MOTDim d Martingale Optimal Transport in Higher Hadrien De March Dimension Optimal transport Formulation of the problems Optimal Hadrien De March transport in practice Martingale CMAP, Ecole Polytechnique optimal transport Difference with classical Under the direction of Nizar Touzi transport State of the art in MoT June 29, 2016 Complete duality in higher dimension Reformulation of the problem Results Work to be done
Outline MOTDim d Hadrien De 1 Optimal transport March Formulation of the problems Optimal Optimal transport in practice transport Formulation of the problems Optimal 2 Martingale optimal transport transport in practice Difference with classical transport Martingale optimal State of the art in MoT transport Difference with classical transport 3 Complete duality in higher dimension State of the art in MoT Reformulation of the problem Complete duality in Results higher dimension Reformulation of 4 Work to be done the problem Results Work to be done
Table of Contents MOTDim d Hadrien De 1 Optimal transport March Formulation of the problems Optimal Optimal transport in practice transport Formulation of the problems Optimal 2 Martingale optimal transport transport in practice Difference with classical transport Martingale optimal State of the art in MoT transport Difference with classical transport 3 Complete duality in higher dimension State of the art in MoT Reformulation of the problem Complete duality in Results higher dimension Reformulation of 4 Work to be done the problem Results Work to be done
The Monge optimal transport problem Originally a soil moving problem for building. MOTDim d Hadrien De March Optimal transport Formulation of the problems Optimal transport in practice Martingale Figure: The Monge problem illustrated. optimal transport Difference with classical transport State of the art in MoT Complete duality in higher dimension Reformulation of the problem Results Figure: The cost of moving building brick. Work to be done
Probabilistic optimal transport MOTDim d Kantorovitch has been the one making this problem Hadrien De probabilistic, so that it becomes a linear problem. March Optimal (Ω , E ) = ( R d × R d , B ( R d × R d )) transport Formulation of the problems X and Y the two canonical random variables Ω → R d , Optimal transport in practice X : ( x , y ) �→ x and Y : ( x , y ) �→ y . Martingale P ( µ, ν ) := { P ∈ P ( R d × R d ) / P ◦ X − 1 = µ, P ◦ Y − 1 = ν } optimal transport the set of all coupling probability laws between µ and ν . Difference with classical transport State of the art Definition in MoT Complete The optimal transport problem is: duality in higher dimension P ∈P ( µ,ν ) E P ( c ( X , Y )) Reformulation of P = inf the problem Results Work to be done
The dual problem We define the dual set: MOTDim d Hadrien De ( φ, ψ ) ∈ L 1 ( µ ) × L 1 ( ν ) , � D µ,ν ( c ) = March � ∀ x , y ∈ R d , c ( x , y ) ≥ φ ( x ) + ψ ( y ) Optimal transport Formulation of the problems Optimal Definition transport in practice The dual problem is Martingale optimal transport D := sup µ ( φ ) + ν ( ψ ) Difference with classical transport ( φ,ψ ) ∈D µ,ν ( c ) State of the art in MoT Complete duality in higher Remark dimension Reformulation of the problem If ( φ, ψ ) ∈ D µ,ν ( c ) and P ∈ P ( µ, ν ) then Results Work to be done E P [ c ( X , Y )] ≥ P ≥ D ≥ P [ φ ( X ) + ψ ( Y )] = µ ( φ ) + ν ( ψ )
Kantorovitch Duality The parallel study of these two problems is justified by the MOTDim d following theorem: Hadrien De March Theorem Optimal transport If the cost c is lower semicontinuous and dominated by Formulation of the problems a ⊕ b ∈ L ( µ ) ⊕ L ( ν ) , then Optimal transport in practice There is duality: P = D . Martingale There are optimizers ( φ ∗ , ψ ∗ ) ∈ D µ,ν ( c ) for D and optimal transport P ∗ ∈ P ( µ, ν ) for P . Difference with classical transport There is a Borel support Γ ⊂ R 2 d such that P ∈ P ( µ, ν ) is State of the art in MoT concentrated on Γ if and only if it is optimal. Complete duality in higher dimension Proof Reformulation of the problem E P ∗ [ c ( X , Y ) − φ ∗ ( X ) − ψ ∗ ( Y )] = P − D = 0 Results Work to be done and c ( X , Y ) − φ ∗ ( X ) − ψ ∗ ( Y ) ≥ 0. ✷
Useful cost functions MOTDim d Hadrien De ( x , y ) �→ d ( x , y ) where d ( · , · ) is a distance. March ( x , y ) �→ | x − y | 2 Optimal transport ( x , y ) �→ | x − y | Formulation of the problems Optimal transport in practice Definition Martingale optimal We define the p -Wasserstein distance for p ≥ 1: for transport µ, ν ∈ P ( R d ), Difference with classical transport State of the art � 1 in MoT � p W p ( µ, ν ) := P ∈P ( µ,ν ) E P [ | X − Y | p ] Complete inf duality in higher dimension Reformulation of the problem ( P ( R d ) , W p ) is a Polish space. Results Work to be done
The Monge-Ampere Equation How to compute the optimal transport in practice ? Equation MOTDim d on the dual optimizer: Hadrien De March Theorem Optimal transport µ ( dx ) = f ( x ) dx and ν ( dy ) = g ( y ) dy Formulation of the problems Optimal y �→ ∂ x c ( x , y ) is injective transport in practice Martingale Then P ∗ [ Y = T ( X )] = 1 with T ( x ) = ∂ x c ( x , · ) − 1 ( φ ( x )) optimal transport Difference with f ( x ) classical Det ( − D 2 φ ( x )+ ∂ xx c ( x , T ( x ))) = | Det ( ∂ x , y c ( x , T ( x ))) | transport State of the art g ( T ( x )) in MoT Complete duality in higher If c ( x , y ) = − x · y , we get the Monge-Ampere equation: dimension Reformulation of the problem f ( x ) Results Det ( − D 2 φ ( x )) = Work to be g ( T ( x )) done
Armadillo to ball optimal transport MOTDim d Example: Optimal transport from Armadillo to ball. Hadrien De March Optimal transport Formulation of the problems Optimal transport in practice Martingale optimal transport Difference with classical transport State of the art in MoT Complete duality in higher dimension Reformulation of the problem Results Work to be done
Table of Contents MOTDim d Hadrien De 1 Optimal transport March Formulation of the problems Optimal Optimal transport in practice transport Formulation of the problems Optimal 2 Martingale optimal transport transport in practice Difference with classical transport Martingale optimal State of the art in MoT transport Difference with classical transport 3 Complete duality in higher dimension State of the art in MoT Reformulation of the problem Complete duality in Results higher dimension Reformulation of 4 Work to be done the problem Results Work to be done
Change in the Primal and the Dual Problems MOTDim d Definition Hadrien De The martingale optimal transport problem and its dual are: March Optimal E P ( c ( X , Y )) P = sup transport Formulation of P ∈M ( µ,ν ) the problems Optimal transport in practice and Martingale D := ( φ,ψ, h ) ∈D µ,ν ( c ) µ ( φ ) + ν ( ψ ) inf optimal transport Difference with classical transport State of the art in MoT With M ( µ, ν ) := { P ∈ P ( µ, ν ) / E P [ Y | X ] = X a . s . } and Complete duality in higher � ( φ, ψ, h ) ∈ L 1 ( µ ) × L 1 ( ν ) × L ( R d , R d ) , dimension D µ,ν ( c ) = Reformulation of the problem Results � ∀ x , y ∈ R d , c ( x , y ) ≤ φ ( x ) + ψ ( y )+ h ( x ) · ( y − x ) Work to be done
Application to finance: robust superhedging problem MOTDim d S t is the price vector of d tradable underlyings. Hadrien De March We are at time 0 and we consider a payoff c ( S 1 , S 2 ). Optimal We can buy vanilla payoffs φ ( S 1 ) and ψ ( S 2 ). transport Formulation of Vanilla prices given by the implied measures µ and ν . the problems Optimal transport in One delta hedge at time 1. Costs h ( S 1 ) · S 1 at time 1 and practice pays h ( S 1 ) · S 2 at time 2. Martingale optimal transport We want to cover perfectly the payoff with the Difference with classical superhedging. transport State of the art in MoT c ( S 1 , S 2 ) ≤ φ ( S 1 ) + ψ ( S 2 ) + h ( S 1 ) · ( S 2 − S 1 ) Complete duality in higher dimension Problem: cheapest price of the superhedging Reformulation of the problem Results price ( φ ( S 1 ) + ψ ( S 2 ) + h ( S 1 ) · ( S 2 − S 1 )) = µ ( φ ) + ν ( ψ ) Work to be done
Complete Duality in dimension 1 MOTDim d Hadrien De March (Beiglbock-Nutz-Touzi 15) showed that in dimension 1 Optimal Need to consider the M ( µ, ν )-quasi sure problem. transport Formulation of the problems Need to use a ”convex moderator” to extend µ ( φ ) + ν ( ψ ) Optimal transport in practice There is duality P = D for non negative measurable costs. Martingale optimal The dual problem has an optimizer ( φ, ψ, h ). transport Difference with R is decomposed in ”irreducible components” convex and classical transport stable by M ( µ, ν ) on which pointwise duality holds. State of the art in MoT There is a monotone set Γ that supports any optimal Complete duality in probability for the primal problem. higher dimension Reformulation of the problem Results Work to be done
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