on singular control problems with state constraints and regime switching this paper investigates a singular stochastic control problem for a multi dimensional regime switching diffusion process confined in an unbounded domain the objective is to maximize the total expected discounted rewards from exerting the singular control such a formulation stems from application areas such as optimal harvesting multiple species and optimal dividends payments schemes in random environments with the aid of weak dynamic programming principle and an exponential transformation we characterize the value function to be the unique constrained viscosity solution of a certain system of coupled nonlinear quasi variational inequalities several examples are analyzed in detail to demonstrate the main results

weak dynamic programming principle for viscosity solutions we prove a weak version of the dynamic programming principle for standard stochastic control problems and mixed control stopping problems which avoids the technical difficulties related to the measurable selection argument in the markov case our result is tailor made for the derivation of the dynamic programming equation in the sense of viscosity solutions

singular optimal stochastic controls ii dynamic programming the dynamic programming principle for a multidimensional singular stochastic control problem is established in this paper when assuming lipschitz continuity on the data it is shown that the value function is continuous and is the unique viscosity solution of the corresponding hamilton jacobi bellman equation

on the consistency of bayes estimates on etudie les proprietes de frequence des regles de bayes avec une attention particuliere a la consistence

optimal control of the risk process in a regime switching environment this paper is concerned with cost optimization of an insurance company the surplus of the insurance company is modeled by a controlled regime switching diffusion in which the regime switching mechanism provides the fluctuations of the random environment the goal is to find an optimal control that minimizes the total cost up to a stochastic exit time a weaker sufficient condition than that of fleming and soner 2006 section v 2 for the continuity of the value function is obtained further the value function is shown to be a viscosity solution of a hamilton jacobi bellman equation

de finetti s coherence and statistical inference on donne la definition d une inference coherente en conformite avec une theorie des probabilites conditionnelles derivee du principe de coherence de de finetti et on effectue une comparaison critique d une telle definition avec celle proposee par heath lane et sudderth

on competitive lotka volterra model in random environments focusing on competitive lotka volterra model in random environments this paper uses regime switching diffusions to model the dynamics of the population sizes of n different species in an ecosystem subject to the random changes of the external environment it is demonstrated that the growth rates of the population sizes of the species are bounded above moreover certain long run average limits of the solution are examined from several angles a partial stochastic principle of competitive exclusion is also derived finally simple examples are used to demonstrate our findings

optimal consumption and portfolio with both fixed and proportional transaction costs we consider a market model with one risk free and one risky asset in which the dynamics of the risky asset are governed by a geometric brownian motion in this market we consider an investor who consumes from the bank account and who has the opportunity at any time to transfer funds between the two assets we suppose that these transfers involve a fixed transaction cost k 0 independent of the size of the transaction plus a cost proportional to the size of the transaction the objective is to maximize the cumulative expected utility of consumption over a planning horizon we formulate this problem as a combined stochastic control impulse control problem which in turn leads to a nonlinear quasi variational hamilton jacobi bellman inequality qvhjbi we prove that the value function is the unique viscosity solution of this qvhjbi finally numerical results are presented

stock trading an optimal selling rule trading in stock markets consists of three major steps select a stock purchase a number of shares and eventually sell them to make a profit the timing to buy and sell is extremely crucial a selling rule can be specified by two preselected levels a target price and a stop loss limit this paper is concerned with an optimal selling rule based on the model characterized by a number of geometric brownian motions coupled by a finite state markov chain such a policy can be obtained by solving a set of two point boundary value differential equations moreover the corresponding expected target period and probability of making money and that of losing money are derived analytic solutions are obtained in one and two dimensional cases finally a numerical example is considered to demonstrate the effectiveness of our method

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

on optimal harvesting problems in random environments this paper investigates the optimal harvesting strategy for a single species living in random environments whose population growth is given by a regime switching diffusion harvesting acts as a stochastic control on the size of the population the objective is to find a harvesting strategy which maximizes the expected total discounted income from harvesting up to the time of extinction of the species the income rate is allowed to be state and environment dependent this is a singular stochastic control problem with both the extinction time and the optimal harvesting policy depending on the initial condition one aspect of receiving payments up to the random time of extinction is that small changes in the initial population size may significantly alter the extinction time when using the same harvesting policy consequently one no longer obtains continuity of the value function using standard arguments for either regular or singular control problems having a fixed time horizon this paper introduces a new sufficient condition under which the continuity of the value function for the regime switching model is established further it is shown that the value function is a viscosity solution of a coupled system of quasi variational inequalities the paper also establishes a verification theorem and based on this theorem an varepsilon optimal harvesting strategy is constructed under certain conditions on the model two examples are analyzed in detail

optimal harvesting of a stochastic delay tri trophic food chain model with levy jumps abstract in this paper the optimal harvesting of a stochastic delay tri trophic food chain model with levy jumps is considered we introduce two kinds of environmental perturbations in this model one is called white noise which is continuous and is described by a stochastic integral with respect to the standard brownian motion and the other one is jumping noise which is modeled by a levy process under some mild assumptions the critical values between extinction and persistent in the mean of each species are established the sufficient and necessary criteria for the existence of optimal harvesting policy are established and the optimal harvesting effort and the maximum of sustainable yield are also obtained we utilize the ergodic method to discuss the optimal harvesting problem the results show that white noises and levy noises significantly affect the optimal harvesting policy while time delays is harmless for the optimal harvesting strategy in some cases at last some numerical examples are introduced to show the validity of our results

comparative testing of experts we show that a simple reputation style test can always identify which of two experts is informed about the true distribution the test presumes no prior knowledge of the true distribution achieves any desired degree of precision in some xed nite time and does not use counterfactual predictions our analysis capitalizes on a result due to fudenberg and levine 1992 on the rate of convergence of supermartingales we use our setup to shed some light on the apparent paradox that a strategically motivated expert can ignorantly pass any test we point out that this paradox arises because in the single expert setting any mixed strategy for nature over distributions is reducible to a pure strategy this eliminates any meaningful sense in which nature can randomize comparative testing reverses the impossibility result because the presence of an expert who knows the realized distribution eliminates the reducibility of nature s compound lotteries

optimal switching with constraints and utility maximization of an indivisible market this work focuses on optimal switching with constraints our motivation stems from utility maximization of an indivisible market the dynamic programming approach is used the value function is characterized as the unique viscosity solution of a quasi variational inequality the unbounded domain introduces new challenges by studying the sample paths of the diffusion at the boundary a sufficient condition for the continuity of the value function is provided yielding the desired characterization not only are the results of this work applicable to the utility maximization problem but also they can be used for general optimal switching problems with finite regimes

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

bayesian representation of stochastic processes under learning de finetti revisited a probability distribution governing the evolution of a stochastic process has infinitely many bayesian representations of the form d among these a natural representation is one whose components s are learnable one can approximate by conditioning on observation of the process and sufficient for prediction s predictions are not aided by conditioning on observation of the process we show the existence and uniqueness of such a representation under a suitable asymptotic mixing condition on the process this representation can be obtained by conditioning on the tail field of the process and any learnable representation that is sufficient for prediction is asymptotically like the tail field representation this result is related to the celebrated de finetti theorem but with exchangeability weakened to an asymptotic mixing condition and with his conclusion of a decomposition into i i d component distributions weakened to components that are learnable and sufficient for prediction

optimal harvesting of a stochastic mutualism model with levy jumps in this letter a stochastic mutualism model with levy jumps and harvesting is considered under some simple assumptions sufficient and necessary criteria for the existence of optimal harvesting policy are established the optimal harvesting effort and the maximum of sustainable yield are also obtained the effects of random noises on the optimal harvesting of the model are discussed and some numerical simulations are introduced to illustrate the main results

merging and testing opinions we study the merging and the testing of opinions in the context of a prediction model in the absence of incentive problems opinions can be tested and rejected regardless of whether or not data produces consensus among bayesian agents in contrast in the presence of incentive problems opinions can only be tested and rejected when data produces consensus among bayesian agents these results show a strong connection between the testing and the merging of opinions they also relate the literature on bayesian learning and the literature on testing strategic experts 1 introduction data can produce consensus among bayesian agents who initially disagree it can also test and reject opinions we relate these two critical uses of data in a model where agents may strategically misrepresent what they know in each period either 0 or 1 is observed let p and q be two probability measures on 0 1 1 such that q is absolutely continuous with respect to p if p and q are additive then as shown by blackwell and dubins 1962 the conditional probabilities of p and q merge in the sense that the two posteriors become uniformly close as the amount of observations increases q almost surely so repeated applications of bayes rule lead to consensus among bayesian agents provided that their opinions were initially compatible now consider savage s axiomatization of subjective probability he proposed postulates that characterize a preference relation over bets in terms of a nonatomic finitely additive probability p call such p for short an opinion savage s framework allows for finitely additive probabilities that are not additive in particular the conclusions of the blackwell and dubins theorem hold for some but not all opinions this flexibility makes savage s

optimal harvesting strategies for stochastic competitive lotka volterra ecosystems this work develops optimal harvest strategies for lotka volterra systems so as to establish economically ecologically and environmentally reasonable strategies for populations subject to the risk of extinction to better reflect reality a continuous time markov chain is used to model the random environment the underlying systems are thus controlled regime switching diffusions that belong to the class of singular control problems starting with a model having multiple species we construct upper bounds for the value functions prove the finiteness of the harvesting value and derive properties of the value functions then we construct explicit chattering harvesting strategies and the corresponding lower bounds for the value functions by using the idea of harvesting only one species at a time we further show that this is a reasonable candidate for the best lower bound that one can expect moreover in some cases the lower bounds provide a good approximation of the value functions

singular optimal stochastic controls i existence we apply the compactification method to study the control problem where the state is governed by an ito stochastic differential equation allowing both classical and singular control the problem is reformulated as a martingale problem on an appropriate canonical space after the relaxed form of the classical control is introduced under some mild continuity hypotheses on the data it is shown by purely probabilistic arguments that an optimal control for the problem exists the value function is shown to be borel measurable

compatible measures and merging two measures and i are updated as more information arrives if with probability 1 the predictions of future events according to both measures become close as time passes we say that i merges to blackwell and dubins blackwell d l dubins 1962 merging of opinions with increasing information ann math statist 38 882 886 showed that if is absolutely continuous with respect to i then i merges to restricting the definition to prediction of near future events and to a full sequence of times yields the new notion of almost weak merging awm presented here we introduce a necessary and sufficient condition and show many cases with no absolute continuity that exhibit awm we show for instance that the fact that i is diffused around implies awm

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

zeros of a random algebraic polynomial with coefficient means in geometric progression this paper provides the mathematical expectation for the number of real zeros of an algebraic polynomial with non identical random coefficients we assume that the coefficients aj n 1j 0 of the polynomial t x a0 a1x a2x2 an 1xn 1 are normally distributed with mean e aj j 1 where 0 and constant non zero variance it is shown that the behaviour of the random polynomial is independent of the variance on the interval 1 1 it differs however for the cases of 1 on the intervals 1 and 1 we find the expected number of real zeros is governed by an interesting relationship between the means of the coefficients and their common variance our result is consistent with those of previous works for identically distributed coefficients in that the expected number of real zeros for 0 is half of that for 0

on the average number of real roots of a random algebraic equation on donne un estimateur du nombre espere d instants ou le polynome prend la valeur reelle k k n etant pas forcement nul

numerical methods for optimal harvesting strategies in random environments under partial observations this work is concerned with optimal harvesting problems in random environments in contrast to the existing literature the markov chain is hidden and can only be observed in a gaussian white noise in our work we first use the wonham filter to estimate the markov chain from the observable evolution of the given process so as to convert the original problem to a completely observable one then we treat the resulting optimal control problem because the problem is virtually impossible to solve in closed form our main effort is devoted to developing numerical approximation algorithms to approximate the value function and optimal strategies markov chain approximation methods are used to construct a discrete time controlled markov chain convergence of the algorithm is proved by weak convergence method and suitable scaling a numerical example is provided to demonstrate the results

prediction optimization and learning in repeated games consider a two player discounted repeated game in which each player optimizes with respect to a prior belief about his opponent s repeated game strategy one would like to argue that if beliefs are cautious then each player s best response will be in the support loosely speaking of his opponent s belief and that therefore players will learn as the game unfolds to predict the continuation path of play if this conjecture were true a convergence result due to kalai and lehrer would imply that the continuation path of the repeated game would asymptotically resemble that of a nash equilibrium one would thus have constructed a theory in which nash equilibrium behavior is a necessary long run consequence of optimization by cautious players this paper points out an obstacle to such a theory loosely put in many repeated games if players optimize with respect to beliefs that satisfy a diversity condition termed neutrality then each player will choose a strategy that his opponent was certain would not be played

connections between optimal stopping and singular stochastic control i monotone follower problems the stochastic control problem of tracking a brownian motion by a nondecreasing process monotone follower is related to a question of optimal stopping direct probabilistic arguments are employed to show that the two problems are equivalent and that both admit optimal solutions

investment consumption models with transaction fees and markov chain parameters this paper considers an infinite horizon investment consumption model in which a single agent consumes and distributes his wealth in two assets a bond and a stock the problem of maximization of the total utility from consumption is treated state amount allocated in assets and control consumption rates of trading constraints are present it is shown that the value function is the unique viscosity solution of a system of variational inequalities with gradient constraints

optimal dividend strategies of two collaborating businesses in the diffusion approximation model in this paper we consider the optimal dividend payment strategy for an insurance company that has two collaborating business lines the surpluses of the business lines are modeled by diffusion processes the collaboration between the two business lines permits that money can be transferred from one line to another with or without proportional transaction costs while money must be transferred from one line to another to help both business lines keep running before simultaneous ruin of the two lines eventually occurs