nonparametric bayes conditional distribution modeling with variable selection this article considers a methodology for flexibly characterizing the relationship between a response and multiple predictors goals are 1 to estimate the conditional response distribution addressing the distributional changes across the predictor space and 2 to identify important predictors for the response distribution change both within local regions and globally we first introduce the probit stick breaking process psbp as a prior for an uncountable collection of predictor dependent random distributions and propose a psbp mixture psbpm of normal regressions for modeling the conditional distributions a global variable selection structure is incorporated to discard unimportant predictors while allowing estimation of posterior inclusion probabilities local variable selection is conducted relying on the conditional distribution estimates at different predictor points an efficient stochastic search sampling algorithm is proposed for posterior computation the methods are illustrated through simulation and applied to an epidemiologic study

order based dependent dirichlet processes in this article we propose a new framework for bayesian nonparametric modeling with continuous covariates in particular we allow the nonparametric distribution to depend on covariates through ordering the random variables building the weights in the stick breaking representation we focus mostly on the class of random distributions that induces a dirichlet process at each covariate value we derive the correlation between distributions at different covariate values and use a point process to implement a practically useful type of ordering two main constructions with analytically known correlation structures are proposed practical and efficient computational methods are introduced we apply our framework through mixtures of these processes to regression modeling the modeling of stochastic volatility in time series data and spatial geostatistical modeling

continuous time red and black how to control a diffusion to a goal a player starts at x in 0 1 and tries to reach 1 the process xt t 0 of his positions moves according to a diffusion process or more generally an ito process whose infinitesimal parameters are chosen by the player at each instant of time from a set depending on his current position to maximize the probability of reaching 1 the player should choose the parameters so as to maximize 2 at least when the maximum is achieved by bounded measurable functions this implies that bold timid play is optimal for subfair superfair continuous time red and black furthermore in superfair red and black the strategy which maximizes the drift coefficient of log xt minimizes the expected time to reach 1

a semi parametric bayesian approach to the instrumental variable problem we develop a bayesian semi parametric approach to the instrumental variable problem we assume linear structural and reduced form equations but model the error distributions non parametrically a dirichlet process prior is used for the joint distribution of structural and instrumental variable equations errors our implementation of the dirichlet process prior uses a normal distribution as a base model it can therefore be interpreted as modeling the unknown joint distribution with a mixture of normal distributions with a variable number of mixture components we demonstrate that this procedure is both feasible and sensible using actual and simulated data sampling experiments compare inferences from the non parametric bayesian procedure with those based on procedures from the recent literature on weak instrument asymptotics when errors are non normal our procedure is more efficient than standard bayesian or classical methods

bayesian semiparametric median regression modeling median regression models become an attractive alternative to mean regression models when employing flexible families of distributions for the errors classical approaches are typically algorithmic with desirable properties emerging asymptotically however nonparametric error models may be most attractive in the case of smaller sample sizes where parametric specifications are difficult to justify hence a bayesian approach enabling exact inference given the observed data may be appealing in this context there is little bayesian work we develop two fully bayesian modeling approaches employing mixture models for the errors in a median regression model the associated families of error distributions allow for increased variability skewness and flexible tail behavior the first family is semiparametric with extra variability captured nonparametrically through mixing and skewness handled parametrically the second family a fully nonparametric one includes all unimodal densities on the real line with

optimal stopping under uncertainty in drift and jump intensity this paper studies the optimal stopping problem in the presence of model uncertainty ambiguity we develop a numerically implementable method to solve this problem in a general setting allowing

reflected solutions of backward sde s and related obstacle problems for pde s we study reflected solutions of one dimensional backward stochastic differential equations the reflection keeps the solution above a given stochastic process we prove uniqueness and existence both by a fixed point argument and by approximation via penalization we show that when the coefficient has a special form then the solution of our problem is the value function of a mixed optimal stopping optimal stochastic control problem we finally show that when put in a markovian framework the solution of our reflected bsde provides a probabilistic formula for the unique viscosity solution of an obstacle problem for a parabolic partial differential equation

permanence and global attractivity of a discrete multispecies lotka volterra competition predator prey systems in this paper we propose a discrete multispecies lotka volterra competition predator prey systems for general non autonomous case sufficient conditions which ensure the permanence and the global stability of the system are obtained for periodic case sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system are obtained

on the multidimensional controller and stopper games we consider a zero sum stochastic differential controller and stopper game in which the state process is a controlled diffusion evolving in a multidimensional euclidean space in this game the controller affects both the drift and diffusion terms of the state process and the diffusion term can be degenerate under appropriate conditions we show that the game has a value and the value function is the unique viscosity solution to an obstacle problem for a hamilton jacobi bellman equation

bayesian semiparametric regression in the presence of conditionally heteroscedastic measurement and regression errors summary we consider the problem of robust estimation of the regression relationship between a response and a covariate based on sample in which precise measurements on the covariate are not available but error prone surrogates for the unobserved covariate are available for each sampled unit existing methods often make restrictive and unrealistic assumptions about the density of the covariate and the densities of the regression and the measurement errors for example normality and for the latter two also homoscedasticity and thus independence from the covariate in this article we describe bayesian semiparametric methodology based on mixtures of b splines and mixtures induced by dirichlet processes that relaxes these restrictive assumptions in particular our models for the aforementioned densities adapt to asymmetry heavy tails and multimodality the models for the densities of regression and measurement errors also accommodate conditional heteroscedasticity in simulation experiments our method vastly outperforms existing methods we apply our method to data from nutritional epidemiology

monotonic limit theorem of bsde and nonlinear decomposition theorem of doob meyers type we have obtained the following limit theorem if a sequence of rcll supersolutions of a backward stochastic differential equations bsde converges monotonically up to y t with e sup t y t 2 then y t itself is a rcll supersolution of the same bsde theorem 2 4 and 3 6 we apply this result to the following two problems 1 nonlinear doob meyer decomposition theorem 2 the smallest supersolution of a bsde with constraints on the solution y z the constraints may be non convex with respect to y z and may be only measurable with respect to the time variable t this result may be applied to the pricing of hedging contingent claims with constrained portfolios and or wealth processes

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

existence of periodic solutions in predator prey with watt type functional response and impulsive effects abstract in this paper the existence of a periodic solution and the permanence of a periodic watt type predator prey system where the prey disperses in patchy environment with two patches are investigated with the continuation theorem of coincidence degree theory we obtain the existence of a positive periodic solution for such a system we assume the watt type functional response within patch dynamics and provide a sufficient condition to guarantee the predator and prey species to be permanent furthermore we give numerical analysis to confirm our theoretical results it will be useful to ecosystem control

game of singular stochastic control and strategic exit we investigate a game of singular control and strategic exit in a model of competitive market share control in the model each player can make irreversible investments to increase his market share which is modeled as a diffusion process in addition each player has an option to exit the market at any point in time we formulate a verification theorem for best responses of the game and characterize markov perfect equilibria mpe under a set of verifiable assumptions we find a class of mpes with a rich structure in particular each player maintains up to two disconnected intervals of singular control regions one of which plays a defensive role and the other plays an offensive role we also identify a set of conditions under which the outcome of the game may be unique despite the multiplicity of the equilibria

permanence and global stability of nonautonomous lotka volterra system with predator prey and deviating arguments by using comparison theorem and constructing suitable lyapunov functional some sufficient conditions for the permanence and global asymptotic stability of the general nonautonomous lotka volterra system with predator prey and deviating arguments are obtained

the semiparametric bernstein von mises theorem in a smooth semiparametric estimation problem the marginal posterior for the parameter of interest is expected to be asymptotically normal and satisfy frequentist criteria of optimality if the model is endowed with a suitable prior it is shown that under certain straightforward and interpretable conditions the assertion of le cam s acclaimed but strictly parametric bernstein von mises theorem univ california publ statist 1 1953 277 329 holds in the semiparametric situation as well as a consequence bayesian point estimators achieve efficiency for example in the sense of hajek s convolution theorem z wahrsch verw gebiete 14 1970 323 330 the model is required to satisfy differentiability and metric entropy conditions while the nuisance prior must assign nonzero mass to certain kullback leibler neighborhoods ghosal ghosh and van der vaart ann statist 28 2000 500 531 in addition the marginal posterior is required to converge at parametric rate which appears to be the most stringent condition in examples the results are applied to estimation of the linear coefficient in partial linear regression with a gaussian prior on a smoothness class for the nuisance

dynamic behaviors of the impulsive periodic multi species predator prey system the dynamic behaviors of an impulsive periodic predator prey model with n preys and m predators are studied in this paper by constructing a suitable lyapunov function and using the comparison theorem of impulsive differential equation sufficient conditions which ensure the permanence and global attractivity of the system are obtained at the same time a set of criteria which guarantee that some species in the system are permanent and globally attractive while the remaining species are driven to extinction is obtained our results show that for the multi species predator prey community impulsivity is one of the important reasons that can change the long time behaviors of species

smoothly mixing regressions this paper extends the conventional bayesian mixture of normals model by permitting state probabilities to depend on observed covariates the dependence is captured by a simple multinomial probit model a conventional and rapidly mixing mcmc algorithm provides access to the posterior distribution at modest computational cost this model is competitive with existing econometric models as documented in the paper s illustrations the first illustration studies quantiles of the distribution of earnings of men conditional on age and education and shows that smoothly mixing regressions are an attractive alternative to nonbayesian quantile regression the second illustration models serial dependence in the s p 500 return and shows that the model compares favorably with arch models using out of sample likelihood criteria

on viscosity solutions of path dependent pdes in this paper we propose a notion of viscosity solutions for path dependent semi linear parabolic pdes this can also be viewed as viscosity solutions of non markovian backward sdes and thus extends the well known nonlinear feynman kac formula to non markovian case we shall prove the existence uniqueness stability and comparison principle for the viscosity solutions the key ingredient of our approach is a functional ito calculus recently introduced by dupire functional ito calculus 2009 preprint

permanence in nonautonomous lotka volterra system with predator prey the nonautonomous lotka volterra system with m predators and n preys is considered in this paper average conditions are obtained for permanence and global attractivity in the system it is shown that our results are generalization of those of the periodic system

variational inequalities for combined control and stopping we study the variational inequality associated with the combined stochastic control problem and establish the existence of a unique viscosity solution without uniform ellipticity an application to the quasi variational inequality is given

asymptotically efficient estimation in the presence of heteroskedasticity of unknown form in a multiple regression model the residual variance is an unknown function of the explanatory variables and estimated by nearest neighbor nonparametric regression the resulting weighted least squares estimator of the regression coefficients is shown to be adaptive in the sense of having the same asymptotic distribution to first order as estimators based on knowledge of the actual variance function or a finite parameterization of it a similar result was established by carroll 1982 using kernel estimation and under substantially more restrictive conditions on the data generating process than ours extensions to various other models seem to be possible

optimal stopping in levy models for nonmonotone discontinuous payoffs we give short proofs of general theorems about optimal entry and exit problems in levy models when payoff streams may have discontinuities and be nonmonotone as applications we consider exit and entry problems in the theory of real options and an entry problem with an embedded option to exit

on a stochastic logistic equation with impulsive perturbations a stochastic logistic model with impulsive perturbations is proposed and investigated first we give a new definition of a solution of an impulsive stochastic differential equation isde which is more convenient for use than the existing one using this definition we show that our model has a global and positive solution and obtain its explicit expression then we establish the sufficient conditions for extinction non persistence in the mean weak persistence persistence in the mean and stochastic permanence of the solution the critical value between weak persistence and extinction is obtained in addition the limit of the average in time of the sample path of the solution is estimated by two constants afterwards the lower growth rate and the upper growth rate of the solution are estimated finally sufficient conditions for global attractivity are established

additive cubic spline regression with dirichlet process mixture errors abstract the goal of this article is to develop a flexible bayesian analysis of regression models for continuous and categorical outcomes in the models we study covariate or regression effects are modeled additively by cubic splines and the error distribution that of the latent outcomes in the case of categorical data is modeled as a dirichlet process mixture we employ a relatively unexplored but attractive basis in which the spline coefficients are the unknown function ordinates at the knots we exploit this feature to develop a proper prior distribution on the coefficients that involves the first and second differences of the ordinates quantities about which one may have prior knowledge we also discuss the problem of comparing models with different numbers of knots or different error distributions through marginal likelihoods and bayes factors which are computed within the framework of chib 1995 as extended to dpm models by basu and chib 2003 the techniques are illustrated with simulated and real data

approximation of conditional densities by smooth mixtures of regressions this paper shows that large nonparametric classes of conditional multivariate densities can be approximated in the kullback leibler distance by different specifications of finite mixtures of normal regressions in which normal means and variances and mixing probabilities can depend on variables in the conditioning set covariates these models are a special case of models known as mixtures of experts in statistics and computer science literature flexible specifications include models in which only mixing probabilities modeled by multinomial logit depend on the covariates and in the univariate case models in which only means of the mixed normals depend flexibly on the covariates modeling the variance of the mixed normals by flexible functions of the covariates can weaken restrictions on the class of the approximable densities obtained results can be generalized to mixtures of general location scale densities rates of convergence and easy to interpret bounds are also obtained for different model specifications these approximation results can be useful for proving consistency of bayesian and maximum likelihood density estimators based on these models the results also have interesting implications for applied researchers

dynamic programming and pricing of contingent claims in an incomplete market the problem of pricing contingent claims or options from the price dynamics of certain securities is well understood in the context of a complete financial market this paper studies the same problem in an incomplete market when the market is incomplete prices cannot be derived from the absence of arbitrage since it is not possible to replicate the payoff of a given contingent claim by a controlled portfolio of the basic securities in this situation there is a price range for the actual market price of the contingent claim the maximum and minimum prices are studied using stochastic control methods the main result of this work is the determination that the maximum price is the smallest price that allows the seller to hedge completely by a controlled portfolio of the basic securities a similar result is obtained for the minimum price which corresponds to the purchase price

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

the rich domain of ambiguity explored ellsberg and others suggested that decision under ambiguity is a rich empirical domain with many phenomena to be investigated beyond the ellsberg urns we provide a systematic empirical investigation of this richness by varying the uncertain events the outcomes and combinations of both although ambiguity aversion is prevailing we also find systematic ambiguity seeking confirming insensitivity we find that ambiguity attitudes depend on the kind of uncertainty the source but not on the kind of outcomes ambiguity attitudes are closer to rationality ambiguity neutrality for natural uncertainties than for artificial ellsberg urn uncertainties this also appears from the reductions of monotonicity violations and of insensitivity ambiguity attitudes have predictive power across different outcomes and sources of uncertainty with individual specific components our rich domain serves well to test families of weighting functions for fitting ambiguity attitudes we find that two parameter families capt

asymptotic normality of semiparametric and nonparametric posterior distributions in this article it is shown that the marginal semiparametric and nonparametric posterior distributions for a parameter of interest behave like an ordinary parametric posterior distribution this in practice provides support of the utility of marginal semiparametric and nonparametric posterior distributions in particular the marginal semiparametric and nonparametric posterior distributions are asymptotically normal and centered at the corresponding maximum likelihood estimates mles or posterior means with covariance matrix the inverse of the fisher information additionally the semiparametric and nonparametric mles for the parameter of interest and the marginal posterior means are asymptotically normal and centered at the true parameter with the same covariance matrix the results are a semiparametric version and a nonparametric version of the parametric bayesian central limit theorem that establish a connection between the semiparametric and the nonparametric bayesian inference and their frequentis

global dynamics of the periodic logistic system with periodic impulsive perturbations this paper studies the global behaviors of the periodic logistic system with periodic impulsive perturbations the results of d d bainov and p s simeonov 1993 are extended and dynamics different from the corresponding continuous system are found it is shown that the system may have a unique positive periodic solution which is globally asymptotically stable or go extinct when the two periods are rational dependent when they are rational independent the system has no periodic solutions however still has a global attractor or go extinct under some conditions

on the robust optimal stopping problem we study a robust optimal stopping problem with respect to a set cal p of mutually singular probabilities this can be interpreted as a zero sum controller stopper game in which the stopper is trying to maximize its payoff while an adverse player wants to minimize this payoff by choosing an evaluation criteria from cal p we show that the upper snell envelope overline z of the reward process y is a supermartingale with respect to an appropriately defined nonlinear expectation underline mathscr e and overline z is further an underline mathscr e martingale up to the first time tau when overline z meets y consequently tau is the optimal stopping time for the robust optimal stopping problem and the corresponding zero sum game has a value although the result seems similar to the one obtained in the classical optimal stopping theory the mutual singularity of probabilities and the game aspect of the problem give rise to major technical hurdles which we circum

an anova model for dependent random measures we consider dependent nonparametric models for related random probability distributions for example the random distributions might be indexed by a categorical covariate indicating the treatment levels in a clinical trial and might represent random effects distributions under the respective treatment combinations we propose a model that describes dependence across random distributions in an analysis of variance anova type fashion we define a probability model in such a way that marginally each random measure follows a dirichlet process dp and use the dependent dirichlet process to define the desired dependence across the related random measures the resulting probability model can alternatively be described as a mixture of anova models with a dp prior on the unknown mixing measure the main features of the proposed approach are ease of interpretation and computational simplicity because the model follows the standard anovastructure interpretation and inference parallels conventions for anova mode

on the existence of optimal controls the optimal control problem where the state is governed by an ito stochastic differential equation possibly just an ordinary differential equation is formulated in martingale terms under a coercivity condition which is weaker than compactness of the control set a convexity condition and mild continuity hypotheses on the data it is shown by the direct method that optimal controls exist hard and soft constraints are allowed in the absence of soft constraints it is shown that there exists an optimal control that is a function only of the present time and state i e the synthesis problem has a solution the main tool here is krylov s markovian selection theorem

utility maximization with discretionary stopping utility maximization problems of mixed optimal stopping control type are considered which can be solved by reduction to a family of related pure optimal stopping problems sufficient conditions for the existence of optimal strategies are provided in the context of continuous time ito process models for complete markets the mathematical tools used are those of optimal stopping theory continuous time martingales convex analysis and duality theory several examples are solved explicitly including one which demonstrates that optimal strategies need not always exist

existence and global attractivity of an almost periodic ecological model the almost periodic lotka volterra model with m predators and n preys is considered in this paper by constructing suitable lyapunov function some sufficient conditions are obtained for the existence and global attractivity of a unique positive almost periodic solution of this model examples show that our criteria are new general and easily verifiable

on a periodic multi species ecological model a periodic predator prey model with m predators and n preys is proposed in this paper which can be seen as the modification of the traditional lotka volterra model by using comparison theorem the ultimately bounded region of the system is obtained by using comparison theorem and brouwer fixed point theorem sufficient conditions with guarantee the existence of a positive periodic solution of the system is obtained finally by constructing a suitable lyapunov function some sufficient conditions are obtained for the existence of a unique globally attractive periodic solution of system the results obtained here generalized the main results of j d zhao w c chen global asymptotic stability of a periodic ecological model applied mathematics and computation 147 3 2004 881 892

martingale approach to stochastic differential games of control and stopping we develop a martingale approach for studying continuous time stochastic differential games of control and stopping in a non markovian framework and with the control affecting only the drift term of the state process under appropriate conditions we show that the game has a value and construct a saddle pair of optimal control and stopping strategies crucial in this construction is a characterization of saddle pairs in terms of pathwise and martingale properties of suitable quantities

permanence in nonautonomous multi species predator prey system with feedback controls a nonautonomous multi species predator prey system with feedback controls is proposed in this paper where the competition among the predator species and among the prey species is considered also some important factors such as the effect of toxins and the age structure are also considered average conditions are obtained for permanence and global attractivity in the system the results obtained here generalized the main results of zhao and jiang j d zhao j f jiang permanence in nonautonomous lotka volterra system with predator prey appl math computat 152 2004 99 109

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions part i the case of bounded stochastic evolutions on etudie des equations elliptiques degenerees d ordre 2 non lineaires de la forme f d 2 u du u x 0 dans h ou h est un espace de hilbert separable x un point generique dans h u une fonction de h r

global attractivity of the periodic lotka volterra system a lotka volterra periodic model withm predators andn preys is studied in this paper a set of easily verifiable sufficient conditions that guarantee the existence uniqueness and global attractivity of the positive periodic solutions is obtained finally a suitable example is given to illustrate that the conditions of the main theorem are feasible

optimal stopping with multiple priors we develop a theory of optimal stopping under knightian uncertainty a suitable martingale theory for multiple priors is derived that extends the classical dynamic programming or snell envelope approach to multiple priors we relate the multiple prior theory to the classical setup via a minimax theorem in a multiple prior version of the classical model of independent and identically distributed random variables we discuss several examples from microeconomics operation research and finance for monotone payoffs the worst case prior can be identified quite easily with the help of stochastic dominance arguments for more complex payoff structures like barrier options model ambiguity leads to stochastic changes in the worst case beliefs

bayesian nonparametric calibration and combination of predictive distributions abstractwe introduce a bayesian approach to predictive density calibration and combination that accounts for parameter uncertainty and model set incompleteness through the use of random calibration functionals and random combination weights building on the work of ranjan and gneiting we use infinite beta mixtures for the calibration the proposed bayesian nonparametric approach takes advantage of the flexibility of dirichlet process mixtures to achieve any continuous deformation of linearly combined predictive distributions the inference procedure is based on combination gibbs and slice sampling we provide some conditions under which the proposed probabilistic calibration converges in terms of weak posterior consistency to the true underlying density for both cases of iid and markovian observations this calibration property improves upon the earlier calibration approaches we study the methodology in simulation examples with fat tails and multimodal densities and apply it to density forecasts of dail

a general stochastic maximum principle for optimal control problems the maximum principle for nonlinear stochastic optimal control problems in the general case is proved the control domain need not be convex and the diffusion coefficient can contain a control variable

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions iii uniqueness of viscosity solutions for general second order equations we prove here the uniqueness and existence of viscosity solutions for a general class of fully nonlinear second order equations in an infinite dimensional hilbert space the uniqueness is proved using the so called inf and sup convolutions which allow some reduction to finite dimensional problems

kernel stick breaking processes we propose a class of kernel stick breaking processes for uncountable collections of dependent random probability measures the process is constructed by first introducing an infinite sequence of random locations independent random probability measures and beta distributed random weights are assigned to each location predictor dependent random probability measures are then constructed by mixing over the locations with stick breaking probabilities expressed as a kernel multiplied by the beta weights some theoretical properties of the process are described including a covariate dependent prediction rule a retrospective markov chain monte carlo algorithm is developed for posterior computation and the methods are illustrated using a simulated example and an epidemiological application

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

forward backward stochastic differential equations and quasilinear parabolic pdes this paper studies under some natural monotonicity conditions the theory existence and uniqueness a priori estimate continuous dependence on a parameter of forward backward stochastic differential equations and their connection with quasilinear parabolic partial differential equations we use a purely probabilistic approach and allow the forward equation to be degenerate

existence uniqueness and stability of positive periodic solution for a nonlinear prey competition model with delays a nonlinear periodic predator prey model with m preys and n m predators and delays is proposed in this paper which can be seen as the modification of the traditional lotka volterra prey competition model sufficient conditions which guarantee the existence of a unique globally attractive positive periodic solution of the system are obtained

solving forward backward stochastic differential equations explicitly a four step scheme in this paper we investigate the nature of the adapted solutions to a class of forward backward stochastic differential equations sdes for short in which the forward equation is non degenerate we prove that in this case the adapted solution can always be sought in an ordinary sense over an arbitrarily prescribed time duration via a direct four step scheme using this scheme we further prove that the backward components of the adapted solution are determined explicitly by the forward components via the solution of a certain quasilinear parabolic pde system moreover the uniqueness of the adapted solutions over an arbitrary time duration as well as the continuous dependence of the solutions on the parameters can all be proved within this unified framework some special cases are studied separately in particular we derive a new form of the integral representation of the clark haussmann ocone type for functionals or functions of diffusions in which the conditional expectation is no longer needed

bayesian modeling of joint and conditional distributions in this paper we study a bayesian approach to exible modeling of conditional distributions the approach uses a exible model for the joint distribution of the dependent and independent variables and then extracts the conditional distributions of interest from the estimated joint distribution we use a nite mixture of multivariate normals fmmn to estimate the joint distribution the conditional distributions can then be assessed analytically or through simulations the discrete variables are handled through the use of latent variables the estimation procedure employs an mcmc algorithm we provide a characterization of the kullback leibler closure of fmmn and show that the joint and conditional predictive densities implied by fmmn model are consistent estimators for a large class of data generating processes with continuous and discrete observables the method can be used as a robust regression model with discrete and continuous dependent and independent variables and as a bayesian alternative to semi and non parametric models such as quantile and kernel regression in experiments the method compares favorably with classical nonparametric and alternative bayesian methods

the periodic solutions for general periodic impulsive population systems of functional differential equations and its applications in this paper the general periodic impulsive population systems of functional differential equations are investigated by using the method of poincare map and horn s fixed point theorem we prove that the ultimate boundedness of all solutions implies the existence of periodic solutions as applications of this result the existence of positive periodic solutions for the general periodic impulsive kolmogorov type population dynamical systems are discussed we further prove that as long as the system is permanent there must exist at least one positive periodic solution in addition the permanence and existence of positive periodic solutions are discussed for the periodic impulsive single species logistic models and the periodic impulsive n species lotka volterra competitive models with delays

viscosity solutions of fully nonlinear parabolic path dependent pdes part ii the main objective of this paper and the accompanying one viscosity solutions of fully nonlinear parabolic path dependent pdes part ii 2012 preprint is to provide a notion of viscosity solutions for fully nonlinear parabolic path dependent pdes our definition extends our previous work ann probab 2014 42 204 236 focused on the semilinear case and is crucially based on the nonlinear optimal stopping problem analyzed in stochastic process appl 2014 124 3277 3311 we prove that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions and satisfies a stability property and a partial comparison result the latter is a key step for the well posedness results established in viscosity solutions of fully nonlinear parabolic path dependent pdes part ii 2012 preprint we also show that the value processes of path dependent stochastic control problems are viscosity solutions of the corresponding path dependent dynamic programming equations

wellposedness of second order backward sdes we provide an existence and uniqueness theory for an extension of backward sdes to the second order while standard backward sdes are naturally connected to semilinear pdes our second order extension is connected to fully nonlinear pdes as suggested in cheridito et al commun pure appl math 60 7 1081 1110 2007 in particular we provide a fully nonlinear extension of the feynman kac formula unlike cheridito et al in commun pure appl math 60 7 1081 1110 2007 the alternative formulation of this paper insists that the equation must hold under a non dominated family of mutually singular probability measures the key argument is a stochastic representation suggested by the optimal control interpretation and analyzed in the accompanying paper soner et al in dual formulation of second order target problems arxiv 1003 6050 2009

optimal investment for worst case crash scenarios a martingale approach we investigate the optimal portfolio problem under the threat of a financial market crash in a multidimensional jump diffusion framework we set up a nonprobabilistic crash model and consider an investor that seeks to maximize crra utility in the worst possible crash scenario we recast the problem as a stochastic differential game with the help of the fundamental notion of indifference strategies we completely solve the portfolio problem using martingale arguments

on a nonlinear nonautonomous predator prey model with diffusion and distributed delay in this paper a nonlinear nonautonomous predator prey model with diffusion and continuous distributed delay is studied where all the parameters are time dependent the system which is composed of two patches has two species the prey can diffuse between two patches but the predator is confined to one patch we first discuss the uniform persistence and global asymptotic stability of the model after that by constructing a suitable lyapunov functional some sufficient conditions for the existence of a unique almost periodic solution of the system are obtained an example shows the feasibility of our main results

the controller and stopper game for a linear diffusion consider a process x x t 0 t 1 with values in the interval i 0 1 absorption at the boundary points of i and dynamics dx t t dt t dw t x 0 x the values t t are selected by a controller from a subset of 0 1 that depends on the current position x t for every t 0 at any stopping rule of his choice a second player called stopper can halt the evolution of the process x upon which he receives from the controller the amount e u x here 2 0 1 is a discount factor and u 0 1 is a continuous reward function under appropriate conditions on this function and on the contoller s set of choices it is shown that the two players have a saddle point of optimal strategies these can be described fairly explicitly by reduction to a suitable problem of optimal stopping whose maximal expected reward v coincides with the value of the game v sup inf x e e u x inf x sup e e u x

global attractivity in an almost periodic multi species nonlinear ecological model abstract a nonlinear almost periodic predator prey model with n preys and m predators is studied in this paper which can be seen as the modification of the traditional multi species lotka volterra predator prey model for general nonautonomous case by using the differential inequality theory we obtain the sufficient conditions which guarantee the uniform persistence and nonpersistence of the system after that by constructing a suitable lyapunov function some sufficient conditions are obtained which ensure the global attractivity of the system for almost periodic case by constructing a suitable lyapunov function sufficient conditions which guarantee the existence of an unique globally attractive positive almost periodic solution of the system are obtained examples together with their numeric simulations show the feasibility of our main results

bayesian regression with heteroscedastic error density and parametric mean function abstract in this paper we consider bayesian estimation of restricted conditional moment models with the linear regression as a particular example a common practice in the bayesian literature for linear regression and other semi parametric models is to use flexible families of distributions for the errors and to assume that the errors are independent from covariates however a model with flexible covariate dependent error distributions should be preferred for the following reason assuming that the error distribution is independent of predictors might lead to inconsistent estimation of the parameters of interest when errors and covariates are dependent to address this issue we develop a bayesian regression model with a parametric mean function defined by a conditional moment condition and flexible predictor dependent error densities sufficient conditions to achieve posterior consistency of the regression parameters and conditional error densities are provided in experiments the proposed method compares favorably with classical and alternative bayesian estimation methods for the estimation of the regression coefficients and conditional densities

semiparametric bayesian inference in autoregressive panel data models this paper develops bayesian methods for inference in dynamic panel data models with individual effects and applies them to study longitudinal data on earnings from the panel study of income dynamics psid we study semiparametric versions of commonly used random effects autoregressive models in which the distribution of the disturbances is not restricted to fall in a parametric class to model the unknown distributions without resorting to strong parametric assumptions we draw upon recent advances in the theory and computation of nonparametric bayesian models using dirichlet process priors the overall approach can be viewed as an application of the general semiparametric bayesian approach of west muller and escobar 1994 which in turn makes use of results on bayesian density estimation by escobar 1994 and escobar and west 1995 most conventional panel data methods e g anderson and hsiao 1981 macurdy

global attractivity of the almost periodic solution of a delay logistic population model with impulses in this paper we study the existence of almost periodic solutions of a delay logistic model with fixed moments of impulsive perturbations by using a comparison theorem and constructing a suitable lyapunov functional a set of sufficient conditions for the existence and global attractivity of a unique positive almost periodic solution is obtained as applications some special models are studied our new results improve and generalize former results

optional decomposition of supermartingales and hedging contingent claims in incomplete security markets letm x be the family of all equivalent local martingale measuresq for some locally boundedd dimensional processx andv be a positive process the main result of the paper theorem 2 1 states that the processv is a supermartingale whateverq m x if and only if this process admits the following decomposition v t whereh is an integrand forx andc is an adapted increasing process we call such a representationoptional because in contrast to the doob meyer decomposition it generally exists only with an adapted optional processc we apply this decomposition to the problem of hedging european and american style contingent claims in the setting ofincomplete security markets

extinction in nonautonomous t periodic competitive lotka volterra system a nonautonomous t periodic competitive lotka volterra system of n species is considered it is shown that if the coefficients are t periodic continuous and satisfy certain inequalities then any solution with strictly positive initial conditions has the property that all but one of its components vanish while the remaining component approaches the canonical solution of a certain logistic differential equation

bayesian regression with nonparametric heteroskedasticity abstract this paper studies large sample properties of a semiparametric bayesian approach to inference in a linear regression model the approach is to model the distribution of the regression error term by a normal distribution with the variance that is a flexible function of covariates the main result of the paper is a semiparametric bernstein von mises theorem under misspecification even when the distribution of the regression error term is not normal the posterior distribution of the properly recentered and rescaled regression coefficients converges to a normal distribution with the zero mean and the variance equal to the semiparametric efficiency bound

global asymptotic stability of a periodic ecological model the periodic lotka volterra model with m predators and n preys is considered in this paper under the suppose that the intrinsic growth rate of the prey species may be negative while the total intrinsic growth rate in a period are positive sufficient conditions are obtained for existence and global asymptotic stability of the periodic solution of this model

stochastic perron s method and elementary strategies for zero sum differential games we develop here the stochastic perron method in the framework of two player zero sum differential games we consider the formulation of the game where both players play symmetrically feedback strategies as opposed to the elliott kalton formulation prevalent in the literature the class of feedback strategies we use is carefully chosen so that the state equation admits strong solutions and the technicalities involved in the stochastic perron method carry through in a rather simple way more precisely we define the game over elementary strategies which are well motivated by intuition within this framework the stochastic perron method produces a viscosity subsolution of the upper isaacs equation dominating the upper value of the game and a viscosity supersolution of the upper isaacs equation lying below the upper value of the game using a viscosity comparison result we obtain that the upper value is the unique and continuous viscosity solution of the upper isaacs equation an identical statement hol

functional ito calculus and stochastic integral representation of martingales we develop a non anticipative calculus for functionals of a continuous semimartingale using an extension of the ito formula to path dependent functionals which possess certain directional derivatives the construction is based on a pathwise derivative introduced by b dupire for functionals on the space of right continuous functions with left limits we show that this functional derivative admits a suitable extension to the space of square integrable martingales this extension denes a weak derivative which is shown to be the inverse of the ito integral and which may be viewed as a non anticipative lifting of the malliavin derivative these results lead to a constructive martingale representation formula for ito processes by contrast with the clark haussmann ocone formula this representation only involves nonanticipative quantities which may be computed pathwise

portfolio risk minimization and differential games abstract a risk minimization problem is considered in a continuous time markovian regime switching financial model modulated by a continuous time finite state markov chain we interpret the states of the chain as different states of an economy a particular form of convex risk measure which includes the entropic risk measure as a particular case as a measure of risk and an optimal portfolio is determined by minimizing the convex risk measure of the terminal wealth we explore the state of the art of the stochastic differential game to formulate the problem as a markovian regime switching version of a two player zero sum stochastic differential game a novel feature of our model is that we provide the flexibility of controlling both the diffusion risk and the regime switching risk a verification theorem for the hamilton jacobi bellman hjb solution of the game is provided

filtration consistent nonlinear expectations and related g expectations from a general definition of nonlinear expectations viewed as operators preserving monotonicity and constants we derive under rather general assumptions the notions of conditional nonlinear expectation and nonlinear martingale we prove that any such nonlinear martingale can be represented as the solution of a backward stochastic equation and in particular admits continuous paths in other words it is a g martingale

controlling a process to a goal in finite time a player starts at x in 0 1 and seeks to reach 1 by time t0 the process xt 0 t t0 of the player s positions is a diffusion process or an ito process whose infinitesimal parameters are chosen by the player at each instant of time from a set depending on the current position the probability of reaching 1 by time t0 is maximized if the player can and does choose the parameters so that and 2 are maximized at least when these maxima are sufficiently regular this result implies that bold play is optimal for subfair continuous time red and black and roulette when there is a limit on playing time

retrospective markov chain monte carlo methods for dirichlet process hierarchical models inference for dirichlet process hierarchical models is typically performed using markov chain monte carlo methods which can be roughly categorized into marginal and conditional methods the former integrate out analytically the infinite dimensional component of the hierarchical model and sample from the marginal distribution of the remaining variables using the gibbs sampler conditional methods impute the dirichlet process and update it as a component of the gibbs sampler since this requires imputation of an infinite dimensional process implementation of the conditional method has relied on finite approximations in this paper we show how to avoid such approximations by designing two novel markov chain monte carlo algorithms which sample from the exact posterior distribution of quantities of interest the approximations are avoided by the new technique of retrospective sampling we also show how the algorithms can obtain samples from functionals of the dirichlet process the marginal and the conditional methods are compared and a careful simulation study is included which involves a non conjugate model different datasets and prior specifications

exact and approximate nash equilibria in discounted markov stopping games with terminal redemption abstract this work is concerned with a class of discrete time zero sum games with two players and markov transitions on a denumerable space the general structure of the game is as follows before the game starts player i receives a bond from player ii and at each decision time player ii can stop the system paying a terminal bond to player i or if the game is not halted player i selects an action to drive the system and receives a running reward bond from player ii at termination player i redeems the accumulated bonds and the performance of a pair of decision strategies is measured by the expectation of the value at time zero of the bonds accumulated by player i under standard continuity compactness conditions the following conclusions are established i it is shown that this stopping game has a value function which is characterized by an equilibrium equation and ii such a result is used to obtain the existence of a nash equilibrium also iii the method of successive approximations is used to construct approximate nash equilibria for the game

bayesian curve fitting using multivariate normal mixtures summary problems of regression smoothing and curve fitting are addressed via predictive inference in a flexible class of mixture models multidimensional density estimation using dirichlet mixture models provides the theoretical basis for semi parametric regression methods in which fitted regression functions may be deduced as means of conditional predictive distributions these bayesian regression functions have features similar to generalised kernel regression estimates but the formal analysis addresses problems of multivariate smoothing parameter estimation and the assessment of uncertainties about regression functions naturally computations are based on multidimensional versions of existing markov chain simulation analysis of univariate dirichlet mixture models

second order backward stochastic differential equations and fully nonlinear parabolic pdes for a d dimensional diffusion of the form dxt xt dt xt dwt and continuous functions f and g we study the existence and uniqueness of adapted processes y z and a solving the second order backward stochastic differential equation 2bsde dyt f t xt yt zt t dt z dxt t 0 t dzt at dt t dxt