improved estimation of the disturbance variance in a linear regression model abstract in this paper we consider estimation of the disturbance variance in a linear regression model a new class of improved estimators is obtained by extending results dating to stein 1964 these estimators dominate the ordinary least squares estimator under squared error loss

on the reachable set of inflated singularly perturbed differential equations singularly perturbed differential equations with slow and fast subsystems are under consideration it is well known that the invariant probability measures of the unperturbed fast subsystems produce a limiting system for the slow subsystem unfortunately the limiting system is non smooth in general and might be too big since it also generates trajectories that are not related to the singularly perturbed system however if the flows produced by the unperturbed fast subsystems are chain transitive an inflation of the singularly perturbed system can be used in order to approximate all trajectories of the limiting system moreover it turns out that the reachable sets of the limiting system are contained in the reachable sets of the slightly inflated singularly perturbed system

on existence of limit occupational measures set of a controlled stochastic differential equation we establish that under certain conditions the set of occupational measures as well as the set of mathematical expectations of occupational measures generated by the admissible controls and the corresponding solutions of a controlled stochastic differential equation csde converge with the time horizon tending to infinity to a set called limit occupational measures set loms and we show that this limit set coincides with the set of stationary marginal distributions of the csde we also demonstrate the applicability of our results for averaging of singularly perturbed csde

ergodicity of the 2d navier stokes equations with degenerate stochastic forcing the stochastic 2d navier stokes equations on the torus driven by degenerate noise are studied we characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic in particular our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in l 2 0 t 2 unlike previous works this class is independent of the viscosity and the strength of the noise the two main tools of our analysis are the asymptotic strong feller property introduced in this work and an approximate integration by parts formula the first when combined with a weak type of irreducibility is shown to ensure that the dynamics is ergodic the second is used to show that the first holds un

pre testing for linear restrictions in a regression model with spherically symmetric disturbances abstract in this paper we derive the exact risk under quadratic loss of pre test estimators of the prediction vector and of the error variance of a linear regression model with spherically symmetric disturbances the pre test in question is one of the validity of a set of exact linear restrictions on the model s coefficient vector we demonstrate how the known results for the model with normal disturbances can be extended to this broader case we also show that the critical value of unity results in a minimum of the risk of the pre test estimator of the error variance to illustrate the results we assume multivariate student t regression disturbances and numerically evaluate the derived expressions

estimation of the error variance after a preliminary test of homogeneity in a regression model with spherically symmetric disturbances abstract in this paper we consider the risk under quadratic loss of an estimator of the error variance after a pre test for homogeneity of the variances in the two sample linear regression model we investigate the effects on risk of assuming normal disturbances when in fact the error distribution is spherically symmetric we also broaden the standard assumption that the never pool variance estimators are based on the least squares principle to illustrate the results numerically we consider two families of variance mixing distributions one of which results in multivariate student t disturbances our results show that in some situations we should always pre test even if the error variances are equal and we provide the optimal test critical value the evaluations also show that using the least squares technique to form the never pool estimators may not always be the preferred strategy

ergodicity of stochastic 2d navier stokes equation with levy noise abstract in this paper we deal with the 2d navier stokes equation perturbed by a levy noise force whose white noise part is non degenerate and that the intensity measure of poisson measure is finite existence and uniqueness of invariant measure for this equation is obtained two main properties of the markov semigroup associated with this equation are proved in other words strong feller property and irreducibility hold in the same space

ergodic and mixing properties of the boussinesq equations with a degenerate random forcing we establish the existence uniqueness and attraction properties of an ergodic invariant measure for the boussinesq equations in the presence of a degenerate stochastic forcing acting only in the temperature equation and only at the largest spatial scales the central challenge is to establish time asymptotic smoothing properties of the markovian dynamics corresponding to this system towards this aim we encounter a lie bracket structure in the associated vector fields with a complicated dependence on solutions this leads us to develop a novel hormander type condition for infinite dimensional systems demonstrating the sufficiency of this condition requires new techniques for the spectral analysis of the malliavin covariance matrix

optimal critical values of pre tests when estimating the regression error variance analytical findings under a general loss structure this paper re visits the problem of estimating the regression error variance in a linear multiple regression model after preliminary hypothesis tests for either linear restrictions on the coefficients or homogeneity of variances there is an extensive literature that discusses these problems particularly in terms of the sampling properties of the pre test estimators using various loss functions as the basis for risk analysis in this paper a unified framework for analysing the risk properties of these estimators is developed under a general class of loss structures that incorporates virtually all first order differentiable losses particular consideration is given to the choice of critical values for the pre tests analytical results indicate that an level substantially higher than those normally used may be appropriate for optimal risk properties under a wide range of loss functions the paper also generalizes some known analytical results in the pre test literature and proves other results only previously shown numerically

averaging method for one sided lipschitz differential inclusions with generalized solutions we investigate the convergence of the solution set for singularly perturbed differential inclusions using an averaging technique the limits of the fast solutions are considered as generalized solutions i e radon probability measures of the degenerate system exploiting in addition the one sided lipschitz condition we prove the existence of a limit of the solution set in an appropriate topology

spectral gaps in wasserstein distances and the 2d stochastic navier stokes equations we develop a general method to prove the existence of spectral gaps for markov semigroups on banach spaces unlike most previous work the type of norm we consider for this analysis is neither a weighted supremum norm nor an lp type norm but involves the derivative of the observable as well and hence can be seen as a type of 1 wasserstein distance this turns out to be a suitable approach for infinite dimensional spaces where the usual harris or doeblin conditions which are geared toward total variation convergence often fail to hold in the first part of this paper we consider semigroups that have uniform behavior which one can view its the analog of doeblin s condition we then proceed to study situations where the behavior is not so uniform but the system has a suitable lyapunov structure leading to a type of harris condition we finally show that the latter condition is satisfied by the two dimensional stochastic navier stokes equations even in situations where the forcing is extremely de generate using the convergence result we show that the stochastic navier stokes equations invariant measures depend continuously on the viscosity and the structure of the forcing

averaging problem for weakly coupled nonexpansive control systems this paper investigates the limit behavior of a singularly perturbed control system with two state variables which are weakly coupled the main novelty of our averaging approach lies in the fact that the limit dynamic may depend the initial condition of the fast system while in the literature this problem has been usually addressed under conditions that ensure that the limit dynamic is independent to this initial condition our study is based on a suitable nonexpansivity condition on the fast system which generalized dissipativity or stability properties of the fast dynamics

linear programming approach to deterministic long run average problems of optimal control we establish that deterministic long run average problems of optimal control are asymptotically equivalent to infinite dimensional linear programming problems lpps and we establish that these lpps can be approximated by finite dimensional lpps the solutions of which can be used for construction of the optimal controls general results are illustrated with numerical examples

discontinuous control problems for non convex dynamics and near viability for singularly perturbed control systems abstract the aim of this paper is to study two classes of discontinuous control problems without any convexity assumption on the dynamics in the first part we characterize the value function for the mayer problem and the supremum cost problem using viscosity tools and the notion of e viability near viability these value functions are given with respect to discontinuous cost functionals in the second part we obtain results describing the e viability near viability of singularly perturbed control systems

singular perturbations in ergodic control of diffusions ergodic control of a nondegenerate diffusion with two time scales is studied in the limiting case as the time scale separation increases to infinity it is shown that the limit problem is another ergodic control problem for the slow time scale component alone with its dynamics averaged over the controlled invariant probability measures for the fast component these measures in turn can be treated as the effective control variable

exponential mixing for stochastic pdes the non additive case we establish a general criterion which ensures exponential mixing of parabolic stochastic partial differential equations spde driven by a non additive noise which is white in time and smooth in space we apply this criterion on two representative examples 2d navier stokes ns equations and complex ginzburg landau cgl equation with a locally lipschitz noise due to the possible degeneracy of the noise doob theorem cannot be applied hence a coupling method is used in the spirit of kuksin and shirikyan j math pures appl 1 567 602 2002 and mattingly commun math phys 230 421 462 2002

estimating the error variance in regression after a preliminary test of restrictions on the coefficients abstract maximum likelihood estimation of the variance of the disturbances in a linear regression is considered in the context of exact linear restrictions on the coefficient vector the exact risk function for the associated preliminary test estimator of this variance is derived and evaluated numerically on the basis of relative quadratic loss our results indicate that in practical situations the risk of the pre test estimator differs only slightly from that of the naive estimator which ignores the restrictions without testing applying the coefficient restrictions without testing their validity is not recommended

viscosity solutions methods for singular perturbations in deterministic and stochastic control viscosity solutions methods are used to pass to the limit in some penalization problems for first order and second order degenerate parabolic hamilton jacobi bellman equations this characterizes the limit of the value functions of singularly perturbed optimal control problems for deterministic systems and for controlled degenerate diffusions the results apply to cases where the usual order reduction method does not give the correct limit and to systems with fast state variables depending nonlinearly on the control some connections with ergodic control and periodic homogenization are discussed

characterization of the optimal trajectories for the averaged dynamics associated to singularly perturbed control systems abstract the aim of this paper is to study singularly perturbed control systems firstly we provide linearized formulation version for the calculus of the value function associated with the averaged dynamics secondly we obtain necessary and sufficient conditions in order to identify the optimal trajectory of the averaged system

lower semicontinuous solutions of hamilton jacobi bellman equations the value function of mayer s problem arising in optimal control is investigated and lower semicontinuous solutions of the associated hamilton jacobi bellman equation are defined in three equivalent ways under quite weak assumptions about the control system the value function is the unique solution moreover it is stable with respect to perturbations of the control system and the cost it coincides with the viscosity solution whenever it is continuous

a coupling approach to randomly forced nonlinear pde s ii we consider a class of discrete time random dynamical systems and establish the exponential convergence of its trajectories to a unique stationary measure the result obtained applies in particular to the 2d navier stokes system and multidimensional complex ginzburg landau equation with random kick force

exponential convergence for the stochastically forced navier stokes equations and other partially dissipative dynamics we prove that the two dimensional navier stokes equations possess an exponentially attracting invariant measure this result is in fact the consequence of a more general harris like ergodic theorem applicable to many dissipative stochastic pdes and stochastic processes with memory a simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting to analyze the iterated map a general doeblin like theorem is proven one of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the non markovian processes

ergodicity and exponential mixing of the real ginzburg landau equation with a degenerate noise abstract in this paper we establish the existence uniqueness and attraction properties of an invariant measure for the real ginzburg landau equation in the presence of a degenerate stochastic forcing acting only in four directions the main challenge is to establish time asymptotic smoothing properties of the markovian dynamics corresponding to this system to achieve this we propose a condition which only requires four noises

duality in linear programming problems related to deterministic long run average problems of optimal control it has been established recently that under mild conditions deterministic long run average problems of optimal control are asymptotically equivalent to infinite dimensional linear programming problems lpps and that these lpps can be approximated by finite dimensional lpps in this paper we introduce the corresponding infinite and finite dimensional dual problems and study duality relationships we also investigate the possibility of using solutions of finite dimensional lpps and their duals for numerical construction of the optimal controls in periodic optimization problems the construction is illustrated with a numerical example

suboptimization of singularly perturbed control systems a technique different from the boundary layer method is developed to deal with singularly perturbed optimal control problems the technique is applicable in particular in the case when the optima

value functions for differential games and control systems with discontinuous terminal cost this paper deals with mayer s problem for control systems and differential games with discontinuous terminal cost there are two main results in the paper the first one says that the value function for control systems can be characterized as the unique solution in suitable sense to the hamilton jacobi bellman equation without any regularity assumptions on the terminal cost for differential games satisfying isaacs s minmax condition the second main result says that the value function is the unique solution to the hamilton jacobi isaacs equation when the terminal cost is semicontinuous this allows to prove the existence of the value under isaacs s condition this paper extends some results already well known in the continuous case

ergodic theory of infinite dimensional systems with applications to dissipative parabolic pdes we consider a class of randomly perturbed dynamical systems satisfying conditions which reflect the properties of general nonlinear dissipative parabolic pdes results on invariant measures and their exponential mixing properties are proved and applications to 2d navier stokes systems are included

averaging method for discontinuous mayer s problem of singularly perturbed control systems abstract in the present paper we prove that the optimal cost of a mayer s problem for a control system with singular perturbation does converge to the optimal cost of a mayer s problem associated with control system obtained by averaging the main novelty of our result lies on the fact that we did not require any continuity assumption on the final cost

on singular perturbations for differential inclusions on the infinite interval abstract we consider a differential inclusion subject to a singular perturbation i e part of the derivatives are multiplied by a small parameter 0 we show that under some stability and structural assumptions every solution of the singularly perturbed inclusion comes close to a solution of the degenerate inclusion obtained for 0 when tends to 0 the goal of the present paper is to provide a new result of tikhonov type on the time interval 0

exponential ergodicity for stochastic burgers and 2d navier stokes equations abstract it is shown that transition measures of the stochastic navier stokes equation in 2 d converge exponentially fast to the corresponding invariant measures in the distance of total variation as a corollary we obtain the existence of spectral gap for a related semigroup obtained by a sort of ground state transformation analogous results are proved for the stochastic burgers equation