sequencing and routing in multiclass queueing networks part ii workload relaxations part ii continues the development of policy synthesis techniques for multiclass queueing networks based upon a linear fluid model the following are shown a relaxation of the fluid model based on workload leads to an optimization problem of lower dimension an analogous workload relaxation is introduced for the stochastic model these relaxed control problems admit pointwise optimal solutions in many instances a translation to the original fluid model is almost optimal with vanishing relative error as the network load rho approaches one it is pointwise optimal after a short transient period provided a pointwise optimal solution exists for the relaxed control problem a translation of the optimal policy for the fluid model provides a policy for the stochastic network model that is almost optimal in heavy traffic over all solutions to the relaxed stochastic model again with vanishing relative error the regret is of order log 1 rho

some potential theory for reflecting brownian motion in holder and lipschitz domains bounds are found on the transition densities and green functions for brownian motion with normal reflection in holder and lipschitz domains for lipschitz domains reflecting brownian motion and boundary local time are constructed a harnack inequality valid up to the boundary is proved a probabilistic solution to the neumann problem is given and the kuramochi boundary is identified

asymptotic irrelevance of initial conditions for skorohod reflection mapping on the nonnegative orthant a reflection map induced by the deterministic skorohod problem on the nonnegative orthant is applied to a vector valued function x on the nonnegative real line and then to a x where a is a nonnegative constant vector a question that was posed over 15 years ago is under what conditions does the difference between the two resulting regulated functions converge to zero for any choice of a as time diverges this in turn implies that if one imposes enough stochastic structure that ensures that the reflection map applied to a multidimensional process x converges in distribution then it will also converge in distribution when it is applied to x where is any almost surely finite valued random vector that may even depend on the process x in this paper we obtain a useful equivalent characterization of this property as a result we are able to identify a natural sufficient condition in terms of the given data x and the constant routing matrix a similar necessary condition is also indicated a particular implication of our analysis is that under additional stochastic assumptions asymptotic irrelevance of the initial condition does not require the existence of a stationary distribution as immediate corollaries of our and earlier results we conclude that under the natural stability conditions a reflected levy process as well as a markov additive process has a unique stationary distribution and converges in distribution to this stationary distribution for every initial condition extensions of the sufficient condition are then developed for reflection maps with drift and routing coefficients that may be time and state dependent some implications to multidimensional insurance models are briefly discussed

stability of constrained markov modulated diffusions a family of constrained diffusions in a random environment is considered constraint set is a polyhedral cone and coefficients of the diffusion are governed by in addition to the system state a finite state markov process that is independent of the driving noise such models arise as limit objects in the heavy traffic analysis of generalized jackson networks gjn with markov modulated arrival and processing rates we give sufficient conditions which in particular includes a requirement on the regularity of the underlying skorohod map for positive recurrence and geometric ergodicity when the coefficients only depend on the modulating markov process i e they are independent of the system state a complete characterization for stability and transience is provided the case where the pathwise skorohod problem is not well posed but the underlying reflection matrix is completely s is treated as well as consequences of geometric ergodicity various results such as exponential integrability of invariant measures and central limit results clt for fluctuations of long time averages of process functionals about their stationary values are obtained conditions for stability are formulated in terms of the averaged drift where the average is taken with respect to the stationary distribution of the modulating markov process finally steady state distributions of the underlying gjn are considered and it is shown that under suitable conditions such distributions converge to the unique stationary distribution of the constrained random environment diffusion

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

mouvement brownien cones et processus stables letw wt t 0 denote a two dimensional brownian motion starting at 0 and for 0 4 here we construct a measure a local time supported onh for 4 2 the brownian motionw time changed by the inverse of this local time is shown to be a two dimensional stable process with index 2 2 this results extends spitzer s construction of the cauchy process which is recovered by taking 2 a formula which describes the behaviour ofw before a timet h is established and applied to the proof of a conjecture of burdzy we also obtain a two dimensional version of the famous theorem of levy concerning the maximum process of linear brownian motion precisely for 0 2 letst denote the vertex of the smallest wedge of the typez c which contains the path ofw up to timet the processst wt is shown to be a reflected brownian motion in the wedgec with oblique reflection on the sides finally we investigate various extensions of the previous results to brownian motion inrd d 3 letc be the cone associated with an open subset of the spheresd 1 and leth be defined ash above sufficient conditions are given forh to contain nonzero times in terms of the first eigenvalue of the dirichlet laplacian on

a subsidy surplus model and the skorokhod problem in an orthant we consider the deterministic skorokhod problem in an orthant of the formzw t w t b u yw u zw u du r u yw u zw u dyw u with yw i nondecreasing and yw i not increasing while zw i 0 this can be viewed as a subsidy surplus model in an interdependent economy existence of a unique solution is established under fairly general conditions viz with r satisfying a uniform spectral radius condition comparison result for sp vis a vis the usual partial order on the orthant is studied we show that the more inward looking the reflection vectors and the drift the larger the values ofyw will be but the values ofzw will be smaller in addition to showing that the leontief type output is a feasible subsidy connection between sp and minimality of feasible subsidies is discussed consequently it is suggested that sp may be taken as a continuous time feedback form analogue of open leontief model in the stochastic case y t z t turns out to be a strong markov process ifw arises from alevy process relevance of the comparison result to recurrence positive recurrence ofz process is pointed out

multidimensional reflected brownian motions having exponential stationary distributions on considere la distribution stationnaire d un mouvement brownien reflechi dans un domaine d dimensionnel g on etudie deux classes de processus de ce type

optimal reflection of diffusions and barrier options pricing under constraints we introduce a new class of control problems in which the gain depends on the solution of a stochastic differential equation sde reflected at the boundary of a bounded domain along directions which are controlled by a bounded variation process we provide a pde characterization of the associated value function this study is motivated by applications in mathematical finance where such equations are related to the pricing of barrier options under portfolio constraints

the effects of dynamic feedbacks on ls and mm estimator accuracy in panel data models the finite sample behavior is analyzed of particular least squares ls and a range of generalized method of moments mm estimators in panel data models with individual effects and both a lagged dependent variable regressor and another explanatory variable the latter may be affected by lagged feedbacks from the dependent variable too asymptotic expansions indicate how the order of magnitude of bias of mm estimators tends to increase with the number of moment conditions exploited they also provide analytic evidence on how the bias of the various estimators depends on the feedbacks and on other model characteristics such as prominence of individual effects and correlation between observed and unobserved heterogeneity simulation results corroborate the theoretical findings and reveal that in small samples of models with dynamic feedbacks none of the techniques examined dominates regarding bias and mean squared error over all parametrizations examined

pseudo jordan domains and reflecting brownian motions the manifold metric between two points in a planar domain is the minimum of the lengths of piecewisec1 curves in the domain connecting these two points we define a bounded simply connected planar region to be a pseudo jordan domain if its boundary under the manifold metric is topologically homeomorphic to the unit circle it is shown that reflecting brownian motionx on a pseudo jordan domain can be constructed starting at all points except those in a boundary subset of capacity zero x has the expected skorokhod decomposition under a condition which is satisfied when g has finite 1 dimensional lower minkowski content

on the stochastic regularity of distorted brownian motions we systematically develop general tools to apply fukushima s absolute continuity condition these tools comprise methods to obtain a hunt process on a locally compact separa ble metric state space whose transition function has a density w r t the reference measure and methods to estimate drift potentials comfortably we then apply our results to distorted brownian motions and construct weak solutions to singular stochastic differential equations i e equations with possibly unbounded and discontinuous drift and reflect ion terms which may be the sum of countably many local times the solutions can start from any point of the explicitly specified state space we consider different kind of weights like muckenhoupt a2 weights and weights with moderate growth at singularities as well as different kind of multiple boundary conditions our approach leads in particular to the construction and explicit identification of countably skew reflected and normally reflected brownian motions with singu lar drift in bounded and unbounded multi dimensional domains

directional derivatives of oblique reflection maps given an oblique reflection map and functions documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document psi chi in mathcal d lim end document the space of k valued functions that have finite left and right limits at every point the directional derivative documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document nabla chi gamma psi end document of along evaluated at is defined to be the pointwise limit as documentclass aastex usepackage amsbsy usepackage amsfonts u

a skorohod type lemma and a decomposition of reflected brownian motion we consider two dimensional reflected brownian motions in sharp thorns pointed downward with horizontal vectors of reflection we present a decomposition of the process into a brownian motion and a process which has bounded variation away from the tip of the thorn the construction is based on a new skorohod type lemma

a dirichlet process characterization of a class of reflected diffusions for a class of stochastic differential equations with reflection for which a certain l p continuity condition holds with p 1 it is shown that any weak solution that is a strong markov process can be decomposed into the sum of a local martingale and a continuous adapted process of zero p variation when p 2 this implies that the reflected diffusion is a dirichlet process two examples are provided to motivate such a characterization the first example is a class of multidimensional reflected diffusions in polyhedral conical domains that arise as approximations of certain stochastic networks and the second example is a family of two dimensional reflected diffusions in curved domains in both cases the reflected diffusions are shown to be dirichlet processes but not semimartingales

relationship between generalized rough sets based on binary relation and covering rough set theory is a powerful tool for dealing with uncertainty granularity and incompleteness of knowledge in information systems this paper systematically studies a type of generalized rough sets based on covering and the relationship between this type of covering based rough sets and the generalized rough sets based on binary relation firstly we present basic concepts and properties of this kind of rough sets then we investigate the relationships between this type of generalized rough sets and other five types of covering based rough sets the major contribution in this paper is that we establish the equivalency between this type of covering based rough sets and a type of binary relation based rough sets through existing results in binary relation based rough sets we present axiomatic systems for this type of covering based lower and upper approximation operations in addition we explore the relationships among several important concepts such as minimal description reduction representative covering exact covering and unary covering in covering based rough sets investigation of this type of covering based will benefit to our understanding of other types of rough sets based on covering and binary relation

on the diffusion approximation to two parallel queues with processor sharing a two dimensional queuing model is considered in which the server in the second queue helps the server in the first queue during periods when the second queue is empty the system is analyzed in the heavy traffic limit and explicit approximate solutions are obtained to the resulting diffusion equations using singular perturbation methods approximate asymptotic formulas are obtained for the stationary distribution of the number of customers as well as for some first passage time problems associated with the busy period it is shown that these formulas reduce to the asymptotic expansions of the exact solutions when the latter are available

functional central limit theorem for brownian particles in domains with robin boundary condition abstract we rigorously derive non equilibrium space time fluctuation for the particle density of a system of reflected diffusions in bounded lipschitz domains in r d the particles are independent and are killed by a time dependent potential which is asymptotically proportional to the boundary local time we generalize the functional analytic framework introduced by kotelenez 20 21 to deal with time dependent perturbations our proof relies on dirichlet form method rather than the machineries derived from kotelenez s sub martingale inequality our result holds for any symmetric reflected diffusion for any bounded lipschitz domain and for any dimension d 1

brownian motion in a wedge with oblique reflection this work is concerned with the existence and uniqueness of a strong markov process that has continuous sample paths and the following additional properties i the state space is an infinite two dimensional wedge and the process behaves in the interior of the wedge like an ordinary brownian motion ii the process reflects instantaneously at the boundary of the wedge the angle of reflection being constant along each side iii the amount of time that the process spends at the comer of the wedge is zero i e the set of times for which the process is at the comer has lebesgue measure zero hereafter let be the angle of the wedge 0 2 let 1 and 2 be the angles of reflection on the two sides of the wedge measured from the inward normals the positive angles being toward the corner 1 2 and set 1 2 the question of existence and uniqueness is recast as a submartingale problem in the style used by stroock and varadhan diffusion processes with boundary conditions comm pure appl math 24 1971 pp 147 225 for diffusions on smooth domains with smooth boundary conditions it is shown that no solution exists if 2 in this case there is a unique continuous strong markov process satisfying i ii above it reaches the corner of the wedge almost surely and it remains there if 2 however then there is a unique continuous strong markov process statisfying i iii it is shown that starting away from the corner this process does not reach the corner of the wedge if 0 and does reach the corner if 0 2 the general theory of multi dimensional diffusions does not apply to the above problem because in general the boundary of the state space is not smooth and there is a discontinuity in the direction of reflection at the corner for some values of the process arises from diffusion approximations to storage systems and queueing networks i the state space is an infinite two dimensional wedge and the process behaves in the interior of the wedge like an ordinary brownian motion ii the process reflects instantaneously at the boundary of the wedge and the angle of reflection being constant along each side iii the amount of time that the process spends at the corner of the wedge is zero i e the set of times for which the process is at the corner has lebesgue measure zero

relational interpretations of neighborhood operators and rough set approximation operators this paper presents a framework for the formulation interpretation and comparison of neighborhood systems and rough set approximations using the more familiar notion of binary relations a special class of neighborhood systems called 1 neighborhood systems is introduced three extensions of pawlak approximation operators are analyzed properties of neighborhood and approximation operators are studied and their connections are examined

ergodic control for constrained diffusions characterization using hjb equations recently in a budhiraja siam j control optim 42 2003 pp 532 558 an ergodic control problem for a class of diffusion processes constrained to take values in a polyhedral cone was considered the main result of that paper was that under appropriate conditions on the model there is a markov control for which the infimum of the cost function is attained in the current work we characterize the value of the ergodic control problem via a suitable hamilton jacobi bellman hjb equation the theory of existence and uniqueness of classical solutions for pdes in domains with corners and reflection fields which are oblique discontinuous and multivalued on corners is not available we show that the natural hjb equation for the ergodic control problem admits a unique continuous viscosity solution which enables us to characterize the value function of the control problem the existence of a solution to this hjb equation is established via the classical vanishing discount argument the key step is proving the precompactness of the family of suitably renormalized discounted value functions in this regard we use a recent technique introduced in v s borkar stochastic process appl 103 2003 pp 293 310 of using the athreya ney nummelin pseudoatom construction for obtaining a coupling of a pair of embedded discrete time controlled markov chains

jump diffusion over feature space for object recognition we present a dynamical model for a population of tests in pattern recognition taking a preprocessed initialization of a feature set we apply a stochastic algorithm based on an efficiency criterion and a gaussian noise to recursively build and improve the feature space this algorithm simulates a markov chain which estimates a probability distribution mathbb p on the set of features the features are structured as binary trees and we show that such random forests are a good way to represent the evolution of the feature set we then obtain properties on the dynamic of the features space before applying this algorithm to practical examples such as face detection and microarray analysis lastly we identify the weak limit of our process as a jump diffusion process defined using the skorokhod map over simplices

asymptotically optimal controls for time inhomogeneous networks a framework is introduced for the identification of controls for single class time varying queueing networks that are asymptotically optimal in the so called uniform acceleration regime a related but simpler first order or fluid control problem is first formulated for a class of performance measures that satisfy a certain continuity property it is then shown that any sequence of policies whose performances approach the infimum in the fluid control problem is asymptotically optimal for the original network problem examples of performance measures with this property are described and simulations implementing asymptotically optimal policies are presented the use of directional derivatives of the reflection map for solving fluid control problems is also illustrated this work serves to complement a large body of literature on asymptotically optimal controls for time homogeneous networks

skorokhod decomposition of reflected diffusions on bounded lipschitz domains with singular non reflection part let overline g subset mathbb r d be a compact set with interior g let l1 g dx 0 dx a e on g and m dx let a aij be symmetric and globally uniformly strictly elliptic on g let be such that mathcal e r f g frac 1 2 sum i j 1 d int g a ij partial i f partial j g dm f g in c infty overline g is closable in l2 g m with closure r d r the latter is fulfilled if satisfies the hamza type condition or i l1loc g dx 1 i d conservative non symmetric diffusion processes xt related to the extension of a generalized dirichlet form mathcal e r f g sum i 1 d int g rho 1 overline b i partial i f g dm f g in d mathcal e r b where rho 1 overline b 1 overline b d in l 2 g mathbb r d m satisfies sum i 1 d int g overline b i partial i f dx 0 quad rm for all f in c infty overline g are constructed and analyzed if g is a bounded lipschitz domain h1 1 g and aij d r a skorokhod decomposition for xt is given this happens through a local time that is uniquely associated to the smooth measure 1 tr 0 d where tr denotes the trace and the surface measure on g

on the effect of mean nonstationarity in dynamic panel data models abstract in this paper we investigate the effect of mean nonstationarity on the first difference generalized method of moments fd gmm estimator in dynamic panel data models we find that when data is mean nonstationary and the variance of individual effects is significantly larger than that of disturbances the fd gmm estimator performs quite well we demonstrate that this is because the correlation between the lagged dependent variable and instruments gets larger owing to the unremoved individual effects i e instruments become strong this implies that under mean nonstationarity the fd gmm estimator does not always suffer from the weak instruments problem even when data is persistent

convex duality and the skorokhod problem ii in this paper we consider skorokhod problems on polyhedral domains with a constant and possibly oblique constraint direction specified on each face of the domain and with a corresponding cone of constraint directions at the intersection of faces in part one of this paper we used convex duality to develop new methods for the construction of solutions to such skorokhod problems and for proving lipschitz continuity of the associated skorokhod maps the main alternative approach to skorokhod problems of this type is the reflection mapping technique introduced by harrison and reiman 8 in this part of the paper we apply the theory developed in part one to show that the reflection mapping technique of 8 is restricted to a slight generalization of the class of problems originally considered in 8 we further illustrate the power of the duality approach by applying it to two other classes of skorokhod problems those with normal directions of constraint and a new class that arises from a model of processor sharing in communication networks in particular we prove existence of solutions to and lipschitz continuity of the skorokhod maps associated with each of these skorokhod problems

the time series and cross section asymptotics of dynamic panel data estimators in this paper we derive the asymptotic properties of within groups wg gmm and liml estimators for an autoregressive model with random effects when both t and n tend to infinity gmm and liml are consistent and asymptotically equivalent to the wg estimator when t n 0 the fixed t results for gmm and liml remain valid but wg although consistent has an asymptotic bias in its asymptotic distribution when t n tends to a positive constant the wg gmm and liml estimators exhibit negative asymptotic biases of order 1 t 1 n and 1 2n t respectively in addition the crude gmm estimator that neglects the autocorrelation in first differenced errors is inconsistent as t n c 0 despite being consistent for fixed t finally we discuss the properties of a random effects pseudo mle with unrestricted initial conditions when both t and n tend to infinity

an explicit formula for the skorokhod map on 0 a the skorokhod map is a convenient tool for constructing solutions to stochastic differential equations with reflecting boundary conditions in this work an explicit formula for the skorokhod map 0 a on 0 a for any a 0 is derived specifically it is shown that on the space d 0 of right continuous functions with left limits taking values in r 0 a a o 0 where a d 0 d 0 is defined by a t t sups 0 t s a infu s t u and 0 d 0 d 0 is the skorokhod map on 0 which is given explicitly by 0 t t sup s 0 t s in addition properties of a are developed and comparison properties of 0 a are established

rough sets induced by ideals in lattices this paper is to study the rough sets within the context of lattices we study the special properties of the rough sets which can be constructed by means of the congruences determined by ideals of lattice also the properties of the generalized rough sets with respect to ideals of lattice are investigated finally we give an example of their application in formal concept analysis

traps for reflected brownian motion consider an open set d r d d 2 and a closed ball b d let e x tb denote the expectation of the hitting time of b for reflected brownian motion in d starting from x d we say that d is a trap domain if supx e x tb a domain d is not a trap domain if and only if the reflecting brownian motion in d is uniformly ergodic we fully characterize the simply connected planar trap domains using a geometric condition and give a number of less complete results for d 2

concentration for multidimensional diffusions and their boundary local times we prove that probability laws of certain multidimensional semimartingales which includes time inhomogenous diffusions under suitable assumptions satisfy quadratic transportation cost inequality under the uniform metric from this we derive concentration properties of lipschitz functions of process paths that depend on the entire history in particular we estimate concentration of boundary local time of reflected brownian motions on a polyhedral domain we work out explicit applications of consequences of measure concentration for the case of brownian motion with rank based drifts

sdes with oblique reflection on nonsmooth domains in this paper we consider stochastic differential equations with reflecting boundary conditions for domains that might have corners and for which the allowed directions of reflection at a point on the boundary of the domain are possibly oblique the main results are strong existence and uniqueness for solutions of such equations a key ingredient is a family of relatively regular functions appropriate to the given domain and directions of reflection two cases are treated in the paper in the first case the direction of reflection is single valued and varies smoothly and the main new feature is that the boundary of the domain may be nonsmooth in the second case the domain is taken to be the intersection of a finite number of domains with relatively smooth boundary and at the resulting corner points more than one oblique direction is allowed

the reflection map with discontinuities we study the multidimensional reflection map on the spacesd 0 t r k andd 0 8 r k of right continuous r k valued functions on 0 t or 0 8 with left limits endowed with variants of the skorohod 1956 m1 topology the reflection map was used with the continuous mapping theorem by harrison and reiman 1981 and reiman 1984 to establish heavy traffic limit theorems with reflected brownian motion limit processes for vector valued queue length waiting time and workload stochastic processes in single class open queueing networks since brownian motion and reflected brownian motion have continuous sample paths the topology of uniform convergence over bounded intervals could be used for those results variants of them1 topologies are needed to obtain alternative discontinuous limits approached gradually by the converging processes as occurs in stochastic fluid networks with bursty exogenous input processes e g with on off sources having heavy tailed on periods or off periods having infinite variance we show that the reflection map is continuous at limits without simultaneous jumps of opposite sign in the coordinate functions provided that the productm1 topology is used as a consequence the reflection map is continuous with the productm1 topology at all functions that have discontinuities in only one coordinate at a time that continuity property also holds for more general reflection maps and is sufficient to support limit theorems for stochastic processes in most applications we apply the continuity of the reflection map to obtain limits for buffer content stochastic processes in stochastic fluid networks

formulation and estimation of dynamic models using panel data abstract this paper presents a statistical analysis of time series regression models for longitudinal data with and without lagged dependent variables under a variety of assumptions about the initial conditions of the processes being analyzed the analysis demonstrates how the asymptotic properties of estimators of longitudinal models are critically dependent on the manner in which samples become large by expanding the number of observations per person holding the number of people fixed or by expanding the number of persons holding the number of observations per person fixed the paper demonstrates which parameters can and cannot be identified from data produced by different sampling plans

on reflecting diffusion processes and skorokhod decompositions letg be ad dimensional bounded euclidean domain h1 g the set off in l2 g such that f defined in the distribution sense is in l2 g reflecting diffusion processes associated with the dirichlet spaces h1 g on l2 g dx are considered in this paper where a aij is a symmetric bounded uniformly ellipticd d matrix valued function such thata ij h1 g for eachi j and h1 g is a positive bounded function ong which is bounded away from zero a skorokhod decomposition is derived for the continuous reflecting markov processes associated with h1 g having starting points ing under a mild condition which is satisfied when g has finite d 1 dimensional lower minkowski content

another look at the instrumental variable estimation of error components models this article develops a framework for efficient iv estimators of random effects models with information in levels which can accommodate predetermined variables our formulation clarifies the relationship between the existing estimators and the role of transformations in panel data models we characterize the valid transformations for relevant models and show that optimal estimators are invariant to the transformation used to remove individual effects we present an alternative transformation for models with predetermined instruments which preserves the orthogonality among the errors finally we consider models with predetermined variables that have constant correlation with the effects and illustrate their importance with simulations

an ergodic control problem for constrained diffusion processes existence of optimal markov control an ergodic control problem for a class of constrained diffusion processes is considered the goal is the almost sure minimization of long term cost per unit time the main result of the paper is that there exists an optimal markov control for the considered problem it is shown that under the assumption of regularity of the skorohod map and the assumption that the drift vector field takes values in a certain cone of stability the class of controlled diffusion processes considered have strong uniform in control stability properties the role of the boundary is critical in obtaining the stability and ergodic control results for the class of controlled constrained diffusion processes considered in this work since the domains are unbounded and the corresponding unconstrained diffusions are typically transient these stability properties are key in obtaining appropriate tightness estimates once these estimates are available the remaining work lies in identifying weak limits of a certain family of occupation measures in this regard an extension to the echeverria weiss kurtz characterization of invariant measures of markov processes to the case of constrained controlled processes considered in this paper is proved this characterization result is also crucially used in proving the compactness of the family of invariant measures of markov processes corresponding to all possible markov controls

stationary distributions for diffusions with inert drift consider reflecting brownian motion in a bounded domain in mathbb r d that acquires drift in proportion to the amount of local time spent on the boundary of the domain we show that the stationary distribution for the joint law of the position of the reflecting brownian motion and the value of the drift vector has a product form moreover the first component is uniformly distributed on the domain and the second component has a gaussian distribution we also consider more general reflecting diffusions with inert drift as well as processes where the drift is given in terms of the gradient of a potential

explicit solution to a robust queueing control problem we consider the robust optimal control of a law of large numbers approximation of a stochastic network the robust control problem is formulated as a differential game with one player choosing the policies that determine service and routing assignments and the other choosing quantities such as the arrival and service rates subject to constraints the cost to be minimized by the first player and maximized by the second is the time until the origin is reached an explicit formula is given for the value function and some of its basic properties are studied

a numerical scheme for invariant distributions of constrained diffusions reflected diffusions in polyhedral domains are commonly used as approximate models for stochastic processing networks in heavy traffic stationary distributions of such models give useful information on the steady state performance of the corresponding stochastic networks and thus it is important to develop reliable and efficient algorithms for numerical computation of such distributions in this work we propose and analyze a monte carlo scheme based on an euler type discretization of the reflected stochastic differential equation using a single sequence of time discretization steps which decrease to zero as time approaches infinity appropriately weighted empirical measures constructed from the simulated discretized reflected diffusion are proposed as approximations for the invariant probability measure of the true diffusion model almost sure consistency results are established that in particular show that weighted averages of polynomially growing continuous functionals evaluated on the discretized simulated system converge a s to the corresponding integrals with respect to the invariant measure proofs rely on constructing suitable lyapunov functions for tightness and uniform integrability and characterizing almost sure limit points through an extension of echeverria s criteria for reflected diffusions regularity properties of the underlying skorohod problems play a key role in the proofs rates of convergence for suitable families of test functions are also obtained a key advantage of monte carlo methods is the ease of implementation particularly for high dimensional problems a numerical example

a large deviation result for a class of dirichlet processes let 0 be a constant and let 1 q x be a bounded measurable function we consider the large deviation problem for the diffusion process with generator we are going to prove that its density function satisfies mathop lim limits t to 0 t log h y t x frac left x y right 2 2

open queueing networks in heavy traffic this paper presents heavy traffic limit theorems for the queue length and sojourn time processes associated with open queueing networks these limit theorems state that properly normalized sequences of queue length and sojourn time processes converge weakly to a certain diffusion as the network traffic intensity converges to unity the limit diffusion is reflected brownian motion on the nonnegative orthant this process behaves like brownian motion on the interior of its state space and reflects instantaneously on the boundaries the reflection direction is a constant for each boundary hyperplane

martin kuramochi boundary and reflecting symmetric diffusion in this paper we will give sufficient conditions for the existence of the reflecting diffusion process on a locally compact space in constructing reflecting diffusion process we consider the corresponding martin kuramochi boundary as the reflecting barrier and introduce the notion of strong u caccioppoli set our method covers reflecting diffusion processes with diffusion coefficient degenerating on the boundary

on the robin problem in fractal domains research partially supported by national science foundation nsf grants dms 0244737 and dms 0303310

singular control with state constraints on unbounded domain we study a class of stochastic control problems where a cost of the form e 0 e s l x s ds h y o s d y s is to be minimized over control processes y whose increments take values in a cone y of r p keeping the state process x x b gy in a cone x of r k k p here x e x b is a brownian motion with drift b and covariance e g is a fixed matrix and y is the radon nikodym derivative dy d y let 1 2 trace d 2 b d where d denotes the gradient solutions to the corresponding dynamic programming pde f l v sup g df h 0 y g 1 on x are considered with a polynomial growth condition and are required to be supersolution up to the boundary corresponding to a state constraint boundary condition on 3x under suitable conditions on the problem data including continuity and nonnegativity of l and h and polynomial growth of l our main result is the unique viscosity sense solvability of the pde by the control problem s value function in appropriate classes of functions in some cases where uniqueness generally fails to hold in the class of functions that grow at most polynomially e g when h 0 our methods provide uniqueness within the class of functions that in addition have compact level sets the results are new even in the following special cases 1 the one dimensional case k p 1 x y r 2 the first order case 0 3 the case where l and h are linear the proofs combine probabilistic arguments and viscosity solution methods our framework covers a wide range of diffusion control problems that arise from queueing networks in heavy traffic

sur une integrale pour les processus a alpha variation bornee dans cet article l auteur obtient une formule d ito et l existence de temps locaux pour certains processus de dirichlet a partir des proprietes d une integrale faisant intervenir un processus a variation bornee et un autre a variation bornee cette integrale est definie comme limite de sommes de riemann

deterministic and stochastic differential inclusions with multiple surfaces of discontinuity we consider a class of deterministic and stochastic dynamical systems with discontinuous drift f and solutions that are constrained to live in a given closed domain g in mathbb r n according to a constraint vector field d specified on the boundary partial g of the domain specifically we consider equations of the form phi theta eta u quad dot theta t in f phi t quad mbox a e t for u in an appropriate class of functions where is the constraining term in the skorokhod problem specified by g d and f is the set valued upper semicontinuous envelope of f the case g mathbb r n when there is no constraining mechanism and u is absolutely continuous corresponds to the well known setting of differential inclusions di we provide a general sufficient condition for uniqueness of solutions and lipschitz continuity of the solution map in the form of existence of a lyapunov set here we assume i g is convex and admits the representation g cup i overline c i where c i i in mathbb i is a finite collection of disjoint open convex polyhedral cones in mathbb r n each having its vertex at the origin ii f b f c is a vector field defined on g such that b assumes a constant value on each of the given cones and f c is lipschitz continuous on g and iii d is an upper semicontinuous cone valued vector field that is constant on each face of g we also provide existence results under much weaker conditions where no lyapunov set condition is imposed for stochastic differential equations sde possibly reflected that have drift coefficient f and a lipschitz continuous possibly degenerate diffusion coefficient we establish strong existence and pathwise uniqueness under appropriate conditions our approach yields new existence and uniqueness results for both di and sde even in the case g mathbb r n the work has applications in the study of scaling limits of stochastic networks

reaction diffusion on metric graphs from 3d to 1d abstract we provide stochastic foundations for the analysis of a class of reaction diffusion systems using as an example the known temporal analysis of products tap experiments showing how to effectively obtain explicit solutions to the associated equations by approximating the 3 dimensional domain of diffusion u the reactor by 1 dimensional network models in a typical tap experiment a pulse of reactant gas of species a is injected into u which is filled with chemically inert material permeable to gas diffusion particles of catalyst are placed amid this inert medium forming active sites where the reaction a b may occur on part of the boundary of u designated as the exit a mixture of a and b can escape we study the problem of determining the molar fraction of product gas in the mixture after u is fully evacuated this fraction is identified with the reaction probability that is the probability of a single diffusing molecule reacting before leaving u specifically we are interested in how this probability depends on the reaction rate constant k after giving a stochastic formulation of the problem and the boundary value problem whose solution is this probability we study a class of domains called fat graphs comprising a network of thin tubes with active sites at junctures the main result of the paper is that in the limit as the thin tubes approach curves in 3 dimensional space reaction probability converges to functions of the point of gas injection that can be computed explicitly in terms of a rescaled parameter k by this 3d to 1d reduction the simpler processes on metric graphs can be used as model systems for more realistic 3 dimensional configurations this is illustrated with analytic and numerical examples one example in particular illustrates an important application of our method finding the catalyst configuration that maximizes reaction probability at a given reaction rate constant

switched diffusion processes and systems of elliptic equations a dirichlet space approach the switched diffusion process associated with a weakly coupled system of elliptic equations is studied via a dirichlet space approach and is applied to prove the existence theorem of the cauchy initial problem for the system a representation theorem for the solution of the dirichlet boundary value problem and a generalised skorohod decomposition for the reflecting switched diffusion process are obtained

the skorohod oblique reflection problem in time dependent domains the deterministic skorohod problem plays an important role in the construction and analysis of diffusion processes with reflection in the form studied here the multidimensional skorohod problem w

a uniqueness result for p monotone viscosity solutions of hamilton jacobi equations in bounded domains we consider a class of hamilton jacobi equations h x du x 0 with no u dependence and with continuity properties consistent with recent applications in queueing theory continuous viscosity solutions are considered in a compact polyhedral domain with oblique derivative neumann type boundary conditions comparison and uniqueness results are presented which use monotonicity of h x p in the p variable for values of p in the appropriate sub and superdifferential sets of the solution u x several examples illustrate the results

on positive recurrence of constrained diffusion processes let g c r k be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces g i i 1 n where n i and d i denote the inward normal and direction of constraint associated with g i respectively stability properties of a class of diffusion processes constrained to take values in g are studied under the assumption that the skorokhod problem defined by the data n i d i i 1 n is well posed and the skorokhod map is lipschitz continuous explicit conditions on the drift coefficient b of the diffusion process are given under which the constrained process is positive recurrent and has a unique invariant measure define c i d i i 0 i 1 n then the key condition for stability is that there exists 0 and a bounded subset a of g such that for all x g a b x c and dist b x c where c denotes the boundary of c

stochastic differential equations for multi dimensional domain with reflecting boundary in this paper we prove that there exists a unique solution of the skorohod equation for a domain inr d with a reflecting boundary condition we remove the admissibility condition of the domain which is assumed in the work 4 of lions and sznitman we first consider a deterministic case and then discuss a stochastic case

stability and paracontractivity of discrete linear inclusions abstract we study stability properties of a finite set of n n matrices such as paracontractivity bv and left convergent products lcps stability and their relations to each other the conjecture on equivalence of paracontractivity and lcp stability is proved moreover we prove the equivalence of the uniform bv stability and the property of vanishing length of steps of any trajectory of

workload models for stochastic networks value functions and performance evaluation this paper concerns control and performance evaluation for stochastic network models structural properties of value functions are developed for controlled brownian motion cbm and deterministic fluid workload models leading to the following conclusions outside of a null set of network parameters the following hold i the fluid value function is a smooth function of the initial state under further minor conditions the fluid value function satisfies the derivative boundary conditions that are required to ensure it is in the domain of the extended generator for the cbm model exponential ergodicity of the cbm model is demonstrated as one consequence ii the fluid value function provides a shadow function for use in simulation variance reduction for the stochastic model the resulting simulator satisfies an exact large deviation principle while a standard simulation algorithm does not satisfy any such bound iii the fluid value function provides upper and lower bounds on performance for the cbm model this follows from an extension of recent linear programming approaches to performance evaluation

martin boundary and integral representation for harmonic functions of symmetric stable processes abstract martin boundaries and integral representations of positive functions which are harmonic in a bounded domain d with respect to brownian motion are well understood unlike the brownian case there are two different kinds of harmonicity with respect to a discontinuous symmetric stable process one kind are functions harmonic in d with respect to the whole process x and the other are functions harmonic in d with respect to the process x d killed upon leaving d in this paper we show that for bounded lipschitz domains the martin boundary with respect to the killed stable process x d can be identified with the euclidean boundary we further give integral representations for both kinds of positive harmonic functions also given is the conditional gauge theorem conditioned according to martin kernels and the limiting behaviors of the h conditional stable process where h is a positive harmonic function of x d in the case when d is a bounded c 1 1 domain sharp estimate on the martin kernel of d is obtained

on the oblique derivative problem for diffusion processes and diffusion equations with holder continuous coefficients on a c2 domain in a euclidean space we consider the oblique derivative problem for a diffusion equation and assume the coefficients of the diffusion and boundary operators are holder continuous we then prove the uniqueness of diffusion processes and fundamental solutions corresponding to the problem for the purpose obtaining a stochastic representation of some so lutions to the problem plays a key role in our situation a difficulty arises from the absence of a fundamental solution with c2 smoothness up to the boundary it is overcome by showing some stability of a fundamental solution and a dif fusion process respectively under approximation of the domain in particular the stability of the fundamental solution is verified through construction it is done by applying the parametrix method twice to a parametrix with explicit expression

reflected brownian motion on an orthant we consider a k dimensional diffusion process z whose state space is the nonnegative orthant on the interior of the orthant z behaves like a kdimensional brownian motion with arbitrary covariance matrix and drift vector at each of the k 1 dimensional hyperplanes that form the boundary of the orthant z reflects instantaneously in a direction that is constant over that hyperplane there is no extant theory of multidimensional diffusion that applies to this process because the boundary of its state space is not smooth we adopt an approach that requires a restriction on the directions of reflection but reiman has shown that this restriction is met by all diffusions z arising as heavy traffic limits in open k station queuing networks our process z is defined as a path to path mapping of k dimensional brownian motion from this construction it follows that z is a continuous markov process and a semimartingale using the latter property we obtain a change of variable formula from which one can develop a complete analytical theory for the process z

boundary trace of reflecting brownian motions we establish a uniform dimensional result for normally reflected brownian motion rbm in a large class of non smooth domains hausdorff dimensions for the boundary occupation time and the boundary trace of rbm are determined extensions to stable like jump processes and to symmetric reflecting diffusions are also given

estimation of dynamic models with error components abstract observations on n cross section units at t time points are used to estimate a simple statistical model involving an autoregressive process with an additive term specific to the unit different assumptions about the initial conditions are a initial state fixed b initial state random c the unobserved individual effect independent of the unobserved dynamic process with the initial value fixed and d the unobserved individual effect independent of the unobserved dynamic process with initial value random asymptotic properties of the maximum likelihood and covariance estimators are obtained when t and when n the relationship between the pseudo and conditional maximum likelihood estimators is clarified a simple consistent estimator that is independent of the initial conditions and the way in which t or n is also suggested

reflected brownian motion in a cone semimartingale property in this paper the object of study is reflected brownian motion in a cone ind dimensions d 3 with nonconstant oblique reflection on each radial line emanating from the vertex of the cone the basic question considered here is when is this process a semimartingale conditions for the existence and uniqueness of the process for which the vertex is an instantaneous state were given by kwon which is resolved in terms of a real parameter depending on the cone and the direction of reflection it is shown that starting from any point of the cone the process is a semimartingale if 1 0 and not a semimartingale if 2

brownian motion in a wedge with variable skew reflection does planar brownian motion confined to a wedge by skew reflection on the sides approach the vertex of the wedge this question has been answered by varadhan and williams in the case where the direction of reflection is constant on each of the sides but here we address the question when the direction reflected is allowed to vary a necessary condition and a sufficient condition are obtained for the vertex to be reached the conditions are of a geometric nature and the gap between them is quite small

on reflected dirichlet spaces reflecting diffusion processes on smooth domains in euclidean space are well understood silverstein in 22 and 23 developed two variant procedures for constructing the reflected processes for a general class of symmetric hunt processes from a dirichlet space point of view a direct approach is given in this paper and these two variant procedures are shown to yield the same result only the techniques of martingales and ordinary markov processes are used

normal derivative for bounded domains with general boundary let d be a general bounded domain in the euclidean space rn a brownian motion which enters from and returns to the boundary symmetrically is used to define the normal derivative as a functional for f with f vf and a f all in l2 on d the corresponding neumann condition normal derivative 0 is an honest boundary condition for the l2 generator of reflected brownian notion on d a conditioning argument shows that for d and f sufficiently smooth this general definition of the normal derivative agrees with the usual one introduction bt t 0 is standard brownian motion on the euclidean space rn the case when the dimension n 1 is well understood and so we assume always that n 2 d is a bounded open connected subset of rn an important role will be played by a inf t 0 bt is not in d the hitting time for the complement rn d the absorbed process xt t 0 is xt bt for 0 a with a a dead point adjoined to d as an isolated point all functions are understood to vanish at 9 the absorbed transition operators po t 0 with action ptf x exf xt form a strongly continuous symmetric semigroup on l2 the hilbert space of real valued functions on d that are square integrable with respect to lebesgue measure dx the strong generator ao is selfadjoint and negative definite by the spectral theorem as formulated and proved in chapter ix of segal and kunze 1968 the operator a0 1 2 is a well defined positive definite selfadjoint operator the absorbed dirichlet space is the pair fo e where 0 1 f0 domain ao 2 e f g dx a 1 2f x a0 1 2g x f g in f received by the editors june 7 1987 1980 mathematics subject classification 1985 revision primary 60b85 secondary 35a85

an escape time criterion for queueing networks asymptotic risk sensitive control via differential games we consider the problem of risk sensitive control of a stochastic network in controlling such a network an escape time criterion can be useful if one wishes to regulate the occurrence of large buffers and buffer overflow in this paper a risk sensitive escape time criterion is formulated which in comparison to the ordinary escape time criteria penalizes exits that occur on short time intervals more heavily the properties of the risk sensitive problem are studied in the large buffer limit and related to the value of a deterministic differential game with constrained dynamics we prove that the game has value and that the value is the viscosity solution of a pde for a simple network the value is computed demonstrating the applicability of the approach

on the semimartingale representation of reflecting brownian motion in a cusp letc be the symmetric cusp x y 2 x y x x 0 where 1 in this paper we decide whether or not reflecting brownian motion inc has a semimartingale representation here the reflecting brownian motion has directions of reflection that make constant angles with the unit inward normals to the boundary our results carry through for a wide class of asymmetric cusps too

choice inclusive general rough semantics similarity based rough set theory rst involving choice in the formation of approximations was recently introduced by the present author though the theory can be used to develop improved semantics and models of knowledge and belief with ontology application requires a priori concepts of granules and granulation as opposed to the more common a posteriori or not a priori concepts of the same prevalent in the literature in this research we clarify the desirable semantic features of a context for seamless application of the theory to more general situations formalise them and refine the semantics a new axiomatic theory of granules in general rst including hybrid versions involving fuzzy set theories is also developed in the process interesting new applications to human learning are also illustrated in this paper

quenched invariance principles for random walks and elliptic diffusions in random media with boundary via a dirichlet form extension theorem and making full use of two sided heat kernel estimates we establish quenched invariance principles for random walks in random environments with a boundary in particular we prove that the random walk on a supercritical percolation cluster or amongst random conductances bounded uniformly from below in a half space quarter space etc converges when rescaled diffusively to a reflecting brownian motion which has been one of the important open problems in this area we establish a similar result for the random conductance model in a box which allows us to improve existing asymptotic estimates for the relevant mixing time furthermore in the uniformly elliptic case we present quenched invariance principles for domains with more general boundaries

reflected brownian motion in a wedge semimartingale property in this paper the object of study is reflected brownian motion in a two dimensional wedge with constant direction of reflection on each side of the wedge the basic question considered here is when is this process a semimartingale it is first shown that a related process defined by specifying the corner of the wedge to be an absorbing state rather than an instantaneous one is a semimartingale conditions for the existence and uniqueness of the process for which the corner is an instantaneous state were given by vardhan and williams brownian motion in a wedge with oblique reflection comm pure appl math to appear under these conditions it is shown that starting away from the corner the process is a semimartingale if and only if there is a convex combination of the directions of reflection that points into the wedge this equivalence is also shown to hold starting from the corner except in one unresolved case for which the wedge angle exceeds and the directions of reflection are exactly opposed

simple necessary and sufficient conditions for the stability of constrained processes in recent years a new approach has emerged for analyzing the stability properties of constrained stochastic processes in this approach one associates with the stochastic model a deterministic model or a family of deterministic models and under appropriate conditions stability of the stochastic model follows if all solutions of the deterministic model are attracted to the origin in the present work we show that a rather sharpcharacterization for the stability of the deterministic model is possible when it can be represented in terms of what we call a regular skorokhod map let g be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces gi i 1 n where ni and di denote the inward normal and direction of constraint associated with gi suppose that the skorokhod problem defined by the data ni di i 1 n is regular under these conditions the deterministic model mentioned above will correspond to a law of large numbers limit of the original stochas

initial conditions and moment restrictions in dynamic panel data models estimation of the dynamic error components model is considered using two alternative linear estimators that are designed to improve the properties of the standard first differenced gmm estimator both estimators require restrictions on the initial conditions process asymptotic efficiency comparisons and monte carlo simulations for the simple ar 1 model demonstrate the dramatic improvement in performance of the proposed estimators compared to the usual first differenced gmm estimator and compared to non linear gmm the importance of these results is illustrated in an application to the estimation of a labour demand model using company panel data

testing parameters in gmm without assuming that they are identified we propose a generalized method of moments gmm lagrange multiplier statistic i e the k statistic that uses the jacobian at the evaluated parameter value instead of the expected jacobian to obtain its limit behavior we use a novel assumption that brings gmm closer to maximum likelihood and which is easily satisfied the usual asymptotic x2 distribution of the k statistic then holds under a wider set of circumstances like weak and many instrument asymptotics and combinations thereof than the standard full rank case for the jacobian the behavior of the k statistic can be spurious around inflexion points and the maximum of the objective function since the moment conditions are then not satisfied combinations of the k statistic with statistics that test the validity of the moment equations overcome the spurious behavior we conduct a power comparison to test for the risk aversion parameter in a stochastic discount factor model and construct its confidence set for observed consumption growth and asset return series