488 research outputs found
Rectifiability of Optimal Transportation Plans
The purpose of this note is to show that the solution to the Kantorovich
optimal transportation problem is supported on a Lipschitz manifold, provided
the cost is with non-singular mixed second derivative. We use this
result to provide a simple proof that solutions to Monge's optimal
transportation problem satisfy a change of variables equation almost
everywhere
Optimal transportation, topology and uniqueness
The Monge-Kantorovich transportation problem involves optimizing with respect
to a given a cost function. Uniqueness is a fundamental open question about
which little is known when the cost function is smooth and the landscapes
containing the goods to be transported possess (non-trivial) topology. This
question turns out to be closely linked to a delicate problem (# 111) of
Birkhoff [14]: give a necessary and sufficient condition on the support of a
joint probability to guarantee extremality among all measures which share its
marginals. Fifty years of progress on Birkhoff's question culminate in Hestir
and Williams' necessary condition which is nearly sufficient for extremality;
we relax their subtle measurability hypotheses separating necessity from
sufficiency slightly, yet demonstrate by example that to be sufficient
certainly requires some measurability. Their condition amounts to the vanishing
of the measure \gamma outside a countable alternating sequence of graphs and
antigraphs in which no two graphs (or two antigraphs) have domains that
overlap, and where the domain of each graph / antigraph in the sequence
contains the range of the succeeding antigraph (respectively, graph). Such
sequences are called numbered limb systems. We then explain how this
characterization can be used to resolve the uniqueness of Kantorovich solutions
for optimal transportation on a manifold with the topology of the sphere.Comment: 36 pages, 6 figure
Systematic co-occurrence of tail correlation functions among max-stable processes
The tail correlation function (TCF) is one of the most popular bivariate
extremal dependence measures that has entered the literature under various
names. We study to what extent the TCF can distinguish between different
classes of well-known max-stable processes and identify essentially different
processes sharing the same TCF.Comment: 31 pages, 4 Tables, 5 Figure
Hedonic price equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness
Hedonic pricing with quasi-linear preferences is shown to be equivalent to stable matching with transferable utilities and a participation constraint, and to an optimal transportation (Monge-Kantorovich) linear programming problem. Optimal assignments in the latter correspond to stable matchings, and to hedonic equilibria. These assignments are shown to exist in great generality; their marginal indirect payoffs with respect to agent type are shown to be unique whenever direct payoffs vary smoothly with type. Under a generalized Spence-Mirrlees condition (also known as a twist condition) the assignments are shown to be unique and to be pure, meaning the matching is one-to-one outside a negligible set. For smooth problems set on compact, connected type spaces such as the circle, there is a topological obstruction to purity, but we give a weaker condition still guaranteeing uniqueness of the stable match
Root to Kellerer
We revisit Kellerer's Theorem, that is, we show that for a family of real
probability distributions which increases in convex
order there exists a Markov martingale s.t.\ .
To establish the result, we observe that the set of martingale measures with
given marginals carries a natural compact Polish topology. Based on a
particular property of the martingale coupling associated to Root's embedding
this allows for a relatively concise proof of Kellerer's theorem.
We emphasize that many of our arguments are borrowed from Kellerer
\cite{Ke72}, Lowther \cite{Lo07}, and Hirsch-Roynette-Profeta-Yor
\cite{HiPr11,HiRo12}.Comment: 8 pages, 1 figur
Exclusion processes in higher dimensions: Stationary measures and convergence
There has been significant progress recently in our understanding of the
stationary measures of the exclusion process on . The corresponding
situation in higher dimensions remains largely a mystery. In this paper we give
necessary and sufficient conditions for a product measure to be stationary for
the exclusion process on an arbitrary set, and apply this result to find
examples on and on homogeneous trees in which product measures are
stationary even when they are neither homogeneous nor reversible. We then begin
the task of narrowing down the possibilities for existence of other stationary
measures for the process on . In particular, we study stationary measures
that are invariant under translations in all directions orthogonal to a fixed
nonzero vector. We then prove a number of convergence results as
for the measure of the exclusion process. Under appropriate initial conditions,
we show convergence of such measures to the above stationary measures. We also
employ hydrodynamics to provide further examples of convergence.Comment: Published at http://dx.doi.org/10.1214/009117905000000341 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Hedonic price equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness
Hedonic pricing with quasilinear preferences is shown to be equivalent to stable matching with transferable utilities and a participation constraint, and to an optimal transportation (Monge-Kantorovich) linear programming problem. Optimal assignments in the latter correspond to stable matchings, and to hedonic equilibria. These assignments are shown to exist in great generality; their marginal indirect payoffs with respect to agent type are shown to be unique whenever direct payoffs vary smoothly with type. Under a generalized Spence-Mirrlees condition the assignments are shown to be unique and to be pure, meaning the matching is one-to-one outside a negligible set. For smooth problems set on compact, connected type spaces such as the circle, there is a topological obstruction to purity, but we give a weaker condition still guaranteeing uniqueness of the stable match. An appendix resolves an old problem (# 111) of Birkhoff in probability and statistics [5], by giving a necessary and sufficient condition on the support of a joint probability to guarantee extremality among all joint measures with the same marginals.
- …