endpoint estimates for one dimensional oscillatory integral operators abstract the one dimensional oscillatory integral operator associated to a real analytic phase s is given by t f x e i s x y x y f y d y in their fundamental work phong and stein established sharp l 2 estimates for t the goal of this paper is to extend their results to all endpoints in particular we obtain a complete characterization for the mapping properties for t on l p r more precisely we show that t f p f p holds for some 0 if and only if 1 p 1 p lies in the reduced newton polygon of s

some generic invariant factor assignment results using dynamic output feedback abstract working with input output transfer functions in the frequency domain and exploiting a formulation involving generalized sylvester resultants we are able to derive necessary and sufficient conditions for generic invariant factor assignment in several cases using proper dynamic output feedback compensators

models of degenerate fourier integral operators and radon transforms fourier integral operators whose lagrangians project with singularities arise frequently in many areas of analysis and geometry 3 5 9 10 18 20 however there are as yet few analytic tools available for their study and even their simplest regularity properties with respect to the singularities of the lagrangian are still obscure 4 5 13 16 in this paper we study a class of oscillatory integrals ta and fourier integral operators r which can be expected to model the higher order singularities of the lagrangian they have homogeneous polynomial phases in two variables of order n and indeed the case n 3 is the model for lagrangians which project with whitney folds 3 10 11 the main difficulty which sets the higher order degeneracy cases n 4 apart from the the lower ones n 2 3 is that the critical varieties which arise there are usually not smooth manifolds a systematic study of these oscillatory integrals was begun in 14 the main idea in that work was to treat the critical varieties as smooth manifolds away from a lower dimensional subvariety and to keep track of the distance to this lower dimensional subvariety to achieve this we introduced a method of stationary phase which exhibited clearly the dependence on the distance between critical points the method gave sharp bounds on the size of the kernel k x y of tat but no information on its phase and the resulting cancellations it does not seem possible to refine this approach to the phase level and this suggests looking instead for decompositions of the operators ta which can incorporate indirectly the required cancellations the main goal of this paper is to introduce such decompositions these decompositions which we will try to describe momentarily reflect the singular nature of the critical points and are powerful enough to yield sharp bounds for the above oscillatory integrals and radon transforms although the models under consideration are very

direct estimation of the risk neutral factor dynamics of gaussian term structure models this paper proposes panel data tests of gaussian affine term structure models yield curve data for different moments in time are pooled with the factors treated as fixed effects with fixed effects the time series properties of the price of risk can be ignored results of tests with us interest rate data show that the gaussian model is able to capture the cross sectional structure of yields as well as unrestricted factor loadings from principal components analysis however estimates of the mean reversion parameters in a 3 factor model differ significantly when the model is estimated from yield levels or forward differences which is inconsistent with the gaussian model

on the geometric convergence of the gibbs sampler summary the rate of convergence of the gibbs sampler is discussed the gibbs sampler is a monte carlo simulation method with extensive application to computational issues in the bayesian paradigm conditions for the geometric rate of convergence of the algorithm for discrete and continuous parameter spaces are derived and an illustrative exponential family example is given this paper investigates conditions under which the gibbs sampler gelfand and smith 1990 tanner and wong 1987 geman and geman 1984 converges at a geometric rate the main results appear in sections 2 and 3 where geometric convergence results are established with respect to total variation and supremum norms under fairly natural conditions on the underlying distribution for ease of exposition we shall concentrate on the two most commonly encountered situations where the state space is finite or continuous all our results will establish uniform convergence a strong form of geometric convergence under appropriate regularity conditions uniform convergence is a useful property in its own right but also happens to be a sufficient condition for certain ergodic central limit theorems such results are important for estimation in markov chain simulation but will not be considered in detail here our approach is to apply the theory of markov chains to the specific gibbs sampler case in the finite state space case uniform ergodicity is automatic however the situation is more complicated for continuous state spaces where even well behaved underlying distributions can give rise to markov chains which converge slowly or have unbounded kernels we give two results in this context corollary 2 and corollary 3 which establish uniform convergence under different sets of conditions on the underlying density finally we apply these results to an example of a bayesian hierarchical model where regularity conditions for geometric convergence are naturally satisfied here the hierarchical structure of the model is crucial in permitting application of corollary 3

resolution of singularities in two dimensions and the stability of integrals abstract for a small disk d centered at the origin in r 2 a smooth real valued function s x y on d and a positive epsilon we consider the measure of the points in d where s x y as well as oscillatory integral analogues specifically we consider the effect of perturbing s x y on these quantities besides being of intrinsic interest these questions are important in the analysis of fourier transforms of surface supported measures complex and higher dimensional analogues of these questions are also connected to various issues in algebraic and complex geometry for real analytic s x y this question has been investigated for example by karpushkin using versal deformation theory and by phong stein sturm who developed a method often referred to as the method of algebraic estimates in this paper we show how the use of resolution of singularities algorithms in two dimensions along with some one dimensional van der corput type lemmas provides another method for dealing with such questions as a result we prove new estimates and theorems for these and related quantities furthermore since these algorithms apply to all smooth functions the theorems will hold for all smooth functions as opposed to the earlier real analytic results

l p improving estimates for some classes of radon transforms in this paper we give l p l q estimates and the l p regularizing estimate of radon transforms associated to real analytic functions and we also give estimates of the decay rate of the if operator norm of corresponding oscillatory integral operators for l p l q estimates and estimates of the decay rate of the l p operator norm we obtain sharp results except for extreme points however for l p regularity we allow some restrictions on the phase function

bayesian analysis of contingent claim model error abstract this paper formally incorporates parameter uncertainty and model error into the implementation of contingent claim models we make hypotheses for the distribution of errors to allow the use of likelihood based estimators consistent with parameter uncertainty and model error we then write a bayesian estimator which does not rely on large sample properties but allows exact inference on the relevant functions of the parameters option value hedge ratios and forecasts this is crucial because the common practice of frequently updating the model parameters leads to small samples even for simple error structures and the black scholes model the bayesian estimator does not have an analytical solution markov chain monte carlo estimators help solve this problem we show how they extend to some generalizations of the error structure we apply these estimators to the black scholes given recent work using non parametric function to price options we nest the b s in a polynomial expansion of its inputs despite improved in sample fit the expansions do not yield any out of sample improvement over the b s also the out of sample errors though larger than in sample are of the same magnitude this contrasts with the performance of the popular implied tree methods which produce outstanding in sample but disastrous out of sample fit as dumas fleming and whaley 1997 show this means that the estimation method is as crucial as the model itself

from harmonic analysis to arithmetic combinatorics arithmetic combinatorics or additive combinatorics is a fast developing area of research combining elements of number theory combinatorics harmonic analysis and ergodic theory its arguably best known result and the one that brought it to global prominence is the proof by ben green and terence tao of the long standing conjecture that primes contain arbitrarily long arithmetic progressions there are many accounts and expositions of the green tao theorem including the articles by kra 119 and tao 182 in the bulletin the purpose of the present article is to survey a broader highly interconnected network of questions and results built over the decades and spanning several areas of mathematics of which the green tao theorem is a famous descendant an old geometric problem lies at the heart of key conjectures in harmonic analysis a major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles an unexpected argument points harmonic analysts towards additive number theory with consequences that could have hardly been anticipated we will not try to give a comprehensive survey of harmonic analysis combinatorics or additive number theory we will not even be able to do full justice to our specific areas of focus instead referring the reader to the more complete expositions and surveys listed in section 7 our goal here is to emphasize the connections between these areas we will thus concentrate on relatively few problems chosen as much for their importance to their fields as for their links to each other the article is written from the point of view of an analyst who in the course of her work was gradually introduced to the questions discussed here and found them fascinating we hope that the reader will enjoy a taste of this experience

on multilinear oscillatory integrals nonsingular and singular basic questions concerning nonsingular multilinear operators with oscillatory factors are posed and partially answered l p norm inequalities are established for multilinear integral operators of calderon zygmund type which incorporate oscillatory factors e ip where p is a real valued polynomial

local parametric analysis of hedging in discrete time when continuous time portfolio weights are applied to a discrete time hedging problem errors are likely to occur this paper evaluates the overall importance of the discretization induced tracking error it does so by comparing the performance of black scholes hedge ratios against those obtained from a novel estimation procedure namely local parametric estimation in the latter the weights of the duplicating portfolio are estimated by fitting parametric models in this paper black scholes in the neighborhood of the derivative s moneyness and maturity local parametric estimation directly incorporates the error from hedging in discrete time results are shown where the root mean square tracking error is reduced up to 41 for short maturity options the performance can still be improved by combining locally estimated hedge portfolio weights with standard analysis based on historically estimated parameters the root mean square tracking error is thereby reduced by about 18 for long maturity options plots of the locally estimated volatility parameter against moneyness and maturity reveal the biases of the black scholes model when hedging in discrete time in particular there is a sharp smile effect in the relation between estimated volatility and moneyness for short maturity options as well as a significant wave effect in the relation with maturity for deep out of the money options 1997 elsevier science s a

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

uniform estimates for bilinear hilbert transforms and bilinear maximal functions associated to polynomials we study the bilinear hilbert transforms and bilinear maximal functions associated to non flat polynomial curves and obtain uniform l r estimates for r d 1 over d where d is the degree of the polynomial in addition the lower bound of r is sharp up to the end point

the lp lq estimates for degenerate radon transforms in c2 abstract let p be a polynomial in c 2 of bounded degree we obtain some l p l q estimates for the semi translation invariant radon like transform defined by r f z c f p z z d a where is a cutoff function supported in a small neighborhood of the origin here d a d x d y provided that x i y where x y r

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

robust reduced order nonstrictly proper state estimation via the optimal projection equations with petersen hollot bounds a state estimation design problem involving parametric plant uncertainties is considered an error bound suggested by recent work of petersen and hollot is utilized for guaranteeing robust estimation necessary conditions which generalize the optimal projection equations for reduced order state estimation are used to characterize the estimator which minimizes the error bound the design equations thus effectively serve as sufficient conditions for synthesizing robust estimators an additional feature is the presence of a static estimation gain in conjunction with the dynamic kalman estimator i e a nonstrictly proper estimator

hilbert integrals singular integrals and radon transforms i on introduit deux nouvelles classes d operateurs on etudie leur relation et on montre comment on peut les appliquer a l etude des problemes aux valeurs limites

uniform estimates for oscillatory integral operators abstract in this paper we prove the uniform l p boundedness of oscillatory singular integral operators with smooth phase functions which satisfy a certain finite type condition these phase functions include the class of real analytic functions the uniform boundedness does not hold if the finite type condition is removed we also obtained estimates for related operators including an optimal l 2 estimate for operators with singular kernels whose phase functions have non singular hessians

reducing the sensitivity to parameter variations of a minimum order reduced order observer a quantitative upper hound is established which defines the extent to which a minimum order reduced order non adaptive observer may be rendered insensitive to parameter variations that upper bound depends on the number of measurements and the location of questionable parameters within the system matrices an integral squared error objective function is explicitly solved that objective function may be used to assess the sensitivity of the system to parameter errors a numerical example is presented for a continuous observer the results are extended to the discrete ease

achieving robust stability by sensing additional system outputs in this note we investigate the problem of robust stability in the context of linear time invariant multivariable systems which are affected by structured parametric type uncertainty we demonstrate that by generating additional outputs which are then made available for feedback and thus using sensors for more measurements robust closed loop system stability can be achieved we state and prove results that suggest when this is possible and show how it can be done

robust reduced order nonstrictly proper state estimation via the optimal projection equations with guaranteed cost bounds a state estimation design problem involving parametric plant uncertainties is considered an estimation error bound suggested by multiplicative white noise modeling is utilized for guaranteeing robust estimation over a specified range of parameter uncertainties necessary conditions that generalize the optimal projection equations for reduced order state estimation are used to characterize the estimator that minimizes the error bound the design equations thus effectively serve as sufficient conditions for synthesizing robust estimators additional features include the presence of a static estimation gain in conjunction with the dynamic kalman estimator to obtain a nonstrictly proper estimator

regularity of fourier integral operators with caustics the caustics of fourier integral operators are defined as caustics of the corresponding schwartz kernels lagrangian distributions on x y the caustic set c of the canonical relation is characterized as the set of points where the rank of the projection c x x y is smaller than its maximal value dim x x y 1 we derive the l p y l q x estimates on fourier integral operators with caustics of corank 1 such as caustics of type a m 1 m n for the values of p and q outside of a certain neighborhood of the line of duality q p the l p l q estimates are proved to be caustics insensitive we apply our results to the analysis of the blow up of the estimates on the half wave operator just before the geodesic flow forms caustics

radon transforms and finite type conditions the purpose of this paper is to study averaging operators of radon transform type we shall formulate suitable finite type conditions and prove lp sobolev and lplq estimates the results will be essentially sharp for operators associated with families of curves in the plane let xc and t be smooth manifolds dimx3 nl dim2 nr and let m be a submanifold in xc x t with conormal bundle n m we denote by t the codimension of m we shall always assume that the projections lry m x and 7rt m 42 are surjective with rank dirx nl rank dr9 nr this in particular implies that n m c t x o x t o where 0 refers to the zero sections in t x and t 2 respectively this is the usual assumption for the canonical relation associated with lagrangian distributions arising as kernels of fourier integral operators the assumptions on dirx d7rt imply that for fixed x e x y e 2 the sets amx y e t x y e mi m x ye2x x y em i

endpoint l p l q estimates for degenerate radon transforms in mathbb r 2 associated with real analytic functions abstract we consider degenerate radon transforms of the form rf t x int mathbb r f t s x y y psi x y dy where psi is supported in a small neighborhood of the origin under the condition that s is real analytic we prove complete l p l q estimates for r

sharp l p boundedness of oscillatory integral operators with polynomial phases in this paper we shall prove the l p endpoint decay estimates of oscillatory integral operators with homogeneous polynomial phases s in mathbb r times mathbb r as a consequence sharp l p decay estimates are also obtained when polynomial phases have the form s x m 1 y m 2 with m 1 and m 2 being positive integers

sharp l estimates for one dimensional oscillatory integral operators with c phase in this paper sharp l 2 estimates are proved for general local one dimensional oscillatory integral operators with smooth phase the proof involves a coarse resolution of singularities in conjunction with a stopping time argument and extensions of the analytic methods of d h phong and e m stein

robust observers for systems with parameters abstract in many cases linear multivariable models which describe physical systems contain structured parametric type uncertainties specifically it may be that the state space model is expressed in a form which involves parameters we first consider circumstances in which a parameter independent observer can be constructed for the system and suggest conditions that guarantee its existence we also give conditions under which such a robust observer can be subsequently used with additional dynamics in a parameter free configuration for regulation

oscillatory integral operators with homogeneous polynomial phases in several variables abstract we obtain l 2 decay estimates in for oscillatory integral operators t whose phase functions are homogeneous polynomials of degree m and satisfy various genericity assumptions the decay rates obtained are optimal in the case of 2 2 dimensions for any m while in higher dimensions the result is sharp for m sufficiently large the proof for large m follows from essentially algebraic considerations for cubics in 2 2 dimensions the proof involves decomposing the operator near the conic zero variety of the determinant of the hessian of the phase function using an elaboration of the general approach of phong and stein d h phong e m stein models of degenerate fourier integral operators and radon transforms ann of math 2 140 1994 703 722

a bayesian approach to outlier detection and residual analysis summary an approach to detecting outliers in a linear model is developed an outlier is defined to be an observation with a large random error generated by the linear model under consideration outliers are detected by examining the posterior distribution of the random errors an augmented residual plot is also suggested as a graphical aid in finding outliers we propose a precise definition of an outlier in a linear model which appears to lead to simple ways of exploring data for the possibility of outliers the definition is such that if the parameters of the model are known then it is also known which observations are outliers alternatively if the parameters are unknown the posterior distribution can be used to calculate the posterior probability that any observation is an outlier in a linear model with normally distributed random errors ei with mean zero and variance a 2we declare the ith observation to be an outlier if iei i ko for some choice of k the value of k can be chosen so that the prior probability of an outlier is small and thus outliers are observations which are more extreme than is usually expected realizations of normally distributed errors of more than about three standard deviations from the mean are certainly surprising and worth further investigation such outlying observations can occur under the assumed model however and this should be taken into account when deciding what to do with outliers and in choosing k note that ei is the actual realization of the random error not the usual estimated residual i the problem of outliers is studied and thoroughly reviewed by barnett lewis 1984 hawkins 1980 beckman cook 1983 and pettit smith 1985 the usual bayesian approach to outlier detection uses the definition given by freeman 1980 freeman defines an outlier to be any observation that has not been generated by the mechanism that generated the majority of observations in the data set freeman s definition therefore requires that a model for the generation of outliers be specified and is implemented by for example box tiao 1968 guttman dutter freeman 1978 and abraham box 1978 our method differs in that we define outliers as arising from the model under consideration rather than arising from a separate expanded model our approach is similar to that described by zellner moulton 1985 and is an extension of the philosophy

a gaussian approximation scheme for computation of option prices in stochastic volatility models abstract we consider european options on a price process that follows the log linear stochastic volatility model two stochastic integrals in the option pricing formula are costly to compute we derive a central limit theorem to approximate them at parameter settings appropriate to foreign exchange data our formulas improve computation speed by a factor of 1000 over brute force monte carlo making mcmc statistical methods practicable we provide estimates of model parameters from daily data on the swiss franc to euro and japanese yen to euro over the period 1999 2002

harmonic analysis on nilpotent groups and singular integrals i oscillatory integrals abstract this paper is devoted to the study of the operator given by where k is a standard calderon zygmund kernel and p is a real polynomial on r n r n we show t is bounded on lp to itself when 1

regulation of split linear systems over rings coefficient assignment and observers a theory of regulators is developed for finite free split linear systems over a commutative ring k this is achieved by developing a theory of coefficient assignment and observers it is shown that the problem of coefficient assignment can be solved for reachable systems by using dynamic state feedback for strongly observable systems it is shown that observers with arbitrary dynamics can be constructed internal stability of observer and state feedback configuration is considered explicitly some examples are given to illustrate these techniques

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea