multiparameter weights with connections to schauder bases abstract we give several characterizations multiparameter a p t d in terms of strong maximal functions rectangular conjugate functions and rectangular partial sums we also define a way to enumerate z d that corresponds to rectangles we then use these enumerations to give another characterization of rectangular a p t d in terms of trigonometric schauder bases finally we present some applications to principle shift invariant spaces and gabor systems

power penalty method for solving hjb equations arising from finance nonlinear hamilton jacobi bellman hjb equation commonly occurs in financial modeling implicit numerical scheme is usually applied to the discretization of the continuous hjb so as to find its numerical solution since it is generally difficult to obtain its analytic viscosity solution this type of discretization results in a nonlinear discrete hjb equation we propose a power penalty method to approximate this discrete equation by a nonlinear algebraic equation containing a power penalty term under some mild conditions we give the unique solvability of the penalized equation and show its convergence to the original discrete hjb equation moreover we establish a sharp convergence rate of the power penalty method which is of an exponential order with respect to the power of the penalty term we further develop a damped newton algorithm to iteratively solve the lower order penalized equation finally we present a numerical experiment solving an incomplete market optimal investment problem to demonstrate the rates of convergence and effectiveness of the new method we also numerically verify the efficiency of the power penalty method by comparing it with the widely used policy iteration method

nonlinear small data scattering for the wave and klein gordon equation on etudie l operateur diffusion qui appartient a la paire d equations u tt au f u 0 et u tt au 0 a represente l operateur j 1 n 2 x j 2 m 2 m r et f u u 1 u ou r 1

mass concentration phenomena for the l 2 critical nonlinear schrodinger equation in this paper we show that any solution of the nonlinear schr o dinger equation iu t delta u pm u frac 4 n u 0 which blows up in finite time satisfies a mass concentration phenomena near the blow up time our proof is essentially based on the bourgain s one cite mr99f 35184 which has established this result in the bidimensional spatial case and on a generalization of strichartz s inequality where the bidimensional spatial case was proved by moyua vargas and vega cite mr1671214 we also generalize to higher dimensions the results in keraani cite mr2216444 and merle and vega cite mr1628235

on the blow up phenomenon of the critical nonlinear schrodinger equation abstract in this paper we consider the blow up phenomenon of critical nonlinear schrodinger equations in dimension 1d and 2d we define the minimal mass as the l 2 norm necessary to ignite a wave collapse and we stress its role in the blow up mechanism asymptotic compactness properties and l 2 concentration are proved the proof relies on linear and nonlinear profile decompositions

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

nonlinear black scholes equations in finance associated control problems and properties of solutions we study properties of solutions to fully nonlinear versions of the standard black scholes partial differential equation these equations have been introduced in financial mathematics in order to deal with illiquid markets or with stochastic volatility we show that typical nonlinear black scholes equations can be viewed as dynamic programming equation of an associated control problem we establish existence and comparison results and show that the equation induces a convex risk measure on the set of all continuous terminal value claims moreover we study the asymptotic behavior of solutions as market frictions get large finally the pricing of individual contracts relative to a book of derivatives is discussed

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

the global cauchy problem for the non linear klein gordon equation on demontre l unicite de solutions faibles et l existence et l unicite de solutions globales fortement continues dans l espace energie

on multivariate discrete moment problems generalization of the bivariate min algorithm for higher dimensions the objective of the multivariate discrete moment problem mdmp is to find the minimum and or maximum of the expected value of a function of a random vector with a discrete finite support where the probability distribution is unknown but some of the moments are given the moments may be binomial power or of a more general type the mdmp can be formulated as a linear programming problem with a very ill conditioned coefficient matrix hence the lp problem can be solved with difficulty or cannot be solved at all the central results of the field of the mdmp concern the structure of the dual feasible bases these bases on one hand provide us with bounds without any numerical difficulties on the other hand they can be used as an initial basis of the dual simplex method that results in shorter running time and better numerical stability because the first phase can be skipped this paper introduces a new type of mdmp where the bivariate moments up to a certain order m consisting of the first variable and further univariate moments up to the order m j j 1 ldots s are given then we generalize the bivariate min algorithm of madi nagy and prekopa math oper res 29 2004 pp 229 258 for higher dimensions which gives numerous dual feasible bases of the mdmp by the aid of this on one hand we can give useful bounds for mdmps with higher dimensional random vectors even if the usual solvers cannot give acceptable results on the other hand applying our algorithm for the binomial mdmp we can give better bounds for probabilities of boolean functions of event sequences than the recent bounds in the literature these results are illustrated by numerical examples

the structure of finitely generated shift invariant spaces in l2 rd a simple characterization is given of finitely generated subspaces of l2 rd which are invariant under translation by any multi integer and is used to give conditions under which such a space has a particularly nice generating set namely a basis and more than that a basis with desirable properties such as stability orthogonality or linear independence the last property makes sense only for local spaces i e shift invariant spaces generated by finitely many compactly supported functions and special attention is paid to such spaces as an application we prove that the approximation order provided by a given local space is already provided by the shift invariant space generated by just one function with this function constructible as a finite linear combination of the finite generating set for the whole space hence compactly supported this settles a question of some 20 years standing

the global cauchy problem for the critical non linear wave equation abstract we study the global cauchy problem for the non linear wave equation p 1 0 for the critical value p n 2 n 2 in space dimension n 3 we identify a weak space time integrability property stip of the solutions and prove that it is sufficient to ensure the uniqueness of weak solutions the global existence of finite energy solutions with the naturally associated stip and the global existence of regular solutions with some n dependent restrictions on the regularity for spherically symmetric solutions we prove that the previous crucial stip follows from the morawetz inequality actually in a much stronger form than necessary thereby proving that all the previous results hold in the spherically symmetric case

oscillations and concentration effects in semilinear dispersive wave equations let un be a sequence of smooth solutions to a dispersive nonlinear wave equation formula inr1 3 with uniformly compactly supported cauchy data converging weakly to 0 inh1 r3 l2 r3 let vn be the sequence of solutions to the linear wave equation with the same cauchy data we show thatun vngoes strongly to 0 in the energy spacec 0 t h1 c1 0 t l2 if f is a subcritical nonlinearity in the critical casef u u5 we show that this property is equivalent tovn 0 inl 0 t l6 then we give sharp sufficient conditions on microlocal measures associated to the data the proof relies on a microlocal version of p l lions concentration compacity soit un une suite de solutions regulieres d une equation d ondes non lineaire dispersive formula dansr1 3 avec donnees de cauchy supportees dans un compact fixe et convergeant faiblement vers 0 dansh1 r3 l2 r3 soit vn la suite des solutions de l equation des ondes lineare avec les memes donnees de cauchy on montre queun vnconverge fortement vers 0 dans l espace d energiec 0 t h1 c1 0 t l2 sifest une non linearite sous critique dans le cas critiquef u u5 on montre que cette propriete equivaut a la convergence forte devnvers 0 dansl 0 t l6 on en donne alors des conditions suffisantes optimales en termes de mesures microlocales associees aux donnees la demonstration est basee sur une version microlocale de la concentration compacite de p l lions

stabilization and control for the nonlinear schrodinger equation on a compact surface in this paper we study the stabilization property and the exact controllability for the nonlinear schrodinger equation on a two dimensional compact riemannian manifold without boundary we use a pseudo differential dissipation the proofs are based on a result of propagation of singularities and on recent dispersion estimates strichartz type inequalities due to n burq p gerard and n tzvetkov

monge property and bounding multivariate probability distribution functions with given marginals and covariances multivariate probability distributions with given marginals are considered along with linear functionals to be minimized or maximized acting on them the functionals are supposed to satisfy the monge or inverse monge or some higher order convexity property and they may be only partially known existing results in connection with monge arrays are reformulated and extended in terms of linear programming dual feasible bases lower and upper bounds are given for the optimum value as well as for unknown coefficients of the objective function based on the knowledge of some dual feasible bases and corresponding objective function coefficients in the two and three dimensional cases dual feasible bases are obtained for the problem where not only the univariate marginals but also the covariances of the pairs of random variables are known

high frequency approximation of solutions to critical nonlinear wave equations this work is devoted to the description of bounded energy sequences of solutions to the equation 1 u u 4 0 in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i up to remainder terms small in energy norm and in every strichartz norm the proof relies on scattering theory for 1 and on a structure theorem for bounded energy sequences of solutions to the linear wave equation in particular we infer the existence of an a priori estimate of strichartz norms of solutions to 1 in terms of their energy

strichartz estimates and wave equation in a conic singular space consider the metric cone x c y 0 infty r times y with metric g dr 2 r 2h where the cross section y is a compact n 1 dimensional riemannian manifold y h let delta g be the positive friedrichs extension laplacian on x and let delta h be the positive laplacian on y and consider the operator mathcal l v delta g v 0 r 2 where v 0 in mathcal c infty y such that delta h v 0 n 2 2 4 is a strictly positive operator on l 2 y in this paper we prove global in time strichartz estimates without loss regularity for the wave equation associated with the operator mathcal l v it verifies a conjecture in wang remark 2 4 in ann inst fourier 56 1903 1945 2006 for wave equation the range of the admissible pair is sharp and the range is influenced by the smallest eigenvalue of delta h v 0 n 2 2 4 to prove the result we show a sobolev inequality and a boundedness of a generalized riesz transform in this setting in addition as an application we study the well posed theory and scattering theory for energy critical wave equation with small data on this setting of dimension n ge 3

a high order finite difference method for option valuation in this paper we propose the use of an efficient high order finite difference algorithm to price options under several pricing models including the blackscholes model the mertons jumpdiffusion model the hestons stochastic volatility model and the nonlinear transaction costs or illiquidity models we apply a local mesh refinement strategy at the points of singularity usually found in the payoff of most financial derivatives to improve the accuracy and restore the rate of convergence of a non uniform high order five point stencil finite difference scheme for linear models the time stepping is dealt with by using an exponential time integration scheme with carathodoryfejr approximations to efficiently evaluate the product of a matrix exponential with a vector of option prices nonlinear blackscholes equations are solved using an efficient iterative scheme coupled with a richardson extrapolation our numerical experiments clearly demonstrate the high order accuracy of the proposed finite difference method making the latter a method of choice for solving both linear and nonlinear partial differential equations in financial engineering problems

control of the black scholes equation the purpose of this work is to apply the results developped by j y chemin and cl david to the black scholes equation this latter equation being directly linked to the heat equation it enables us to propose a new approach allowing to control properties of the solution by means of a shape parameter

monge properties discrete convexity and applications abstract starting from monge s mass transportation problem we review the role monge properties play in optimization in particular we discuss transportation problems whose cost functions fulfill a monge property monge sequences algebraic monge properties the recognition of permuted monge arrays and multidimensional monge arrays and the connections between monge properties and discrete convexity finally we discuss prekopa s recent approach using monge arrays in bounding multivariate probability distribution functions

minimal initial data for potential navier stokes singularities abstract assuming some initial data u 0 h 1 2 r 3 lead to a singularity for the 3d navier stokes equations we show that there are also initial data with the minimal h 1 2 norm which will produce a singularity

an endpoint balian low theorem for schauder bases abstract we prove two new results on the validity of the balian low theorem in the setting of schauder bases our first main result proves an endpoint balian low theorem for gabor systems that form schauder bases for l 2 r if g l 2 r is compactly supported and 2 g 2 d then the gabor system g g 1 1 cannot be a schauder basis of so called type our second main result proves that the classical balian low theorem for orthonormal bases and riesz bases fails in the setting of schauder bases in particular given 0 there exists g l 2 r such that g g 1 1 is a schauder basis for l 2 r and such that t 3 g t 2 d t and 3 g 2 d

on stability of schauder bases of integer translates abstract we apply the garnett jones distance to the analysis of schauder bases of translates a special role is played by periodization functions p with ln p in the closure of l in bmo t in particular for schauder bases with such periodization functions we study the corresponding coefficient space we also use the garnett jones distance approach to show the stability of bases of translates with respect to convolution powers the case of democratic conditional schauder bases of translates is emphasized as well