A canonical martingale coupling

(1)

Workshop on Optimal Transportation and Appplications

Nicolas JUILLET

Université de Strasbourg

Pisa, November 2012

(2)

A canonical martingale coupling

Outline

1 The martingale transport plans

2 Tools for the martingale transport problem

3 Results

Nicolas JUILLET A canonical martingale coupling

(3)

A canonical martingale coupling The martingale transport plans

Definition: martingale transport plan

A probability measurePonR×Ris termed a martingale transport plan if P=Law(X,Y)where(X,Y)is a two-times martingale process.

Equivalently if(P_x)_x_∈_Ris a disintegration (allias conditional laws, allias Markov kernel) ofP, it has to satisfy

Barycenter(P_x) = Z

ydP_x(y) =x

for(proj^x_#P)-almost everyx.

(4)

Some examples

P=Law(x,Y)wherex=E(Y).

P=∑²_i=1∑³_j=1a_i,jδ₍_x_i_,_y_j₎wherex∈ {−1,1}andy∈ {−2,0,2}and (a_i,j) =

1/4 1/4 0 1/12 1/12 1/3

+t

1/12 −1/6 1/12

−1/12 1/6 −1/12

for somet∈[0,1].

P=Law(X,X+I)where the incrementIis independent fromX. (for instanceX andIare Gaussian)

P=^P¹⁺₂^P² whereP₁,P₂are martingale transport plans.

P=Law(E(Y|

F

^),^Y⁾^{for some}

F

^⊆^σ(^Y⁾^.

(5)

The general problem

The problem Minimize

P7→

Z

c(x,y)dP(x,y) among the martingale transport plans fromµtoν.

For different cost functionscwe would like to know:

How do the minimizers look like?

What are their properties?

Is there a unique minimizer?

(6)

Model theorem

Model theorem in the classical setting

Forµandνin

P

2in the convex order andPa transport plan fromµtoν. The following statements are equivalent:

The planPis optimal for the transport problem withc(x,y) = (y−x)², The planPis concentrated on a monotone setΓ,

The planPis the quantile coupling.

We have proved a theorem similar to this one in the martingale setting.

(7)

A canonical martingale coupling Tools for the martingale transport problem

The convex order

Definition: the convex order We write

µCν

and say thatµis smaller thanνin the convex order if and only if there exists a martingale transport planPwith

proj^x_#P=µ and proj^y_#P=ν.

According to a (non constructive) theorem of Strassen, it is equivalent to assume Z

ϕdµ≤ Z

ϕdν

for every convex functionϕ:R→R^.

(8)

The extended order and the shadows

Proposition - Definition

We writeµ_Eνand say thatµis smaller thanνin the extended order if F_µ^ν:={θ:µCθandθ≤ν}

is not empty.

The partially ordered set(F_µ^ν,C)has a minimum. We call it the shadow ofµin νand denote it byS^ν(µ).

µ

ν γ1

γ2

S^ν(γ₁) =ν1

S^ν−ν¹(γ₂)

γ1

µ

γ2

ν S^ν(γ1) =ν1 S^ν−ν¹(γ2)

Figure:Shadow ofµinνand associativity of the shadow projection.

(9)

The variational lemma

This lemma is a kind ofc-cyclical monotonicity lemma for the martingale setting.

Variational Lemma

LetPbe optimal, there existsΓ⊆R×Rsuch that for any finitely supported measure αwithα(Γ) =1, the minimum ofα⁰7→^Rc(x,y)dα⁰(x,y)over

Competitor(α) =





 α⁰:α⁰

has the same marginals asα

∀x∈R, Z

ydα_x= Z

ydα⁰_x





 is obtained inα.

(10)

A canonical martingale coupling Results

Martingale theorem

Theorem

Forµandνin

P

3in the convex order andPa martingale transport plan fromµtoν. The following statements are equivalent:

The planPis optimal for the martingale transport problem with cost c(x,y) = (y−x)³,

The planPis concentrated on a martingale-monotone setΓ(see the figure), The planPis the left- curtain coupling (i.e., transportsµ_]−∞,_x_]to its shadow)

x x⁰

y⁻ y⁰ y⁺

Figure:This configuration of three points(x,y),(x⁰,y⁻)and(x⁰,y⁺)is forbidden on martingale-monotone setsΓ.

(11)

One example

x x⁰

T₂(x⁰) T₁(x⁰) T₁(x) =T₂(x)

Figure:Optimal transport plan between Gaussian measures.

(12)

Corollary

Corollary forµcontinuous

Ifµis continuous (=no atom), there areT₁,T₂:R→Rsuch that the optimalPis concentrated ongraph(T₁)∪graph(T₂).

The variational lemma is of general use, especially whenµis continuous.

Examples

c(x,y) =−|y−x| c(x,y) =|y−x| c(x,y) = (y−x)ⁿ