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

what is the fractional laplacian a comparative review with new results abstract the fractional laplacian in r d which we write as 2 with 0 2 has multiple equivalent characterizations moreover in bounded domains boundary conditions must be incorporated in these characterizations in mathematically distinct ways and there is currently no consensus in the literature as to which definition of the fractional laplacian in bounded domains is most appropriate for a given application the riesz or integral definition for example admits a nonlocal boundary condition where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional laplacian in contrast the spectral definition requires only the standard local boundary condition these differences among others lead us to ask the question what is the fractional laplacian beginning from first principles we compare several commonly used definitions of the fractional laplacian theoretically through their stochastic interpretations as well as their analytical properties then we present quantitative comparisons using a sample of state of the art methods we discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems radial basis function collocation for the riesz fractional laplacian and nonharmonic lifting for the spectral fractional laplacian in our numerical studies we aim to compare different definitions on bounded domains using a collection of benchmark problems we consider the fractional poisson equation with both zero and nonzero boundary conditions where the fractional laplacian is defined according to the riesz definition the spectral definition the directional definition and the horizon based nonlocal definition we verify the accuracy of the numerical methods used in the approximations for each operator and we focus on identifying differences in the boundary behaviors of solutions to equations posed with these different definitions through our efforts we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications

average optimal singular control and a related stopping problem we observe a 2 brownian motion xt we can increase or decrease the value of the process paying r times the size of increase and l times the size of decrease holding costs are incurred continuously at a rate h zt where zt is the resulting process we consider a convex holding cost function h which is either finite everywhere or finite inside a finite interval and equal to infinity outside this interval the objective is to minimize average per unit of time expected cost it is shown that the optimal behavior is characterized by two constants a and b a b such that the optimal policy is to keep the process inside the interval a b with minimal efforts it is shown that to find the optimal interval a b one must solve a free boundary problem for the second order differential equation we give a probabilistic solution of the free boundary problem in terms of the value of a special game this game is a generalization of the optimal stopping problem there are two players observing the 2

on complementary graph entropy it has been recently discovered that complementary graph entropy characterizes and offers new insights into the minimum asymptotic rate for zero error source coding with decoder side information this paper presents new results that build on and complement this discovery specifically i previously unknown subadditivity properties of complementary graph entropy are derived and ii zero error coding rates are characterized in terms of complementary graph entropy for two multiterminal coding scenarios for both scenarios the rate characterization implies no rate loss relative to point to point source coding

on network coding for sum networks a directed acyclic network is considered where all the terminals need to recover the sum of the symbols generated at all the sources we call such a network a sum network it is shown that there exists a solvably and linear solvably equivalent sum network for any multiple unicast network and thus for any directed acyclic communication network it is also shown that there exists a linear solvably equivalent multiple unicast network for every sum network it is shown that for any set of polynomials having integer coefficients there exists a sum network which is scalar linear solvable over a finite field f if and only if the polynomials have a common root in f for any finite or cofinite set of prime numbers a network is constructed which has a vector linear solution of any length if and only if the characteristic of the alphabet field is in the given set the insufficiency of linear net work coding and unachievability of the network coding capacity are proved for sum networks by using similar known results for communication networks under fractional vector linear network coding a sum network and its reverse network are shown to be equivalent however under nonlinear coding it is shown that there exists a solvable sum network whose reverse network is not solvable

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

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

infinite dimensional linear programming approach to singularstochastic control we consider a multidimensional singular stochastic control problem with state dependent diffusion matrix and drift vector and control cost depending on the position and direction of displacement of the controlled process the objective is to minimize the total expected discounted cost we write an equivalent infinite dimensional linear programming problem on a subspace of the space conjugate to cal c bbb r n times cal c bbb r n times b where b is the unit sphere in bbb r n we write a dual linear program and prove absence of duality gap the dual program characterizes the optimal cost function as a maximal solution to the variational inequality with gradient constraints

marginal proportional hazards models for multiple event time data the wei et al 1989 semiparametric approach to the analysis of multiple event time data assumes that each event time is related to covariates through a proportional hazards model with a completely unspecified baseline hazard function but does not impose any constraint on the joint distribution of different event times as a result of the order restriction it is not clear whether or not event times can simultaneously satisfy their respective marginal proportional hazards assumption while having continuous joint distribution this leads to inability to conduct simulation studies we resolve this issue by constructing parametric models for multiple event times with proper joint density functions and marginal proportional hazards simulation studies are reported that compare efficiencies of the method of wei et al 1989 and of the maximum likelihood approach

a multidimensional stochastic singular control problem via dynkin game and dirichlet form the traditional difficulty with stochastic singular control is characterizing the regularities of the value function and the optimal control policy in this paper a multidimensional singular control problem is considered we found the optimal value function and the optimal control policy of this problem via a dynkin game whose solution is given by the saddle point of the cost function the existence and uniqueness of the solution to this dynkin game are proved through an associated variational inequality problem involving dirichlet form as a consequence the properties of the value function of this dynkin game imply the smoothness of the value function of the stochastic singular control problem in this way we are able to show the existence of a classical solution to this multidimensional singular control problem which was traditionally solved in the sense of viscosity solutions and this enables the application of the verification theorem to prove optimality

source coding and graph entropies a sender wants to accurately convey information to a receiver who has some possibly related data we study the expected number of bits the sender must transmit for one and for multiple instances in two communication scenarios and relate this number to the chromatic and korner 1973 entropies of a naturally defined graph

bivariate survival models induced by frailties abstract when two observed survival times depend via a proportional hazards model on the same unobserved variable called a frailty this common dependence induces an association between the observed times this article considers the class of bivariate survival distributions that can arise in this way extends it to allow negative association and shows that the observable bivariate distribution determines the unobserved frailty distribution up to a scale parameter a cross ratio function easily estimated even from censored data is the key to both the characterization results and inferential procedures and diagnostic plots that are introduced and illustrated by real examples the models considered are the natural counterpart for survival data of the latent variable models that have long been used in factor analysis for continuous data or in latent structure analysis for binary data the main distinguishing feature of survival data is the possibility of censoring which makes the standard methods difficu

integration by parts formulae for wiener measures on a path space between two curves this paper is concerned with the integration by parts formulae for the pinned or the standard wiener measures restricted on a space of paths staying between two curves the boundary measures concentrated on the set of paths touching one of the curves once are specified our approach is based on the polygonal approximations in particular to establish the convergence of boundary terms a uniform estimate is derived by means of comparison argument for a sequence of random walks conditioned to stay between two polygons applying the brascamp lieb inequality the stochastic integrals of wiener type are constructed relative to the three dimensional bessel bridge or the brownian meander

instantaneous control of brownian motion a controller continuously monitors a storage system such as an inventory or bank account whose content z zt t 0 fluctuates as a 2 brownian motion in the absence of control holding costs are incurred continuously at rate h zt at any time the controller may instantaneously increase the content of the system incurring a proportional cost of r times the size of the increase or decrease the content at a cost of l times the size of the decrease we consider the case where h is convex on a finite interval and h outside this interval the objective is to minimize the expected discounted sum of holding costs and control costs over an infinite planning horizon it is shown that there exists an optimal control limit policy characterized by two parameters a and b a b roughly speaking this policy exerts the minimum amounts of control sufficient to keep zt a b for all t 0 put another way the optimal control limit policy imposes on z a lower reflecting barrier at a an

regression analysis of multivariate incomplete failure time data by modeling marginal distributions abstract many survival studies record the times to two or more distinct failures on each subject the failures may be events of different natures or may be repetitions of the same kind of event in this article we consider the regression analysis of such multivariate failure time observations each marginal distribution of the failure times is formulated by a cox proportional hazards model no specific structure of dependence among the distinct failure times on each subject is imposed the regression parameters in the cox models are estimated by maximizing the failure specific partial likelihoods the resulting estimators are shown to be asymptotically jointly normal with a covariance matrix that can be consistently estimated simultaneous inferential procedures are then proposed extensive monte carlo studies indicate that the normal approximation is adequate for practical use the new methods allow time dependent covariates missing observations and arbitrary patterns of censorship they are illustrat

numerical solutions of linear mixed boundary value problems using stochastic representations the application of ito s formula induces some probabilistic representations of solutions of deterministic linear problems with boundary conditions of dirichlet neumann fourier and mixed types these representations are used to establish some easily implementable algorithms which compute an approximate solution by means of simulation of reflected random walks they do not require selected configurations at the neighbourhood of the domain boundary nor a discretization mesh the associated simulation methods are obtained and applied to problems for each class of boundary conditions numerical experiments with distributed source in two or three dimensional geometries and computational results with estimation of error are reported 1997 by john wiley sons ltd

a class of multivanate failure time distributions summary a class of continuous multivariate lifetime distributions is proposed the dependence between individuals in a group is modelled by a group specific quantity which can be interpreted as an unobserved covariate common to the individuals in the group and assumed to follow a positive stable distribution it is possible to include covariates in the model and discuss whether the dependence is still present after specific covariates are taken into account if the conditional hazards are proportional then the hazards in the marginal distributions are also proportional but with different constants of proportionality also the hazard for the minimum in a group is proportional to the marginal hazards if the conditional distributions given the group quantity are weibull then the marginal distributions are also weibull this class can be used to test the hypothesis of independence of litter mates in the proportional hazards model

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

dynkin games via dirichlet forms and singular control of one dimensional diffusions we consider a zero sum game of optimal stopping in which each of the opponents has the right to stop a one dimensional diffusion process there are two types of costs the first is accumulated continuously at the rate h xt where xt is the current position of the process the second is a cost associated with the stopping of the process it is given by the function f1 x for the first player and the function f2 x for the second player where x is the position of the process when the stopping option is exercised we study the solution of the free boundary problem associated with this game via dirichlet forms on the appropriate functional space integrating the value function of the game we get a solution to another free boundary problem which yields the optimal return function for a singular stochastic control problem

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

function computation through a bidirectional relay we consider a function computation problem in a three node wireless network nodes a and b observe two correlated sources x and y respectively and want to compute a function f x y to achieve this nodes a and b send messages to a relay node c at rates r a and r b respectively the relay c then broadcasts a message to a and b at rate r c we allow block coding and study the achievable region of rate triples under both zero error and epsilon error as a preparation we first consider a broadcast network from the relay to a and b a and b have side information x and y respectively the relay node c observes both x and y and broadcasts an encoded message to a and b we want to obtain the optimal broadcast rate such that a and b can recover the function f x y from the received message and their individual side information x and y respectively for this problem we show equivalence between epsilon error and zero error computations this gives a rate characterization for zero error computation as a corollary this also gives a rate characterization for the relay network under zero error for a class of functions called component wise one to one functions when the support set of p xy is full for the relay network the zero error rate region for arbitrary functions is characterized in terms of graph coloring of some suitably defined probabilistic graphs we then give a single letter inner bound to this rate region furthermore we extend the graph theoretic ideas to address the epsilon error problem and obtain a single letter inner bound

a dichotomy of functions f x y of correlated sources x y consider separate encoding of correlated sources x n x l cdots x n y n y l cdots y n for the decoder to reliably reproduce a function f x i y i n i 1 we establish the necessary and sufficient condition for the set of all achievable rates to coincide with the slepian wolf region whenever the probability density p x y is positive for all x y

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

index coding an interference alignment perspective author s maleki h cadambe vr jafar sa abstract the index coding problem is studied from an interference alignment perspective providing new results as well as new insights into and generalizations of previously known results an equivalence is established between the capacity of multiple unicast index coding where each message is desired by exactly one receiver and groupcast index coding where a message can be desired by multiple receivers which settles the heretofore open question of insufficiency of linear codes for the multiple unicast index coding problem by equivalence with groupcast settings where this question has previously been answered necessary and sufficient conditions for the achievability of rate half per message in the index coding problem are shown to be a natural consequence of interference alignment constraints and generalizations to feasibility of rate 1 l 1 per message when each destination desires at least l messages are similarly obtained finally capacity optimal solutions are presented to a series of symmetric index coding problems inspired by the local connectivity and local interference characteristics of wireless networks the solutions are based on vector linear coding 1963 2012 ieee

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

two way source coding with a helper consider the two way rate distortion problem in which a helper sends a common limited rate message to both users based on side information at its disposal we characterize the region of achievable rates and distortions when the markov relation helper user 1 user 2 holds the main insight of the result is that in order to achieve the optimal rate the helper may use a binning scheme as in wyner ziv where the side information at the decoder is the further user namely user 2 we derive these regions explicitly for the gaussian sources with square error distortion analyze a tradeoff between the rate from the helper and the rate from the source and examine a special case where the helper has the freedom to send different messages at different rates to the encoder and the decoder the converse proofs use a technique for verifying markov relations via undirected graphs

on the win ratio statistic in clinical trials with multiple types of event pocock et al 2012 following finkelstein schoenfeld 1999 has popularized the win ratio for analysis of controlled clinical trials with multiple types of outcome event the approach uses pairwise comparisons between patients in the treatment and control groups using a primary outcome say the time to death with ties broken using a secondary outcome say the time to hospitalization in general the observed pairwise preferences and the weight they attach to the component rankings will depend on the distribution of potential follow up time we present expressions for the win and loss probabilities for general bivariate survival models when follow up of all patients is limited to a specified time horizon in the special case of a bivariate lehmann model we show that the win ratio does not depend on this horizon we show how the win ratio may be estimated nonparametrically or from a parametric model extensions to events of three or more types are described application of the method of marginal estimation due to wei et al 1989 to this problem is described

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

distributed computing and the graph entropy region two remote senders observe x and y respectively and can noiselessly send information via a common relay node to a receiver that observes z the receiver wants to compute a function f x y z of these possibly related observations without error we study the average number of bits that need to be conveyed to that end by each sender to the relay and by the relay to the receiver in the limit of multiple instances we relate these quantities to the entropy region of a probabilistic graph with respect to a cartesian representation of its vertex set which we define as a natural extension of graph entropy general properties and bounds for the graph entropy region are derived and mapped back to special cases of the distributed computing setup

an alternative approach to confidence interval estimation for the win ratio statistic summary pocock et al 2012 european heart journal 33 176 182 proposed a win ratio approach to analyzing composite endpoints comprised of outcomes with different clinical priorities in this article we establish a statistical framework for this approach we derive the null hypothesis and propose a closed form variance estimator for the win ratio statistic in all pairwise matching situation our simulation study shows that the proposed variance estimator performs well regardless of the magnitude of treatment effect size and the type of the joint distribution of the outcomes

a class of proportional win fractions regression models for composite outcomes the win ratio is gaining traction as a simple and intuitive approach to analysis of prioritized composite endpoints in clinical trials to extend it from two sample comparison to regression we propose a novel class of semiparametric models that includes as special cases both the two sample win ratio and the traditional cox proportional hazards model on time to the first event under the assumption that the covariate specific win and loss fractions are proportional over time the regression coefficient is unrelated to the censoring distribution and can be interpreted as the log win ratio resulting from one unit increase in the covariate u statistic estimating functions in the form of an arbitrary covariate specific weight process integrated by a pairwise residual process are constructed to obtain consistent estimators for the regression parameter the asymptotic properties of the estimators are derived using uniform weak convergence theory for u processes visual inspection of a score process provides useful clues as to the plausibility of the proportionality assumption extensive numerical studies using both simulated and real data from a major cardiovascular trial show that the regression methods provide valid inference on covariate effects and outperform the two sample win ratio in both efficiency and robustness the proposed methodology is implemented in the r package wr publicly available from the comprehensive r archive network cran

fluctuations for a φ interface model with repulsion from a wall we consider a interface model on a one dimensional lattice with repulsion from a hard wall we suppose that the repulsion is of the form c 1 where c 0 and denotes the height of the interface from the wall we prove convergence of the equilibrium fluctuations around the hydrodynamic limit to the solution of a spde with singular drift if c 0 the system becomes the funaki olla interface model with reflection at the wall whose equilibrium fluctuations converge to the solution of a spde with reflection we give a new proof of this result using the characterization of such solution as the diffusion generated by an infinite dimensional dirichlet form obtained in a previous paper our method is based on a study of integration by parts formulae w r t the equilibrium measure of the interface model and allows to avoid the proof of the so called boltzmann gibbs principle we also obtain convergence of finite dimensional distributions of non equilibrium fluctuations around the stationary hydrodynamic limit 0

a system of infinitely many mutually reflecting brownian balls in ℝd an infinite system of skorohod type equations is studied the unique solution of the system is obtained from a finite case by passing to the limit it is a diffusion process describing a system of infinitely many brownian hard balls and has a gibbs state associated with the hard core pair potential as a reversible measure

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

lossy broadcasting with complementary side information a pair of strings x y put out by a memoryless source needs to be reliably communicated over a memoryless broadcast channel receiver 1 has y as side information and must reconstruct x to within some distortion receiver 2 has x and must reconstruct y to within some distortion the problem is motivated by the broadcast phase downlink of the two way relay channel we characterize reliable communication for gaussian sources with quadratic distortion functions conditionally independent sources deterministic distortion functions and small distortions with hamming distortion functions the last result is obtained by solving a new version of the broadcast problem with steinberg s common reconstruction decoding constraint

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

on the alternative hypotheses for the win ratio we extend the results of luo et al 2015 biometrics 71 139 145 regarding the alternative hypotheses for the win ratio from hazard orders to the upper quadrant stochastic order on the plane this extension substantially widens the range of alternatives against which the win ratio is known to be consistent it incorporates alternatives induced by simple and popular copula models that are left out by the characterization of luo et al 2015 we also discuss how our results may be generalized to win ratios in multivariate and stratified settings

system of interacting particles and nonlinear diffusion reflecting in a domain with sticky boundary we study a system of particles and the nonlinear mckean vlasov diffusion that is its limit for weak interactions in statistical mechanics reflecting in a domain with sticky boundary the interaction takes place in particular in the sojourn condition we show existence and uniqueness for the nonlinear martingale problem by a contraction argument on time change then we construct the system of particles by a limiting procedure and show propagation of chaos towards the nonlinear diffusion

an irreversible investment problem with maintenance expenditure on a finite horizon free boundary analysis abstract this paper concerns a continuous time finite horizon optimal irreversible investment problem with maintenance expenditure of a firm under uncertainty we assume that the firm can make irreversible investments to expand its production capacity and spend maintenance expenditure to achieve better performance of the productivity the objective of the firm is to construct optimal investment and maintenance policies to maximize its expected total profit over a finite horizon mathematically it is a singular stochastic control problem whose value function satisfies a parabolic variational inequality with gradient constraint the problem gives rise to two free boundaries which stand for the optimal investment and maintenance strategies respectively we investigate behaviors of free boundaries study regularities of the value function and give optimal investment and maintenance policies as we know this is a first integral result for an investment maintenance problem with a finite time horizon due to use of partial differential equation pde technique

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

optimal function computation in directed and undirected graphs we consider the problem of information aggregation in sensor networks where one is interested in computing a function of the sensor measurements we allow for block processing and study in network function computation in directed graphs and undirected graphs we study how the structure of the function affects the encoding strategies and the effect of interactive information exchange depending on the application there could be a designated collector node or every node might want to compute the function we begin by considering a directed graph c e on the sensor nodes where the goal is to determine the optimal encoders on each edge which achieve function computation at the collector node our goal is to characterize the rate region in r e i e the set of points for which there exist feasible encoders with given rates which achieve zero error computation for asymptotically large block length we determine the solution for directed trees specifying the optimal encoder and decoder for each edge for general directed acyclic graphs we provide an outer bound on the rate region by finding the disambiguation requirements for each cut and describe examples where this outer bound is tight next we address the scenario where nodes are connected in an undirected tree network and every node wishes to compute a given symmetric boolean function of the sensor data undirected edges permit interactive computation and we therefore study the effect of interaction on the aggregation and communication strategies we focus on sum threshold functions and determine the minimum worst case total number of bits to be exchanged on each edge the optimal strategy involves recursive in network aggregation which is reminiscent of message passing in the case of general graphs we present a cut set lower bound and an achievable scheme based on aggregation along trees for complete graphs we prove that the complexity of this scheme is no more than twice that of the optimal scheme

properties of composite time to first event versus joint marginal analyses of multiple outcomes many clinical studies eg cardiovascular outcome trials investigate the effect of an intervention on multiple event time outcomes the most common method of analysis is a so called composite analysis of a composite outcome defined as the time to the first component event other approaches have been proposed including the win ratio or win difference for ordered outcomes and the application of the wei lachin test herein we assess the influence of the marginal and joint distributions of the component events and their correlation structures on the operating characteristics of these methods for the analysis of multiple events the operating characteristics are investigated using a bivariate exponential model with a shared frailty under which these properties are obtained in closed form while the composite time to first event analysis provides an unbiased test of the hypothesis of equality of the distribution of the time to first event we show that it can provide a biased test of the joint null hypothesis of equal marginal hazards when the correlation of event times differs between groups the same applies to the win ratio however the operating characteristics of the wei lachin or other tests of the joint equality of the marginal hazards are unaffected furthermore when the correlations are equal the wei lachin test is more powerful to detect a difference in marginal hazards than the composite analysis test careful consideration of the properties of the various methods for analysis of composite outcome measures are in order before adopting one as primary analysis in a clinical study

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

an equivalence between network coding and index coding we show that the network coding and index coding problems are equivalent this equivalence holds in the general setting which includes linear and nonlinear codes specifically we present a reduction that maps a network coding instance to an index coding instance while preserving feasibility i e the network coding instance has a feasible solution if and only if the corresponding index coding instance is feasible in addition we show that one can determine the capacity region of a given network coding instance with colocated sources by studying the capacity region of a corresponding index coding instance previous connections between network and index coding were restricted to the linear case

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

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

on zero error source coding with decoder side information let x y be a pair of random variables distributed over a finite product set v spl times w according to a probability distribution p x y the following source coding problem is considered the encoder knows x while the decoder knows y and wants to learn x without error the minimum zero error asymptotic rate of transmission is shown to be the complementary graph entropy of an associated graph thus previous results in the literature provide upper and lower bounds for this minimum rate further these bounds are tight for the important class of perfect graphs the algorithmic aspects of instantaneous code design are considered next it is shown that optimal code design is np hard an optimal code design algorithm is derived polynomial time suboptimal algorithms are also presented and their average and worst case performance guarantees are established

index coding with side information motivated by a problem of transmitting supplemental data over broadcast channels birk and kol infocom 1998 we study the following coding problem a sender communicates with n receivers r1 rn he holds an input x 0 01l n and wishes to broadcast a single message so that each receiver ri can recover the bit xi each ri has prior side information about x induced by a directed graph grain nodes ri knows the bits of a in the positions j i j is an edge of g g is known to the sender and to the receivers we call encoding schemes that achieve this goal indexcodes for 0 1 n with side information graph g in this paper we identify a measure on graphs the minrank which exactly characterizes the minimum length of linear and certain types of nonlinear index codes we show that for natural classes of side information graphs including directed acyclic graphs perfect graphs odd holes and odd anti holes minrank is the optimal length of arbitrary index codes for arbitrary index codes and arbitrary graphs we obtain a lower bound in terms of the size of the maximum acyclic induced subgraph this bound holds even for randomized codes but has been shown not to be tight

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

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

strong feller properties for distorted brownian motion with reflecting boundary condition and an application to continuous n particle systems with singular interactions abstract assuming that r n n 2 is an open relatively compact set with boundary of lebesgue measure zero we prove strong feller properties for a class of distorted brownian motions in with reflecting boundary condition dirichlet form techniques give the existence of a weak solution to the corresponding stochastic differential equation for quasi all starting points in the sense of the associated martingale problem combining this result with the strong feller properties we can construct a weak solution for specified starting points if has c 2 boundary the construction works for all starting points where the drift term is not singular even on the boundary but also for a certain class of sets with less smooth boundary our approach works for all points in where the drift term is not singular and at least some points from our techniques allow very singular drift terms this enables us to construct continuous n particle gradient stochastic dynamics in cuboids r d d n with reflecting boundary condition and singular interactions for d n 2 we can start the stochastic dynamics in all initial configurations having at most one particle in provided is locally smooth there

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