2,541 research outputs found
An optimal matching problem
Given two measured spaces (X,dx), (Y,dy) and a third space Z, given two
functions u(x,z) and v(x,z), we study the problem of finding two maps s from X
to Z and t from Y to Z such that the images s(dx) and t(dy) coincide, and the
integral of u(x,s(x))dx+v(y,t(y))dy is maximal. We give condition on u and v
for which there is a unique solution
A theory of bond portfolios
We introduce a bond portfolio management theory based on foundations similar
to those of stock portfolio management. A general continuous-time zero-coupon
market is considered. The problem of optimal portfolios of zero-coupon bonds is
solved for general utility functions, under a condition of no-arbitrage in the
zero-coupon market. A mutual fund theorem is proved, in the case of
deterministic volatilities. Explicit expressions are given for the optimal
solutions for several utility functions.Comment: Published at http://dx.doi.org/10.1214/105051605000000160 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
A surjection theorem for maps with singular perturbation and loss of derivatives
In this paper we introduce a new algorithm for solving perturbed nonlinear
functional equations which admit a right-invertible linearization, but with an
inverse that loses derivatives and may blow up when the perturbation parameter
goes to zero. These equations are of the form
with , small and given, small and unknown. The main
difference with the by now classical Nash-Moser algorithm is that, instead of
using a regularized Newton scheme, we solve a sequence of Galerkin problems
thanks to a topological argument. As a consequence, in our estimates there are
no quadratic terms. For problems without perturbation parameter, our results
require weaker regularity assumptions on and than earlier ones, such as
those of Hormander. For singularly perturbed functionals, we allow to be
larger than in previous works. To illustrate this, we apply our method to a
nonlinear Schrodinger Cauchy problem with concentrated initial data studied by
Texier-Zumbrun, and we show that our result improves significantly on theirs.Comment: Final version, to appear in Journal of the European Mathematical
Society (JEMS
An implicit function theorem for non-smooth maps between Fr\'echet spaces
We prove an inverse function theorem of Nash-Moser type for maps between
Fr\'echet spaces satisfying tame estimates. In contrast to earlier proofs, we
do not use the Newton method, that is, we do not use quadratic convergence to
overcome the lack of derivatives. In fact, our theorem holds when the map to be
inverted is not C^
Matching for Teams.
We are given a list of tasks Z and a population divided into several groups X j of equal size. Performing one task z requires constituting a team with exactly one member x j from every group. There is a cost (or reward) for participation: if type x j chooses task z, he receives p j (z); utilities are quasi-linear. One seeks an equilibrium price, that is, a price system that distributes all the agents into distinct teams. We prove existence of equilibria and fully characterize them as solutions to some convex optimization problems. The main mathematical tools are convex duality and mass transportation theory. Uniqueness and purity of equilibria are discussed. We will also give an alternative linear-programming formulation as in the recent work of Chiappori et al.Matching; Equilibria; Convex duality; Optimal transportation;
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