Optimal Martingale Transport problem in higher dimensions
Nassif Ghoussoub
University of British Columbia
Based on joint work with
Young-Heon Kim
&
Tongseok Lim (UBC)
September, 2015
Varese
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Main Points of the talk:
I
Comparison between optimal transport problem and optimal
martingale transport problem.
I
The one-dimensional case.
I
Optimal martingale transport problem and Skorohod embeddings
in Brownian motion.
I
Recent progress on the optimal martingale transport problem in
higher dimension.
I
Conjectures / open problems.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Optimal Transport Problem
Given two Borel probability measures µ, ν on Rd , one considers the set
Π(µ, ν) of probability measures π on Rd × Rd whose marginals are µ, ν. They
are called a transport plans from µ to ν.
Given a cost function c : Rd × Rd → R, one needs to characterize the optimal
transfer plans, that is those π ∈ Π(µ, ν) that minimize or maximize
Z
min
c(x, y )dπ(x, y ).
π∈Π(µ,ν)
X ×Y
Motivation:
I
[Economics] find a most cost-effective matching between
resources/agents:
I
[Physics] density functional theory: describe the state of many
correlated particles.
I
[Mathematics] Appropriate distances between measures? Notions of
barycenters of a family of measures?
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Equivalent probabilistic statement
A coupling of µ and ν is a pair of random variables X : Ω → Rd , Y : Ω → Rd
on a probability space (Ω, F, P) such that Law(X ) = µ, Law(Y ) = ν.
An equivalent problem is to study the couplings that minimize the expected
cost
min EP c(X , Y ).
X ∼µ,Y ∼ν
I
Existence of optimal π usually follows easily from standard compactness.
I
When is an optimal measure π unique?
I
When is an optimal measure π concentrated on a graph {(x, T (x))},
where T : X → Y is a map?
Originated by Monge (19th century) and Kantorovitch (1940’s).
I
Brenier ’87, Gangbo ’95, Caffarelli ’96, Gangbo-McCann ’96, Levin ’96,
Ma-Trudinger-Wang ’05, ... :
Assume µ << dx and that c satisfies the twist condition, i.e,:
y 7→ Dx c(x, y ) is injective
(e.g. c(x, y ) = |x − y |2 .)
Then π is concentrated on a graph and is unique.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
A dual problem
There is an important dual problem (Monge-Kantorovich)
Z
Z
Z
min
c(x, y )dπ = sup
β(y )dν −
α(x)dµ; (β, α) ∈ D(c, µ, ν)
π∈Π(µ,ν)
X ×Y
Y
1
X
1
D(c, µ, ν) is the set of (α, β) ∈ L (µ) × L (ν) so that
β(y ) − α(x) ≤ c(x, y ) for all (x, y ) ∈ X × Y .
If c(x, y ) = hx, y i, then α can be taken to be any convex function and
β = −α∗ , the Legendre transform of α (Fenchel-Young inequality).
.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Optimal Martingale Transport Problem
I
Consider now Borel probability measures µ, ν on Rd in convex order;
µ ≤C ν, that is
R
R
φ dµ ≤ Rd φ dν for all φ : Rd → R convex.
Rd
I
Consider the set MT (µ, ν) of probability measures π on Rd × Rd with
marginals µ, ν, but also whose disintegration (πx )x∈Rd satisfy that the
barycenter of each πx is x (martingale constraint).
These are the martingale transport plans.
Recall that the disintegration (πx )x∈Rd of π with respect to the first variable is:
dπ(x, y ) = dπx (y )dµ(x).
The basic problem is to characterize the optimal solutions of the
maximization or minimization problem
Z
min
c(x, y )dπ(x, y ).
π∈MT (µ,ν)
Rd ×Rd
Motivation:
I [Finance] find the maximum / minimum price of the option whose value
depends on the stock π, given the information that can be observed from
the market (the marginals).
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Equivalent probabilistic statement and duality
Consider all couplings X : Ω → Rd , Y : Ω → Rd on a probability space
(Ω, F, P) that form a 1-step martingale, that is
I
Law(X ) = µ, Law(Y ) = ν,
I
E(Y |X ) = X .
Note that for B ⊂ Rd , πx (B) = P(Y ∈ B|X = x).
Study the (1-step) martingales (stocks) (X , Y ) with prescribed marginals,
which maximize / minimize the expected cost (option price)
min
X ∼µ,Y ∼ν,E(Y |X )=X
EP c(X , Y ).
We also have a duality problem
Z
Z
min
c(x, y ) dπ; π ∈ MT (µ, ν) = sup
R2d
Z
βdν −
Rd
αdµ; (α, β) ∈ Dm (c)
Rd
Dm (c) is the set of (α, β) ∈ L1 (µ) × L1 (ν) so that for some γ ∈ Cb (Rd ),
β(y ) − α(x) − γ(x)(y − x) ≤ c(x, y ) for all (x, y ) ∈ Rd × Rd .
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
1-dimensional results
Theorem (Beiglböck-Juillet ’13)
Suppose µ ≤C ν on R and µ << L1 . Let c(x, y ) = |x − y |. Then, there exists
a unique minimizing martingale transport plan π that is concentrated on a set
Γ ⊂ R × R such that |Γx | ≤ 3 for every x ∈ R.
More precisely, π can be decomposed into πstay + πgo , where
πstay = (Id × Id)# (µ ∧ ν) (this measure is concentrated on the diagonal of R2 )
and πgo is concentrated on graph(T1 ) ∪ graph(T2 ) where T1 , T2 are two
decreasing real functions.
Theorem (Hobson-Neuberger ’13)
Suppose µ ≤C ν on R and µ << L1 . Let c(x, y ) = |x − y |. Then, there exists
a unique maximizing martingale transport plan π that is concentrated on a set
Γ ⊂ R × R such that π is concentrated on Γ and |Γx | ≤ 2 for every x ∈ R.
More precisely, π is concentrated on graph(T1 ) ∪ graph(T2 ) where T1 , T2 are
two increasing real functions.
Question What is a right extension of these results to higher dimensions?
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Equivalent problem; a.k.a. the Skorokhod embedding problem
Let (Ω, F, Ft , P) be a probability space with Brownian motion Btx valued in R.
If µ and ν are in convex order on R, then there exists a (randomized) stopping
time τ such that
Law(B0 ) = µ and Law(Bτ ) = ν.
So, given a cost function c : Rd × Rd → R, it suffices to study the optimal
stopping time τ which "embeds" µ to ν, that is:
min
B0 ∼µ,Bτ ∼ν
EP c(B0 , Bτ ).
Problem What can be said on the optimal time? Is it unique? Is it
non-random? Is it a hitting time of a "barrier"?
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
The higher-dimensional case is much richer
I
I
Suppose now that µ and ν are such that
R
R
φ(y ) dµ(y ) ≤ Rd φ(y ) dν(y ) for all φ subharmonic on Rd ,
Rd
then we say that µ ≤SH ν in the subharmonic order.
If µ = δx , then x is the subharmonic barycenter for ν.
The corresponding problem is then
Z
min
c(x, y )dπ(x, y )
π∈SHMT (µ,ν)
I
I
I
I
Rd ×Rd
where SHMT (µ, ν) is set of probability measures π on Rd × Rd with
marginals µ, ν and such that their disintegration (πx )x∈Rd have
subharmonic barycenter at x.
In R1 every convex function is subharmonic and vice versa.
In higher dimension, genuine differences between MT and SHMT
problem emerge; in particular, not every measure in P(Rd ) has a
subharmonic barycenter.
This stringent constraint on πx makes it difficult to vary a given
disintegration (πx )x∈Rd to another, in order to use "calculus of variation".
A similar set of problems if one consider plurisubharmonic functions.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
However, we have the Brownian approach
I
If Btx is an Rd -valued Brownian motion starting at x ∈ Rd , and if f is a
subharmonic function on Rd , then f (Btx )t≥0 is a submartingale.
I
If τ is a stopping time, and ν is the distribution ν of Bτ , where B is
Brownian motion starting at µ, then µ ≤SH ν in the subharmonic order.
I
Vice-versa, if µ ≤SH ν, then there exists a Brownian motion starting at µ
and a randomized stopping time τ such that ν ∼ Bτ .
Theorem (Ghoussoub, Kim & Lim, to appear)
Let µ and ν be radially symmetric probability measures on Rd which are in
subharmonic order. Then, there exists a stopping time τ that minimizes
min
B0 ∼µ,Bτ ∼ν
EP |B0 − Bτ |.
Moreover, it is of barrier-type, and in particular, it is unique.
Same for the case of a plurisubharmonic order.
Deep problem: Can one do without assuming radial symmetry?
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Back to the convex order in higher-dimensions
The first result in higher dimension is due to Lim. He considers the case
when the marginals are radially symmetric.
Theorem (Lim ’14)
Assume that µ and ν are radially symmetric probability measures in convex
order on Rd , and that µ << Ld . Consider the minimization problem. Then:
I
Under any optimal solution π, the common mass µ ∧ ν does not move.
I
WLOG suppose µ ∧ ν = 0. Then for µ a.e. x, the disintegration
(conditional probability) πx is concentrated on two points; say
{T1 (x), T2 (x)}, where T1 (x), T2 (x) lie on the 1-dimensional subspace
spanned by x. In particular, the optimal solution π is unique.
Idea By using radial symmetry, one can use variational methods and show
that the disintegrations (πx )x∈Rd must be supported on the one-dimensional
subspace spanned by x. Then one can reduce the problem to
one-dimensional situation.
Remark Optimal solutions in the maximization problem behave very
differently even in the case of radially symmetric marginals.
Main problem What if µ and ν are non-radial?
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Two conjectures
Let Γ ⊂ Rd × Rd be a support of an optimal martingale measure π.
I
Γx is the transported image of x; Γx := {y ∈ Rd ; (x, y ) ∈ Γ}, that is a
support of πx for each x.
What can be said about the "geometry" of the fiber Γx , for each x?
Assume that c(x, y ) = |x − y |, µ << Ld .
Conjecture 1: If π is an optimal martingale solution for either maximization or
minimization problem. Then, for µ almost every x, Γx is supported on the set
of extreme points) of the closed convex hull of Γx , i.e.,
Γx ⊂ Ext conv(Γx ) .
Conjecture 2: If µ ∧ ν = 0 and π is an optimal martingale solution for the
minimization problem. Then for µ almost every x, the set Γx consists of k + 1
points that form the vertices of a k -dimensional polytope, where k := k (x) is
the dimension of the linear span of Γx and therefore, the minimizing solution
is unique.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Back to optimal transport
Let X , Y be arbitrary sets, and c : X × Y → R be a function. A subset
Γ ⊂ X × Y is said to be c-cyclically monotone if, for any N ∈ N, and any
family (x1 , y1 ), ..., (xN , yN ) of points in Γ, holds the inequality
N
X
c(xi , yi ) ≤
i=1
N
X
c(xi , yi+1 )
i=1
(with the convention yN+1 = y1 ).
A transport plan is said to be c-cyclically monotone if it is concentrated on a
c-cyclically monotone set.
Roughly, it is a plan that cannot be improved: it is impossible to perturb it (in
the sense considered before, by rerouting mass along some cycle) and get
something more economical.
Theorem
A transfer plan π is optimal if and only if its support Γ is c-cyclically monotone.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Dual statement to the c-cyclical monotonicity
If Γ ⊂ X × Y , we let XΓ be the projection of Γ onto the first coordinate space
Rd and YΓ to the second.
Theorem
Γ ⊂ X × Y is c-cyclically monotone if and only if for each finite subset H ⊂ Γ,
there exist a pair of functions αH : XH → R, βH : YH → R, such that
βH (y ) − αH (x) ≤ c(x, y ) ∀x ∈ XH , y ∈ YH ,
βH (y ) − αH (x) = c(x, y ) ∀(x, y ) ∈ H.
Remark This follows from the duality theorem in finite-dimensional linear
programming.
Question Can one find a dual pair (α, β) that works for the whole set Γ?
Answer Yes; for example for the canonical cost c(x, y ) = 21 |x − y |2 , this
follows from Rockafellar’s theorem.
Theorem (Rockafellar ’69)
Suppose that Γ ⊂ Rd × Rd is a cyclically monotone set. Then there exists a
convex function α on Rd such that Γ ⊂ ∂α.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Implication of Rockafellar’s theorem
Let c(x, y ) = 12 |x − y |2 and let Γ ⊂ Rd × Rd be c-cyclically monotone set.
Rockafellar’s theorem yield convex functions (α, β) on Rd such that
1
|x − y |2 ∀x ∈ Rd , y ∈ Rd ,
2
1
β(y ) − α(x) = |x − y |2 ∀(x, y ) ∈ Γ.
2
β(y ) − α(x) ≤
In particular, whenever (x, y ) ∈ Γ, we have
α(x) +
1
1
|x − y |2 ≤ α(x 0 ) + |x 0 − y |2 ∀x 0 ∈ Rd .
2
2
Now if α is differentiable at x, then
∇x (α(x) +
1
|x − y |2 ) = 0
2
⇐⇒
y = x + ∇α(x).
The presence of regular solution for the dual problem implies good properties
on the optimal transport plans.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Monotonicity principle for martingale transport problem
Definition: Say that 2 measures π 1 , π 2 on Rd × Rd are competitors, if they
have the same marginals and Rif the barycenters
R of their respective
disintegrations are the same: Rd y dπx1 (y ) = Rd y dπx2 (y ), ∀x ∈ Rd .
Definition: Say that Γ ⊂ Rd × Rd is martingale-monotone if for any finite
subset H ⊂ Γ and any measure πH supported on H, we have for any
competitor π 0 of πH ,
Z
Z
c(x, y )dπH (x, y ) ≤ c(x, y )dπ 0 (x, y )
Theorem(Monotonicity principle) (Beiglböck-Juillet ’13)
Let π be an optimal solution for martingale transport problem. Then there
exists a martingale-monotone support Γ of π.
An Interpretation of martingale-monotonicity:
• spt(πH ) ⊆ Γ means that πH is a “subplan" of the full transport plan π
• A competitor means that if we change the subplan πH to π 0 , then the
martingale
R structure
R of πH will not be disrupted.
• Now if c dφ > c dψ, then we may modify π to have ψ as its subplan,
achieving less cost, meaning that the current plan π is not a minimizer.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Dual statement of the Monotonicity principle
Let c : Rd × Rd → R be a cost function. Say that a subset G of Rd × Rd
admits a c-dual, or that G is c-dualizable, if there exists a triple {α, β, γ},
α : XG → R, β : YG → R, γ : XG → Rd , such that the following duality relation
(for the minimization problem) holds:
β(y ) − α(x) − γ(x) · (y − x) ≤ c(x, y ) ∀x ∈ XG , y ∈ YG ,
(0.1)
β(y ) − α(x) − γ(x) · (y − x) = c(x, y ) ∀(x, y ) ∈ G.
(0.2)
In the maximization problem, the inequality in (0.1) is reversed.
We will call the triple {α, β, γ} a c-dual for G.
Theorem
Suppose Γ ⊂ Rd × Rd is martingale-monotone. Then any finite subset H ⊂ Γ
admits a c-dual.
Question Can one then find a dual triplet (α, β, γ) for the whole martingale
monotone set Γ? Is there a Rockafellar-type theorem?
Answer No. At least in the maximization case.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Dual problem is NOT attained in general
Example
Let µ = ν be two identical probability measures on the interval [0, 1], then the
only martingale (say π) from µ to itself is the identity transport, hence it is
obviously the solution of the maximization problem with respect to the
distance cost, and its support is Γ = {(x, x) : x ∈ [0, 1]}. If now {α, β, γ} is a
solution to the dual problem, then
β(y ) ≥ |x − y | + γ(x) · (y − x) + α(x) ∀x ∈ [0, 1], ∀y ∈ [0, 1];
β(y ) = |x − y | + γ(x) · (y − x) + α(x) ∀(x, y ) ∈ Γ.
The above relations easily yield that for any 0 < a < b < 1, we have
γ(a) + 2 ≤ γ(b), which means that it is impossible to define a suitable
real-valued function γ for a.e. x in [0, 1].
Remark In the case of the minimization problem, the triplet (0, 0, 0) is a
c-dual.
Conjecture 3: In the minimization problem, the dual problem is always
attained.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Good things happen if duality is attained
Lemma: Suppose Γ ⊂ Rd × Rd admits a dual triplet (α, β, γ). If α and γ are
differentiable at x, then Γx is supported on the set of extreme points of the
closed convex hull of Γx .
Proof: If (x, y ) ∈ Γ, then
|x − y | + γ(x) · (y − x) + α(x) ≤ |x 0 − y | + γ(x 0 ) · (y − x 0 ) + α(x 0 ) ∀x 0
Hence
∇x (|x − y | + γ(x) · (y − x) + α(x))
x −y
+ ∇γ(x) · (y − x) − γ(x) + ∇α(x) = 0.
=
|x − y |
Now suppose that we can find {y , y0 , ..., ys } ⊂ Γx with y = Σsi=0 pi yi ,
Σsi=0 pi = 1, pi > 0. Then we get
s
X x − yi
x −y
=
pi
.
|x − y |
|x − yi |
i=0
But this can hold only if all yi lie on the ray emanating from x.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Once there is a dual, it can be extended & improved
Lemma: Suppose Γ admits a c-dual triplet {α, β, γ}. Set Ω := int(conv (YΓ )).
Then, there exists α : Ω → R, β : Rd → R, and γ : Ω → Rd such that
{α, β, γ} coincides with {α, β, γ} on XΓ , YΓ , XΓ , respectively, (thus, is a c-dual
triplet for Γ), and α is locally Lipschitz. Furthermore, γ is differentiable at
points where α is differentiable.
Proof.
We define the extensions as
α(x) := inf{c ∈ R : ∃a ∈ Rd such that β(y ) − |x − y | ≤ a · (y − x) + c ∀ y ∈ YΓ }
γ(x) := {a ∈ Rd : β(y ) − |x − y | ≤ a · (y − x) + α(x) ∀y ∈ YΓ
β(y ) − |x − y | = a · (y − x) + α(x) ∀y ∈ Γx }
β(y ) := inf {|x − y | + γ(x) · (y − x) + α(x)}.
x∈Ω
Then...
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
The conclusion in the presence of a dual
Theorem
Let π be an optimal solution for martingale transport problem and let Γ be its
support. Suppose that µ << Ld and Γ admits a dual. Then for µ a.e. x,
Γx ⊂ Ext conv(Γx ) .
I
We can find a triplet (α, β, γ) such that α is locally Lipshitz, hence
differentiable for a.e x. Since µ << Ld , it is differentiable for µ a.e x,
hence Conjecture 1 holds.
I
As in the optimal transport case, exploiting the dual side of the problem
can be powerful.
I
However, the dual is not always attained in the martingale transport
case, at least in the maximization problem.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Splitting the problem
We then try to partition the support Γ of a martingale transport into subsets
so that the restricted Martingale transport on each component attains its dual.
Theorem: Decomposition into irreducible components: Let Γ be a
support of a martingale transport. Then, there exists an equivalence relation
∼ on XΓ such that or each x ∈ XΓ , there exists an open convex subset C(x)
of IC(YΓ ) such that
1. x ∼ x 0 if and only if C(x) = C(x 0 );
2. If x x 0 , then C(x) ∩ C(x 0 ) = ∅.
3. If Γ is martingale monotone (hence its finite subsets admit c-duals), then
for each x ∈ X , Γ ∩ (C(x) × Rd ) admits a c-dual.
• Γ ∩ (C(x) × Rd ) is the largest subset of Γ for which one can get a dual.
• But in the minimization problem with distance cost, we suspect that the dual
problem is always attained for the whole optimal support Γ.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Disintegration of martingale plans
Let (µ, ν) be probability measures on Rd in convex order, and π ∈ MT (µ, ν).
Assume further that µ ∧ ν = 0 in the case of minimization problem. Then,
there exists a support Γ of π and a probability measure π̃ on K(Rd ) such that:
1. For each Borel set S ⊂ Rd × Rd , we have
Z
π(S) =
πC (S)d π̃(C),
K(Rd )
where for π̃-a.e. C, πC is a probability supported on ΓC := Γ ∩ (C × Rd ).
2. For π̃-a.e. C, there exist probability measures µC , νC such that (µC , νC )
is in convex order, µC is supported on XC := X ∩ C, νC on YΓC , and
πC ∈ MT (µC , νC );
3. If π is optimal on MT(µ, ν), then for π̃-a.e. C, πC is optimal for
MT (µC , νC ), and its corresponding dual problem is also attained, that is
ΓC := Γ ∩ (C × Rd ) has a c-dual.
4. If µC is absolutely continuous with respect to Lebesgue measure on
V (C), and c(x, y ) = |x − y |, then for µC -almost all x, Γx ⊂ Ext(conv(Γx )).
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
If µ could be disintegrated along the partitions in an a.c. way, we are
done. But...
I
We know that each piece Γ ∩ (C(x) × Rd ) admits a dual.
I
So if the source measure µ were "restricted" (disintegrated) on each
partition in an absolutely continuous way, then one would apply the
previous theorem on
each partition and conclude that, for µ a.e. x,
Γx ⊂ Ext conv(Γx ) .
I
Unfortunately, this is not always the case.
A counterexample (Ambrosio, Kirchheim, and Pratelli) They constructed
a Nikodym set in R3 having full measure in the unit cube.
Yet, it intersects each element of a family of pairwise disjoint open lines only
in one point.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Nikodym set connected with MG transport; unhappy disintegration can
indeed happen
Consider the following obvious inequality,
1
(|x
2
− y | − )2 ≥ 0, hence
1
1
1
|y |2 ≥ |x − y | + x · (y − x) + |x|2 − .
2
2
I
I
I
I
I
(0.3)
1
1
|x|2 − , β (y ) = 2
|y |2 and γ (x) = 1 x, this is a dual
Letting α (x) = 2
relation for the maximization problem.
It shows that every martingale π := (X , Y ) with |X − Y | = a.s. is
optimal with its own marginals X ∼ µ and Y ∼ ν.
Fix > 0 and let X be a random variable whose distribution µ has
uniform density on [−1, 1]3 . Define Y conditionally on X by evenly
distributing the mass along the lines lx in the Nikodym set at distance .
That is Y splits equally in two pieces from x ∈ X along lx with distance .
Then the martingale (X , Y ) is optimal for the maximization problem.
But each equivalence class [x] is the singleton {x}, so the disintegration
of µ along the partitions C(x) is the Dirac mass δx , which is obviously
not absolutely continuous w.r.t. L1 . Hence, the decomposition is not
useful in this case.
Note that the martingale (X , Y ) defined here has codimension 2.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
The codimension ≤ 1 case
Theorem
Let c(x, y ) = |x − y |, and let π ∈ MT (µ, ν) be an optimal solution with a
support Γ. Assume µ << Ld and suppose that for µ a.e. x,
dim(V (C(x))) ≥ d − 1.
Then for µ almost every x ∈ Rd , Γx ⊂ Ext(conv(Γx )).
Idea If each convex partition {C(x)}x∈X has codimension at most 1, then by
mutual disjointness, their relative "direction" cannot be as wild as the Nikodym
set case. In fact we can define a bi-Lipschitz map by which the partitions are
parallelized, so the disintegration of µ can be done in an a.c. way.
Corollary
Let π be a solution of the optimization problem with c(x, y ) = |x − y | and Γ
be its support. Suppose µ << Ld . Then for µ-almost every x, the Hausdorff
dimension of Γx is at most d-1.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Conjecture 2 in the case of a discrete target
Theorem
Let c(x, y ) = |x − y |, suppose µ << Ld and that ν is discrete; i.e. ν is
supported on a countable set. Then for µ a.e. x, Γx consists of exactly d + 1
points which are vertices of a polytope in Rd , and therefore the optimal
solution is unique.
Remark The result holds for both max/min problem. Note that not only the
µ << Ld , but also the "singular" nature of ν help here.
This result along with the radial symmetric case.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
What is unsatisfactory?
I
The codimension ≤ 1 assumption is not "practically checkable", unless
d = 2.
I
We do not have an example of non-existence of a dual for minimization
problem.
I
Even though we have an example of non-dual existence for
maximization problem, still we do not have an example of optimal
solution which is NOT supported on the Choquet boundary; hence the
conjecture may still be valid for the maximization problem as well.
I
For conjecture 2, we have no idea. More precisely, we do not understand
what really distinguishes the min / max problems.
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions
Thank You !
Nassif Ghoussoub
Optimal Martingale Transport problem in higher dimensions