537 research outputs found
Infinite-body optimal transport with Coulomb Cost
We introduce and analyze symmetric infinite-body optimal transport (OT)
problems with cost function of pair potential form. We show that for a natural
class of such costs, the optimizer is given by the independent product measure
all of whose factors are given by the one-body marginal. This is in striking
contrast to standard finite-body OT problems, in which the optimizers are
typically highly correlated, as well as to infinite-body OT problems with
Gangbo-Swiech cost. Moreover, by adapting a construction from the study of
exchangeable processes in probability theory, we prove that the corresponding
-body OT problem is well approximated by the infinite-body problem.
To our class belongs the Coulomb cost which arises in many-electron quantum
mechanics. The optimal cost of the Coulombic N-body OT problem as a function of
the one-body marginal density is known in the physics and quantum chemistry
literature under the name SCE functional, and arises naturally as the
semiclassical limit of the celebrated Hohenberg-Kohn functional. Our results
imply that in the inhomogeneous high-density limit (i.e. with
arbitrary fixed inhomogeneity profile ), the SCE functional converges
to the mean field functional.
We also present reformulations of the infinite-body and N-body OT problems as
two-body OT problems with representability constraints and give a dual
characterization of representable two-body measures which parallels an
analogous result by Kummer on quantum representability of two-body density
matrices.Comment: 22 pages, significant revision
Multi-marginal optimal transport: theory and applications
Over the past five years, multi-marginal optimal transport, a generalization
of the well known optimal transport problem of Monge and Kantorovich, has begun
to attract considerable attention, due in part to a wide variety of emerging
applications. Here, we survey this problem, addressing fundamental theoretical
questions including the uniqueness and structure of solutions. The (partial)
answers to these questions uncover a surprising divergence from the classical
two marginal setting, and reflect a delicate dependence on the cost function.
We go one to describe two applications of the multi-marginal problem.Comment: Typos corrected and minor changes to presentatio
A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options
We consider the problem of superhedging under volatility uncertainty for an
investor allowed to dynamically trade the underlying asset, and statically
trade European call options for all possible strikes with some given maturity.
This problem is classically approached by means of the Skorohod Embedding
Problem (SEP). Instead, we provide a dual formulation which converts the
superhedging problem into a continuous martingale optimal transportation
problem. We then show that this formulation allows us to recover previously
known results about lookback options. In particular, our methodology induces a
new proof of the optimality of Az\'{e}ma-Yor solution of the SEP for a certain
class of lookback options. Unlike the SEP technique, our approach applies to a
large class of exotics and is suitable for numerical approximation techniques.Comment: Published in at http://dx.doi.org/10.1214/13-AAP925 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
60 years of cyclic monotonicity: a survey
The primary purpose of this note is to provide an instructional summary of
the state of the art regarding cyclic monotonicity and related notions. We will
also present how these notions are tied to optimality in the optimal transport
(or Monge-Kantorovich) problem
Optimal transportation for a quadratic cost with convex constraints and applications
We prove existence of an optimal transport map in the Monge-Kantorovich
problem associated to a cost which is not finite everywhere, but
coincides with if the displacement belongs to a given convex
set and it is otherwise. The result is proven for satisfying
some technical assumptions allowing any convex body in and any convex
polyhedron in , . The tools are inspired by the recent
Champion-DePascale-Juutinen technique. Their idea, based on density points and
avoiding disintegrations and dual formulations, allowed to deal with
problems and, later on, with the Monge problem for arbitrary norms
A study of the dual problem of the one-dimensional L-infinity optimal transport problem with applications
The Monge-Kantorovich problem for the infinite Wasserstein distance presents
several peculiarities. Among them the lack of convexity and then of a direct
duality. We study in dimension 1 the dual problem introduced by Barron, Bocea
and Jensen. We construct a couple of Kantorovich potentials which is "as less
trivial as possible". More precisely, we build a potential which is non
constant around any point that the plan which is locally optimal moves at
maximal distance. As an application, we show that the set of points which are
displaced to maximal distance by a locally optimal transport plan is minimal
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