x,y) =˜c(|x−y|)with ˜cstrictly concave, andµandνare mutually singular.

ConditionH(µ,ν,c)also holds if

iii) c(x,y) = ˜c(|x− y|)with ˜c strictly convex and superlinear, and moreoverµ is absolutely continuous with respect to Lebesgue measure.

Whenµ,ν∈ Pp(R^d), fundamental examples are the cost functionc(x,y) =|x−y|^p with p≥2 for case i),p>1 for case iii), andp∈(0, 1)for case ii).

Our main result is

Theorem 1.1. Let(E,Σ,m)be aσ−finite measure space and consider a measurable functionλ∈ E7→(µλ,νλ)∈(P(R^d))²such that for m−almost everyλ, H(µλ,νλ,c)holds, with optimal transport map T_λ:R^d→R^d. Then, there exists a function(λ,x)7→T(λ,x)which is measurable with respect toΣ⊗ B(R^d)and such that m(dλ)−almost everywhere,

T(λ,x) =T_λ(x) µλ(d x)-almost surely.

In particular, T_λ(x) is measurable with respect to the completion of Σ⊗ B(R^d) with respect to m(dλ)µ_λ(d x).

Theorem 1.1 generalizes Theorem 1.2 in[3], where a predictable version of the quadratic trans- port map between a time-varying law and empirical samples of it was constructed. This allowed us to exhibit the convergence rate of a Brownian motion driven interacting particle system towards a white-noise driven nonlinear process. More precisely, we considerednindependent copies(Xⁱ)ⁿ_i=1 of the process inR^d:

Xⁱ_t=X₀ⁱ+ Z t

0

Z

R^d

σ(X_sⁱ−y)Wⁱ(ds,d y) + Z t

0

Z

R^d

b(X_sⁱ−y)P_s(y)d y ds (1) where σ and b satisfy usual Lipschitz assumptions, Xⁱ_t ∼ P_t(y)d y and Wⁱ is an R^d valued space-time white noise on [0,T]×R^d with independent coordinates of covariance measures P_t(y)d y⊗d t. By pushing forward eachWⁱ(d t,d x)through the predictable version of the optimal transport map from P_t to ¹_nPn

j=1δX_t^j, we constructed in the same probability spacen²indepen- dent R^d−Brownian motions(B^ik), suitably correlated with the martingale measures (Wⁱ). This provided us a coupling between the processes(Xⁱ_t)and the particles(X_t^i,n)governed by

X_t^i,n=X₀ⁱ+ 1 pn

Z t

0 n

X

k=1

σ(X_s^i,n−X_s^k,n)d B^ik_s +1 n

Zt

0 n

X

k=1

b(X_s^i,n−X_s^k,n)ds, i=1, . . . ,n,

yielding the estimate W₂²(l aw(X^1,n),l aw(X¹)) ≤ C_T,dn^d+4−2 for the Wasserstein-2 distanceW2 on the space of probability measures on C([0,T],R^d). That is indeed the convergence rate to 0 of

the (squared) expected Wasserstein-2 distanceW₂²(¹_nPn

j=1δ_Xt^j,P_t)inP2(R^d), asngoes to infinity.

(For details and precise statements, see Theorem 1.1 and Corollary 6.2. in[3])

The proof of Theorem 1.1 is developed in Section2.. We firstly establish a type of measurable de- pendence onλof the support of the optimizers. From this result, we can define measurable (w.r.t.

λ) partitions ofE×R^d induced by a dyadic partition ofR^d, and construct bi-measurable discrete approximations of T(λ,x). This approximation procedure was not needed in the simpler case studied in[3], where one of the marginals (the empirical measure) had finite support. In Section 3we address the construction of strong couplings between orthogonal martingale measures.

2 Proof of Theorem 1.1

Let us first state an intermediary result concerning measurability properties of minimizers in the general framework. Its formulation and proof require some notions of set-valued analysis, see e.g.

chapter 14 of[9].

Theorem 2.1. The function assigning to(µ,ν)the set ofR^2d

Ψ(µ,ν):=C l





 [

π∈Π^∗_c(µ,ν)

supp(π)





, (2)

where C l stands for the topological closure, is measurable in the sense of set-valued mappings. That is, for any open setθ inR^2d, its inverse imageΨ⁻¹(θ) ={(µ,ν)∈(P(R^d))²:Ψ(µ,ν)∩θ6=;}is a Borel set in(P(R^d))².

Remark 2.2. In the case of a set-valued mapping taking closed-set values, measurability is equivalent to the fact that inverse images of closed sets are measurable (see[9]).

Proof.The idea of the proof is similar to the one of Theorem 1.3 in[3], where we considered the quadratic cost and the measurable structure induced by the Wasserstein topology. (In the present case, the topology inP(R^d)andP(R^2d)is the usual weak one.)

We observe thatΨwrites as the topological closure of a set-valued composition,

Ψ(µ,ν) =C l U◦S(µ,ν) :=C l





 [

π∈S(µ,ν)

U(π)





, whereSandUare the set-valued mappings respectively defined by

S(µ,ν):= Π^∗_c(µ,ν) and U(π):=supp(π).

Measurability ofΨis equivalent toU◦Sbeing measurable. The latter will be true as soon asSis measurable andU⁻¹(θ)is open for every open setθ(see[9]).

The stability theorem for optimal transference plans of Schachermayer and Teichmann (Theorem 3 in[11]) exactly states that inverse images throughSof closed sets inP(R^2d)are closed sets in (P(R^d))². This, together with the fact that the mappingStakes closed-set values (by lower semi continuity ofI(π)) imply thatSis a measurable set valued mapping.

On the other hand, the inverse image byUof an open setθ ofR^2d is

U⁻¹(θ) ={π∈ P(R^2d):supp(π)∩θ6=;}={π∈ P(R^2d):π(θ)>0}.

It then follows by the Portmanteau Theorem that U⁻¹(θ) is an open set in P(R^2d), and this concludes the proof.

Corollary 2.3. Let(E,Σ)be a measurable space, andλ∈E7→(µ_λ,ν_λ)∈(P(R^d))²a measurable function. We consider the functionΨdefined by (2) and let F be a closed set ofR^d. Then, the set

(λ,x):({x} ×F)∩Ψ(µλ,νλ)6=;

belongs toΣ⊗ B(R^d). In particular, if for allλ∈E,Π^∗c(µλ,νλ) ={πλ}is a singleton, the set F˜:=(λ,x):({x} ×F)∩supp(π_λ)6=;

is measurable.

Proof. Without loss of generality, we assume thatF is nonempty. Let us first show that for any open setθ ofR^2d, the set

G={z∈R^d:({z} ×F)∩θ6=;}

is open. Indeed, for x ∈ G there exists y ∈ F and" > 0 such that B(x,")×B(y,") ⊂ θ. In particular, for allz∈B(x,")one has(z,y)∈θ and soB(x,")⊂G. By definition of measurability, the set-valued mappings(λ,x)→ {x} ×Fand(λ,x)→Ψ(µλ,νλ)−({x} ×F)are thus measurable (see[9], Proposition 14.11(c)). The latter mapping being also closed valued, we conclude that

(λ,x):

Ψ(µλ,νλ)−({x} ×F)

∩ {0} 6=;

is a measurable set, which finishes the proof.

Proof of Theorem 1.1Since any σ-finite measure is equivalent to a finite one, we can assume without loss of generality thatmis finite.

For a fixedk≥1, we denote by(A_n,k)n∈Z^d the partition ofR^din dyadic half-open rectangles of size 2^{−d k}, that is

A_n,k:=

d

Y

i=1

n_i 2^k,n_i+1

2^k

, wheren= (n₁, . . . ,n_d)∈Z^d.

Consider the setsB_n,k={(λ,x)∈E×R^d :({x} ×A_n,k)∩Ψ(µλ,νλ)6=;}, withΨdefined by (2).

Notice that sinceA_n,k=S

j∈N

Qd i=1

n_i 2^k,ⁿⁱ⁺¹

2^k −₂k+j¹

, one has

B_n,k=[

j∈N

(

(λ,x): {x} ×

d

Y

i=1

n_i 2^k,n_i+1

2^k − 1 2^k+j

!

∩Ψ(µλ,νλ)6=; )

,

and soB_n,kis measurable thanks to Corollary 2.3.

Denote now by a_n,k∈A_n,kthe “center” of the set, and define aΣ⊗ B(R^d)−measurable function by

T^k(λ,x) = X

n∈Z^d

a_n,k1B_n,k(λ,x). (3)

For eachλ∈E, letν_λ^kbe the discrete measure defined by pushing forwardµ_λthroughT^k, that is, ν_λ^k(A) =

Z

1T^k(λ,x)∈Aµλ(d x), A∈ B(R^d).

Denote also by ˜E∈Σ a measurable set withm(E˜^c) =0 and such that for all ˜λ∈E,˜ H(µ_λ^˜,ν_λ^˜,c) holds.

By hypothesis, for eachλ∈E˜we have that

µλ(d x)almost surely:1B_n,k(λ,x) =1_{x:Tλ(x)∈A_n,k}. (4) whereT_λhas been defined in the statement of Theorem 1.1. This implies that

ν_λ^k({a_n,k}) = Z

1_Bn,k(λ,x)µ_λ(d x) =µ_λ({x:T_λ(x)∈A_n,k}) =ν_λ(A_n,k) by definition ofT_λ.

We now check that(T^k)k∈Nis a Cauchy sequence inL¹(E×R^d,m(dλ)µ_λ(d x)). Fixk≤k⁰, and for eachn∈Z^d denote by{A_n0,k⁰}n⁰ the unique partition ofA_n,kin dyadic rectangles of size 2^{−d k0}. We then have that

Z

E

Z

R^d

|T^k(λ,x)−T^k0(λ,x)|µλ(d x)m(dλ)

= Z

E

Z

R^d

X

n∈Z^d

X

n⁰:A_n0,k0⊂A_n,k

1_B

n0,k0(λ,x)|a_n,k−a_n0,k⁰|µ_λ(d x)m(dλ)

= Z

E

X

n∈Z^d

X

n⁰:A_n0,k0⊂An,k

|a_n,k−a_n0,k⁰|νλ(A_n0,k⁰)m(dλ)

≤ Z

E

X

n∈Z^d

2^−k X

n⁰:A_n0,k0⊂A_n,k

ν_λ(A_n0,k⁰)m(dλ)

= Z

E

X

n∈Z^d

2^−kνλ(A_n,k)m(dλ)

=2^−k Z

E

νλ(R^d)m(dλ) =2^−km(E),

and the Cauchy property follows sincem(E)<∞.

Let us denote byT the limit in L¹(E×R^d,m(dλ)µ_λ(d x))of the sequenceT^k. Theorem 1.1 will be proved by verifying that for allλ in a set ofΣof full m-measure set, one hasπ_λ(d x,d y) = µ_λ(d x)δT(λ,x)(d y). To that end, it is enough to check that

Z

g(x)f(T(λ,x))µλ(d x) = Z

g(x)f(y)πλ(d x,d y)

for any pair f,g : R^d → R of bounded Lipschitz-continuous functions. We denote by kfk := sup_x6=y ^|f(x_|x−^)−f_y|^(y)|+sup_x|f(x)|andkgktheir corresponding norms.

We have forλ∈E˜that

Z

g(x)f(y)πλ(d x,d y)−

Z

g(x)f(T(λ,x))µλ(d x)

≤ Z

g(x)f(y)πλ(d x,d y)− Z

g(x)f(T^j(λ,x))µλ(d x) +kgkkfk

Z

T^j(λ,x)−T(λ,x) µλ(d x) :=∆j + ∆⁰j.

(5)

SinceR

T^j(λ,x)−T(λ,x)

µλ(d x)converges inL¹(m(dλ))to 0, there is setEˆ∈Σof full measure and a subsequenceT^ji such that lim_i→∞∆⁰j_i=0 for allλ∈E¯:=E˜∩E.ˆ

It is therefore enough to show that for allλ∈E, one has¯ ∆j→0 as jgoes to∞. First, we claim that forλ∈E, one has for any Borel set¯ C⊆R^d and alln∈Z^d,k∈Nthat

Z

1C×A_n,k(x,T^j(λ,x))µλ(d x) =πλ

€C×A_n,kŠ

for all j≥k. (6)

Fix a Borel set D_λof R^d of fullµλmeasure (which might depend on C, k andn) where (4) is everywhere true. Then,

Z

1_C×An,k(x,T^j(λ,x))µλ(d x) = Z

1_{(Dλ∩C)×An,k}(x,T^j(λ,x))µλ(d x)

= Z

1_{(Dλ∩C)×An,k} x, X

m∈Z^d

a_m,j1_Bm,j(λ,x)

! µ_λ(d x)

= Z

1₍D_λ∩C)×A_n,k





x, X

m:a_m,j∈An,k

a_m,j1A_m,j(T_λ(x))





µλ(d x). Remark now that for all j ≥ k, y ∈A_n,k ⇐⇒ P

m:am,j∈An,ka_m,j1A_m,j(y)∈A_n,k. This yields, for all j≥k,

Z

1_C×An,k(x,T^j(λ,x))µλ(d x) = Z

1_{(Dλ∩C)×An,k}(x,T_λ(x))µλ(d x) =πλ

€C×A_n,kŠ

establishing (6). Next, we introduce for eachk∈Nan approximation ofh:R^d →Ra bounded Lipschitz-continuous function by step functions

h^(k)(x):= X

n∈Z^d

h(a_n,k)1_An,k(x), x∈R^d.

One has|h(x)−h^(k)(x)| ≤ khkd2^−kfor allx∈R^d, from which it follows that

∆j≤d2^−k(4kfkkgk) + Z

g^(k)(x)f^(k)(T^j(λ,x))µλ(d x)− Z

g^(k)(x)f^(k)(y)πλ(d x,d y) ≤ d2^−k(4kfkkgk) + X

m,n∈Z^d

|g(a_n,k)||f(a_m,k)|

Z

1A_m,k×A_n,k(x,T^j(λ,x))µλ(d x)−πλ

€A_m,k×A_n,kŠ

for allk∈N. Thanks to (6), the sum identically vanishes for all j≥k. This means that∆j→0 as jgoes to∞, and the proof is complete.

3 Application: strong coupling for orthogonal martingale mea- sures

We now develop an application of Theorem (1.1), which is a generalization of what has been used in [3] to estimate the convergence rate of Landau type interacting particle systems. Let (Ω,F,Ft,P)be a filtered space and consider M an adapted orthogonal martingale measure on R₊×R^d(in the sense of Walsh[15]). Assume that its covariance measure has the formq_t(d a)d k_t, whereq_t(ω,d a)is a predictable random probability measure onR^dwith finite second moment and k_t a predictable increasing process. Let us also consider another predictable random probability measureˆq_t(ω,d a)onR^d with finite second moment.

We want to construct in the same probability space a second martingale measure with covariance measureˆq_t(ω,d a)d k_t, in such a way that in some sense, the distance between the martingale mea- sures is controlled by the Wasserstein distance between their covariance measures. This distance is defined forµ,ν∈ P2(R^d)byW₂²(µ,ν) =inf_π<^µ_νI(π)with the quadratic costc(x,y) =|x−y|². It makes the setP2(R^d)a Polish space, and strengthens the weak topology with the convergence of second moments (see e.g.[8]).

Theorem 3.1. In the previous setting, assume moreover that P(dω)d k_t(ω)a.e. q_t has a density with respect to Lebesgue measure inR^d.

Then, there exists in(Ω,F,Ft,P)a martingale measure M onˆ R₊×R^d with covariance measure ˆ

q_t(d a)d k_t, such that for all S>0and for every predictable functionφ:R₊×Ω×R^d →Rthat is Lipschitz continuous in the last variable withE

RS 0

Rφ²(s,a) q_s(d a) + ˆq_s(d a) d k_s

<∞, one has

E sup

t≤S

‚Z t

0

Z

φ(s,·,a)M(ds,d a)− Z t

0

Z

φ(s,·,a) ˆM(ds,d a)

Œ2!

≤E ZS

0

(K_s^φ)²W₂²(q_s,ˆq_s)d k_s

! , (7) where K_s^φ(ω) is a measurable version of a Lipschitz constant of a 7→ φ(s,ω,a) and W₂² is the quadratic Wasserstein distance inP2(R^d).

Proof.Sinceq_t(ω,d a)has a density for almost every(t,ω), assumption H(q_t(ω,d a),ˆq_t(ω,d a),c) is satisfied. We can therefore apply Theorem 1.1 to(E,Σ,m) = (Ω×R₊,Pr ed,P(dω)d k_t(ω)), wherePr edis the predictableσ−field with respect toFt. Then, there exists a predictable map- pingT :R₊×Ω×R^d→R^dthat form-almost every(t,ω)pushes forwardq_t toˆq_t. Moreover, for a.e.(t,ω), one has

Z

|a−T(t,ω,a)|²q_t(ω,d a) =W₂²(q_t,ˆq_t).

One can thus define a martingale measureMˆ by the stochastic integrals Z t

0

Z

ψ(s,a) ˆM(ds,d a):= Z t

0

Z

ψ(s,T(s,a))M(ds,d a)

for predictable simple functionsψ. Its covariance measure is by constructionqˆ_t(d a)d k_t, and by Doob’s inequality, the left hand side of (7) is less than

E ZS

0

Z

|φ(s,a)−φ(s,T(s,a))|²q_s(d a)d k_s

≤E ZS

0

(K_s^φ)²

‚Z

|a−T(s,a)|²q_s(d a)

Πd k_s

!

≤E ZS

0

(K_s^φ)²W₂²(q_s,ˆq_s)d k_s

! ,

by definition of Tand ofW₂².

Remark 3.2. In general, a coupling satisfying the estimate(7)can be obtained by constructing an or- thogonal martingale measureM˜(d t,d a,d a⁰)onR⁺×R^d×R^dwith covariance measureπt(d a,d a⁰)d k_t, πt being an optimal transference plan between q_t andˆq_t. Then, the marginal martingale measures M˜(d t,d a,R^d) and M˜(d t,R^d,d a⁰) are orthogonal martingale measures with the required covari- ances. Our argument provides a simple way of constructing M˜(d t,d a,d a⁰)when the realization of one of the “marginal” martingale measures is given in advance.

An immediate consequence is

Corollary 3.3. Let the space(Ω,F,Ft,P), the martingale measure M and the random measure q_t(d a)d k_t be as in Theorem 3.1. Assume that(qⁿ_t(ω,d a))n∈N is a sequence of predictable random probability measures onR^d in the same probability space, such that for all t>0,Rt

0 W₂²(q_s,q_sⁿ)d k_s goes to0in L¹(P)when n→ ∞.

Then, there exists a sequence(M_n)of orthogonal martingale measures in(Ω,F,Ft,P)with covari- ance measures(qⁿ_t(d a))such that for all t>0and allφwithksup_s≤tK_s^φkL^∞(P)<∞,Rt

0φ(s,a)Mⁿ(ds,d a) converges toRt

0φ(s,a)M(ds,d a)in L²(P).

Acknowledgements We thank the anonymous referees for pointing out a gap in the proof of Theorem 1.1 and for valuable comments that allowed us to improve the presentation of this work.

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