numerical approximation for functionals of reflecting diffusion processes the aim of this paper is to approximate the expectation of a large class of functionals of the solution x xi of a stochastic differential equation with normal reflection in a piecewise smooth domain of rd this also yields a monte carlo method for solving partial differential problems of parabolic type with mixed boundary conditions the approximation is based on a modified euler scheme for the stochastic differential equation the scheme can be driven by a sequence of bounded independently and identically distributed i i d random variables or when the domain is convex by a sequence of gaussian i i d random variables the order of weak convergence for both cases is given in the former case the order of convergence is 1 2 and it is shown to be exact by an example in the last section numerical tests are presented the behavior of the error as a function of the final time t for fixed values of the discretization step and as a function of the discretization step for fixed values of the final

a splitting step algorithm for reflected stochastic differential equations in r 1 a new algorithm which is based on the splitting step idea and the penalization method for reflected stochastic differential equation rsde in the upper half space r 1 is presented in this paper after some important estimates about rsdes and penalization odes are obtained the local pathwise error of the suggested algorithm are considered and an estimate of local error in the 2pth moment is proved under the assumptions of lipschitz continuity and linear growth some numerical experiments based on matlab programs are given to support the suggested algorithm

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

reflected brownian motion in a cone with radially homogeneous reflection field this work is concerned with the existence and uniqueness of a strong markov process that has continuous sample paths and the following ad ditional properties i the state space is a cone in dimensions d 3 and the process behaves in the interior of the cone like ordinary brownian motion ii the process reflects instantaneously at the boundary of the cone the direction of reflection being fixed on each radial line emanating from the vertex of the cone iii the amount of time that the process spends at the vertex of the cone is zero i e the set of times for which the process is at the vertex has zero lebesgue measure the question of existence and uniqueness is cast in precise mathematical terms as a submartingale problem in the style used by stroock and varadhan for diffusions on smooth domains with smooth boundary conditions the question is resolved in terms of a real parameter a which in general depends in a rather complicated way on the geometric data of the problem i e on the cone and the directions of reflection however a criterion is given for determining whether a 0 it is shown that there is a unique continuous strong markov process satisfying i iii above if and only if a 2 there is a unique continuous strong markov process satisfying i and ii above it reaches the vertex of the cone almost surely and remains there these results are illustrated in concrete terms for some special cases the process considered here serves as a model for comparison with a re flected brownian motion in a cone having a nonradially homogeneous reflection field this is discussed in a subsequent work by kwon

l q optimal control for discrete descriptor systems with a complete set of boundary information this paper presents the linear quadratic optimal control problem for the discrete time descriptor systems exk 1 axk buk where e is a singular matrix the system structure is in general non causal and its boundary conditions information is specified in terms of d initial conditions and n d final conditions where d deg det e a it is shown that the optimal control law depends only on the d dimensional state variables of the causal subsystem and the optimal cost to go is a general quadratic expression of the causal state variables and does not depend on the entire descriptor vector this result constitutes an extension to the non causal system case of larson et al 1978 where the structure of the descriptor system had been assumed causal

feedback and pole placement in descriptor variable systems the effects and uses of applying linear feedback to continuous time descriptor systems are studied structural changes resulting from feeding back the slow and fast parts of the trajectory separately are characterized it is shown that under certain conditions related to controllability the poles of the slow subsystem may be shifted arbitrarily and the impulsive behaviour of the fast subsystem may be eliminated

brownian motion in a wedge with variable reflection existence and uniqueness existence and uniqueness in law of reflecting brownian motion in a wedge is proved the direction of reflection along the sides of the wedge varies in a reasonable fashion except perhaps at the corner

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

roof duality complementation and persistency in quadratic 0 1 optimization the paper is concerned with the primal problem of maximizing a given quadratic pseudo boolean function four equivalent problems are discussed the primal the complementation the discrete rhys lp and the weighted stability problem of a sam graph each of them has a relaxation the roof dual the quadratic complementation the continuous rhys lp and the fractional weighted stability problem of a sam graph the main result is that the four gaps associated with the four relaxations are equal furthermore a solution to any of these problems leads at once to solutions of the other three equivalent ones the four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem the optimal solutions of the roof dual define best linear majorantsp x off having the following persistency property if theith coefficient inp is positive negative thenx i 1 0 in every optimum of the primal problem several characterizations are given for the case where these persistency results cannot be used to fix any variable of the primal on the other hand a class of gap free functions properly including the supermodular ones is exhibited

maxsolver an efficient exact algorithm for weighted maximum satisfiability maximum boolean satisfiability max sat is the optimization counterpart of boolean satisfiability sat in which a variable assignment is sought to satisfy the maximum number of clauses in a boolean formula a branch and bound algorithm based on the davis putnam logemann loveland procedure dpll is one of the most competitive exact algorithms for solving max sat in this paper we propose and investigate a number of strategies for max sat the first strategy is a set of unit propagation or unit resolution rules for max sat we summarize three existing unit propagation rules and propose a new one based on a nonlinear programming formulation of max sat the second strategy is an effective lower bound based on linear programming lp we show that the lp lower bound can be made effective as the number of clauses increases the third strategy consists of a binary clause first rule and a dynamic weighting variable ordering rule which are motivated by a thorough analysis of two existing well known variable orderings based on the analysis of these strategies we develop an exact solver for both max sat and weighted max sat our experimental results on random problem instances and many instances from the max sat libraries show that our new solver outperforms most of the existing exact max sat solvers with orders of magnitude of improvement in many cases

rate of convergeance of a particle method for the solution of a 1d viscous scalar conservation law in a bounded interval in this paper we give a probabilistic interpretation of a viscous scalar conservation law in a bounded interval thanks to a nonlinear martingale problem the underlying nonlinear stochastic process is reflected at the boundary to take into account the dirichlet conditions after proving uniqueness for the martingale problem we show existence thanks to a propagation of chaos result indeed we exhibit a system of n interacting particles the empirical measure of which converges to the unique solution of the martingale problem as n as a consequence the solution of the viscous conservation law can be approximated thanks to a numerical algorithm based on the simulation of the particle system when this system is discretized in time thanks to the euler lepingle scheme d lepingle math comput simulation 38 1995 119 126 we show that the rate of convergence of the error is in o t 1 n where at denotes the time step finally we give numerical results which confirm this theoretical rate

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

efficient branch and bound algorithms for weighted max 2 sat max 2 sat is one of the representative combinatorial problems and is known to be np hard given a set of m clauses on n propositional variables where each clause contains at most two literals and is weighted by a positive real max 2 sat asks to find a truth assignment that maximizes the total weight of satisfied clauses in this paper we propose branch and bound exact algorithms for max 2 sat utilizing three kinds of lower bounds all lower bounds are based on a directed graph that represents conflicts among clauses and two of them use a set covering representation of max 2 sat computational comparisons on benchmark instances disclose that these algorithms are highly effective in reducing the number of search tree nodes as well as the computation time

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

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

probabilistic approach for nonlinear partial differential equations and stochastic partial differential equations with neumann boundary conditions abstract in this study we consider nonlinear partial differential equations pdes and stochastic pdes spdes with neumann boundary conditions on domains that satisfy the lions sznitmann saisho conditions using the convergence result based on the penalization approximation for stochastic differential equations with normal reflections on nonsmooth and nonconvex domains we establish the existence and comparison principle for the viscosity solutions of nonlinear pdes with the neumann conditions associated with the optimal control problem we also obtain for nonlinear spdes and backward spdes with neumann condition the representations of the stochastic viscosity solutions

sweeping by a continuous prox regular set abstract the evolution problem known as sweeping process is considered for a class of nonconvex sets called prox regular or convex assuming essentially that such sets contain in the interior a suitable subset and move continuously w r t the hausdorff distance we prove local and global existence as well as uniqueness of solutions which are continuous functions with bounded variation some examples are presented

resolution for max sat max sat is the problem of finding an assignment minimizing the number of unsatisfied clauses in a cnf formula we propose a resolution like calculus for max sat and prove its soundness and completeness we also prove the completeness of some refinements of this calculus from the completeness proof we derive an exact algorithm for max sat and a time upper bound we also define a weighted max sat resolution like rule and show how to adapt the soundness and completeness proofs of the max sat rule to the weighted max sat rule finally we give several particular max sat problems that require an exponential number of steps of our max sat rule to obtain the minimal number of unsatisfied clauses of the combinatorial principle these results are based on the corresponding resolution lower bounds for those particular problems

a stochastic differential game in the orthrant we study a zero sum stochastic differential game in the nonnegative orthrant the state of the system is governed by controlled reflecting diffusions in the nonnegative orthrant we consider discounted and average payoff evaluation criteria we prove the existence of values and optimal strategies for both payoff criteria

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

reflecting brownian motion in a cusp let c be the cusp x y x 0 x y x where 1 set c 1 x y x 0 y x and c 2 x y x 0 y x we study the existence and uniqueness in law of reflecting brownian motion in c the angle of reflection at c j 0 relative to the inward unit normal is a constant j 2 2 and is positive iff the direction of reflection has a negative first component in all sufficiently small neighborhoods of 0 when 1 2 0 existence and uniqueness in law hold when 1 2 0 existence fails we also obtain results for a large class of asymmetric cusps we make essential use of results of warschawski on the differentiability at the boundary of conformal maps

reflected brownian motion with skew symmetric data in a polyhedral domain this paper is concerned with the characterization and invariant measures of certain reflected brownian motions rbm s in polyhedral domains the kind of rbm studied here behaves like d dimensional brownian motion with constant drift in the interior of a simple polyhedron and is instantaneously reflected at the boundary in directions that depend on the face that is hit under the assumption that the directions of reflection satisfy a certain skew symmetry condition first introduced in harrison williams 9 it is shown that such an rbm can be characterized in terms of a family of submartingales and that it reaches non smooth parts of the boundary with probability zero in 9 a purely analytic problem associated with such an rbm was solved here the exponential form solution obtained in 9 is shown to be the density of an invariant measure for the rbm furthermore if the density is integrable over the polyhedral state space then it yields the unique stationary distribution for the rbm in the proofs of these results a key role is played by a dual process for the rbm and by results in 9 for reflected brownian motions on smooth approximating domains

heavy traffic limits for the g h2 n m queue we establish heavy traffic stochastic process limits for queue length waiting time and overflow stochastic processes in a class ofg gi n m queueing models withn servers andm extra waiting spaces we let the arrival process be general only requiring that it satisfy a functional central limit theorem to capture the impact of the service time distribution beyond its mean within a markovian framework we consider a special class of service time distributions denoted byh2 which are mixtures of an exponential distribution with probabilityp and a unit point mass at 0 with probability 1 p these service time distributions exhibit relatively high variability having squared coefficients of variation greater than or equal to one as in halfin and whitt 1981 heavy traffic limits for queues with many exponential servers oper res 29 567 588 puhalskii and reiman 2000 the multiclassgi ph n queue in the halfin whitt regime adv appl probab 32 564 595 and garnett mandelbaum and reiman 2002 designing a call center with impatient customers manufacturing service oper management 4 208 227 we consider a sequence of queueing models indexed by the number of servers n and letn tend to infinity along with the traffic intensities nso that v n 1 n for 8 8 to treat finite waiting rooms we letm n v n for 0 8 with the specialh2 service time distribution the limit processes are one dimensional markov processes behaving like diffusion processes with different drift and diffusion functions in two different regions above and below zero we also establish a limit for theg m n m m model having exponential customer abandonments

lifschitz tail on hyperbolic space neumann conditions on considere un espace hyperbolique h de dimension n et l operateur laplacien hyperbolique h etant donne une distribution aleatoire de points a l aide d un processus ponctuel de poisson tel que chaque point se situe dans une boule hyperbolique de rayons on etudie la question suivante si est un petit nombre reel combien y a t il de valeurs propres inferieures a pour l operateur 1 2 h dans une boule b de centre o avec des conditions de neumann sur la frontiere et avec des conditions aux limites de dirichlet sur les boules aleatoires formant des intersections avec b si le rayon de b est suffisamment grand

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

continuous dependence results for non linear neumann type boundary value problems abstract we obtain estimates on the continuous dependence on the coefficient for second order non linear degenerate neumann type boundary value problems our results extend previous work of cockburn et al jakobsen and karlsen and gripenberg to problems with more general boundary conditions and domains a new feature here is that we account for the dependence on the boundary conditions as one application of our continuous dependence results we derive for the first time the rate of convergence for the vanishing viscosity method for such problems we also derive new explicit continuous dependence on the coefficients results for problems involving bellman isaacs equations and certain quasilinear equation

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

the submartingale problem for a class of degenerate elliptic operators we consider the degenerate elliptic operator acting on c 2 b functions on 0 d mathcal l f x sum i 1 d a i x x i alpha i frac partial 2 f partial x i 2 x sum i 1 d b i x frac partial f partial x i x where the ai are continuous functions that are bounded above and below by positive constants the bi are bounded and measurable and the alpha i in 0 1 we impose neumann boundary conditions on the boundary of 0 d there will not be uniqueness for the submartingale problem corresponding to mathcal l if we consider however only those solutions to the submartingale problem for which the process spends 0 time on the boundary then existence and uniqueness for the submartingale problem for mathcal l holds within this class our result is equivalent to establishing weak uniqueness for the system of stochastic differential equations rm d x t i sqrt 2a i x t x t i alpha i 2 rm d w i t b i x t rm d t rm d l t x i quad x i t geq 0 where w t i are independent brownian motions and l x i t is a local time at 0 for xi

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

the submartingal problem for brownian motion in a cone with non constant 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 1 the state space is a cone ind dimensions d 3 and the process behaves in the interior of the cone like ordinary brownian motion 2 the process reflects instantaneously at the boundary of the cone the direction of reflection is continuous except at the vertex and has a limit which is fixed on each radial line emanating from the vertex 3 the amount of time that the process spends at the vertex of the cone is zero i e the set of times for which the process is at the vertex has zero lebesgue measure

a new bounding procedure and an improved exact algorithm for the max 2 sat problem given a cnf conjunctive normal form boolean expression with clauses of size at most two the max 2 sat problem asks to find a truth assignment to the boolean variables that makes true the maximum number of clauses in this paper we describe an innovative upper bound computation procedure which is centered around the use of equations and inequalities that are satisfied by all solutions to the problem the procedure is incorporated in a branch and bound algorithm for max 2 sat an initial solution to the problem is provided by an iterated tabu search heuristic we present computational experience on the max 2 sat benchmark instances from the max sat evaluation 2007 the results show that the developed branch and bound algorithm is very competitive with the best previously reported techniques

a multidimensional process involving local time on presente une approche probabiliste a un probleme pose par mckean 1967 pour la construction d un systeme de n particules sur r avec un generateur formel

partial constraint satisfaction abstract a constraint satisfaction problem involves finding values for variables subject to constraints on which combinations of values are allowed in some cases it may be impossible or impractical to solve these problems completely we may seek to partially solve the problem in particular by satisfying a maximal number of constraints standard backtracking and local consistency techniques for solving constraint satisfaction problems can be adapted to cope with and take advantage of the differences between partial and complete constraint satisfaction extensive experimentation on maximal satisfaction problems illuminates the relative and absolute effectiveness of these methods a general model of partial constraint satisfaction is proposed

generalized bsdes and nonlinear neumann boundary value problems we study a new class of backward stochastic differential equations which involves the integral with respect to a continuous increasing process this allows us to give a probabilistic formula for solutions of semilinear partial differential equations with neumann boundary condition where the boundary condition itself is nonlinear we consider both parabolic and elliptic equations