A canonical martingale coupling Workshop on Optimal Transportation - PowerPoint PPT Presentation

A canonical martingale coupling A canonical martingale coupling Workshop on Optimal Transportation and Appplications Nicolas JUILLET Université de Strasbourg Pisa, November 2012 Nicolas JUILLET A canonical martingale coupling

A canonical martingale coupling Outline 1 The martingale transport plans Tools for the martingale transport problem 2 3 Results Nicolas JUILLET A canonical martingale coupling

A canonical martingale coupling The martingale transport plans Definition: martingale transport plan A probability measure P on R × R is termed a martingale transport plan if P = Law ( X , Y ) where ( X , Y ) is a two-times martingale process. Equivalently if ( P x ) x ∈ R is a disintegration (allias conditional laws, allias Markov kernel) of P , it has to satisfy � Barycenter ( P x ) = y d P x ( y ) = x for ( proj x # P ) -almost every x . Nicolas JUILLET A canonical martingale coupling

A canonical martingale coupling The martingale transport plans Some examples P = Law ( x , Y ) where x = E ( Y ) . P = ∑ 2 i = 1 ∑ 3 j = 1 a i , j δ ( x i , y j ) where x ∈ {− 1 , 1 } and y ∈ {− 2 , 0 , 2 } and � � � � 1 / 4 1 / 4 1 / 12 − 1 / 6 1 / 12 0 ( a i , j ) = + t 1 / 12 1 / 12 1 / 3 − 1 / 12 1 / 6 − 1 / 12 for some t ∈ [ 0 , 1 ] . P = Law ( X , X + I ) where the increment I is independent from X . (for instance X and I are Gaussian) P = P 1 + P 2 where P 1 , P 2 are martingale transport plans. 2 P = Law ( E ( Y | F ) , Y ) for some F ⊆ σ ( Y ) . Nicolas JUILLET A canonical martingale coupling

A canonical martingale coupling The martingale transport plans The general problem The problem Minimize � P �→ c ( x , y ) d P ( x , y ) among the martingale transport plans from µ to ν . For different cost functions c we would like to know: How do the minimizers look like? What are their properties? Is there a unique minimizer? Nicolas JUILLET A canonical martingale coupling

A canonical martingale coupling The martingale transport plans Model theorem Model theorem in the classical setting For µ and ν in P 2 in the convex order and P a transport plan from µ to ν . The following statements are equivalent: The plan P is optimal for the transport problem with c ( x , y ) = ( y − x ) 2 , The plan P is concentrated on a monotone set Γ , The plan P is the quantile coupling. We have proved a theorem similar to this one in the martingale setting. Nicolas JUILLET A canonical martingale coupling

A canonical martingale coupling Tools for the martingale transport problem The convex order Definition: the convex order We write µ � C ν and say that µ is smaller than ν in the convex order if and only if there exists a martingale transport plan P with proj x proj y # P = µ # P = ν . and According to a (non constructive) theorem of Strassen, it is equivalent to assume � � ϕ d µ ≤ ϕ d ν for every convex function ϕ : R → R . Nicolas JUILLET A canonical martingale coupling

A canonical martingale coupling Tools for the martingale transport problem The extended order and the shadows Proposition - Definition We write µ � E ν and say that µ is smaller than ν in the extended order if F ν µ := { θ : µ � C θ and θ ≤ ν } is not empty. The partially ordered set ( F ν µ , � C ) has a minimum. We call it the shadow of µ in ν and denote it by S ν ( µ ) . µ µ γ 2 γ 1 γ 1 γ 2 ν ν S ν − ν 1 ( γ 2 ) S ν ( γ 1 ) = ν 1 S ν − ν 1 ( γ 2 ) S ν ( γ 1 ) = ν 1 Figure: Shadow of µ in ν and associativity of the shadow projection. Nicolas JUILLET A canonical martingale coupling

A canonical martingale coupling Tools for the martingale transport problem The variational lemma This lemma is a kind of c -cyclical monotonicity lemma for the martingale setting. Variational Lemma Let P be optimal, there exists Γ ⊆ R × R such that for any finitely supported measure α with α (Γ) = 1, the minimum of α ′ �→ c ( x , y ) d α ′ ( x , y ) over �   has the same marginals as α    α ′ : α ′ Competitor ( α ) = � � y d α ′ ∀ x ∈ R , y d α x = x  is obtained in α . Nicolas JUILLET A canonical martingale coupling

A canonical martingale coupling Results Martingale theorem Theorem For µ and ν in P 3 in the convex order and P a martingale transport plan from µ to ν . The following statements are equivalent: The plan P is optimal for the martingale transport problem with cost c ( x , y ) = ( y − x ) 3 , The plan P is concentrated on a martingale-monotone set Γ ( see the figure ), The plan P is the left- curtain coupling ( i.e., transports µ ] − ∞ , x ] to its shadow ) x ′ x y − y ′ y + Figure: This configuration of three points ( x , y ) , ( x ′ , y − ) and ( x ′ , y + ) is forbidden on martingale-monotone sets Γ . Nicolas JUILLET A canonical martingale coupling

A canonical martingale coupling Results One example x ′ x T 2 ( x ′ ) T 1 ( x ′ ) T 1 ( x ) = T 2 ( x ) Figure: Optimal transport plan between Gaussian measures. Nicolas JUILLET A canonical martingale coupling

A canonical martingale coupling Results Corollary Corollary for µ continuous If µ is continuous (=no atom), there are T 1 , T 2 : R → R such that the optimal P is concentrated on graph ( T 1 ) ∪ graph ( T 2 ) . The variational lemma is of general use, especially when µ is continuous. Examples c ( x , y ) = −| y − x | c ( x , y ) = | y − x | c ( x , y ) = ( y − x ) n Nicolas JUILLET A canonical martingale coupling