On a problem of optimal transport under marginal martingale constraints

By Mathias Beiglbock, Nicolas Juillet. 2012

236 citations recorded by the source corpus.

The basic problem of optimal transportation consists in minimizing the expected costs E[c(X 1 , X 2)] by varying the joint distribution (X 1 , X 2) where the marginal distributions of the random variables X 1 and X 2 are fixed. Inspired by recent applications in mathematical finance and connections with the peacock problem, we study this problem under the additional condition that (X i) i=1,2 is a martingale, that is, E[X 2 |X 1 ] = X 1. We establish a variational principle for this problem which enables us to determine optimal martingale transport plans for specific cost functions. In particular, we identify a martingale coupling that resembles the classic monotone quantile coupling in several respects. In analogy with the celebrated theorem of Brenier, the following behavior can be observed: If the initial distribution is continuous, then this "monotone martingale" is supported by the graphs of two functions T 1 , T 2 : R → R.

Subfields: Applied Mathematics, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/c2281b709f62896cacb591e62c48f48043ea4f97

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Citations in the corpus

Adapted Wasserstein distances and stability in mathematical finance (2019, methodology): “e eld, see the surveys of Hobson [34] and Ob loj [45]. Recently this has been complemented by techniques coming from (martingale) optimal transport, early papers which advance this viewpoint include [35, 14, 26, 15, 19, 24, 22, 13]. The literature on ‘local’ misspecication of volatility in a sense m”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “On the contrary, it is possible to realize as a combination of two transport maps, and optimality results of this kind are proven in [31, 193].”
Multi-marginal optimal transport: theory and applications (2014, background): “In the later variant, for one dimensional marginals, explicit solutions for the two marginal case have been found [5,43] under natural conditions on the cost.”
Complete Duality for Martingale Optimal Transport on the Line (2015, background): “In analogy to the Hoeffding– Fréchet coupling of classical transport, [7] establishes a measure P , the so– called Left-Curtain Coupling, that is optimal in (1.”
Structure of optimal martingale transport plans in general dimensions (2015, methodology): “We note that this result can be seen as a generalization of the decomposition of Beiglböck-Juillet [5] in the one-dimensional case d = 1, where the disintegration comes from restricting the measures μ, ν onto open subintervals of R obtained by examining the potential functions for μ, ν.”
Monotone martingale transport plans and Skorokhod embedding (2017, unknown): “Since the extension of the above to arguments to the more general case is straightforward and the result is known from [5] we do not elaborate.”
Stability of entropic optimal transport and Schrödinger bridges (2021, background): “Related to the present work at least in spirit, there are several areas where analogues of c-cyclical monotonicity have recently lead to breakthroughs, including martingale optimal transport [7], optimal Skorokhod embeddings [5, 8] and weak transport [4].”
Arbitrage and duality in nondominated discrete-time models (2013, background): “The authors thank Mathias Beiglböck and Jan Obłój for their constructive comments.”
Martingale Benamou–Brenier: A probabilistic perspective (2017, background): “While [13, 31] propose a martingale version of Brenier’s monotone transport mapping, our starting point is the Benamou-Brenier continuous time transport problem which we restate here for comparison with the martingale analogues that we will consider subsequently.”
Irreducible convex paving for decomposition of multidimensional martingale transport plans (2017, methodology): “Motivated by the martingale monotonicity principle of Beiglböck & Juillet [3] (see also Zaev [22] for higher dimension and general linear constraints), their strategy is to find a monotone set Γ Ă RdˆRd, where the robust superhedging holds with equality, as a support of the optimal martingale transp”
The geometry of relative arbitrage (2014, background): “ptimal relative arbitrage when the covariance matrix is uncertain. The theory of optimal transport also arises in this context, see, for example, Beiglbock et al. [BHLP13] and Beiglbock and Juillet [BJ14], although the motivation is quite dierent from this paper. 1.2. Notations. Let n2 be the number”
A new class of costs for optimal transport planning (2018, methodology): “1 Around the barycentre constraint As noted in [5], when one selects the barycentre constraint H =Hbar then the quadratic cost c(x, y)= |y− x|2 turns (P) to an easy problem since the functional to be minimised is constant over the admissible transport plans.”
Existence, duality, and cyclical monotonicity for weak transport costs (2018, background): “More recently, variants of this ‘monotonicity priniciple’ have been applied in transport problems for finitely or infinitely many marginals [32, 18, 26, 11, 44], the martingale version of the optimal transport problem [12, 31, 13], the Skorokhod embedding problem [9] and the distribution constrained”
Sampling of probability measures in the convex order and approximation of Martingale Optimal Transport problems (2017, unknown): “right-curtain coupling πrc(dx, dy) = 121[−1,1](x)dx ( 3 4δ 32x− 1 2 (dy) + 1 4δ 32− x 2 (dy) ) introduced by Beiglböck and Juillet ([4], Theorem 1.”
On a mixture of Brenier and Strassen Theorems (2018, methodology): “This optimal transport problem with martingale contraint has been thoroughly studied in [6] for the dimension 1.”