control of singularly perturbed hybrid stochastic systems we study a class of optimal stochastic control problems involving two different time scales the fast mode of the system is represented by deterministic state equations whereas the slow mode of the system corresponds to a jump disturbance process under a fundamental ergodicity property for a class of infinitesimal control systems associated with the fast mode we show that there exists a limit problem which provides a good approximation to the optimal control of the perturbed system both the finite and infinite discounted horizon cases are considered we show how an approximate optimal control law can be constructed from the solution of the limit control problem in the particular case where the infinitesimal control systems possess the so called turnpike property i e characterized by the existence of global attractors the limit control problem can be given an interpretation related to a decomposition approach

on existence of limit occupational measures set of a controlled stochastic differential equation we establish that under certain conditions the set of occupational measures as well as the set of mathematical expectations of occupational measures generated by the admissible controls and the corresponding solutions of a controlled stochastic differential equation csde converge with the time horizon tending to infinity to a set called limit occupational measures set loms and we show that this limit set coincides with the set of stationary marginal distributions of the csde we also demonstrate the applicability of our results for averaging of singularly perturbed csde

optimal portfolio and consumption decisions for a small investor on a finite horizon a general consumption investment problem is considered for an agent whose actions cannot affect the market prices and who strives to maximize total expected discounted utility of both consumption

explicit solution of a general consumption investment problem this paper solves a general consumption and investment decision problem in closed form an investor seeks to maximize total expected discounted utility of consumption there are n distinct risky investments modelled by dependent geometric brownian motion processes and one riskless deterministic investment the analysis allows for a general utility function and general rates of return the model and analysis take into consideration the inherent nonnegativity of consumption and consider bankruptcy so this paper generalizes many of the results of lehoczky sethi and shreve lehoczky j s sethi s shreva 1983 optimal consumption and investment policies allowing consumption constraints and bankruptcy math oper res 8 613 636 the value function is determined explicitly as are the optimal consumption and investment policies the analysis is extended to consider more general risky investments under certain conditions the value functions derived for geometric brownian motion are shown to provide upper and lower bounds on the value functions in the more general context

optimal bankruptcy time and consumption investment policies on an infinite horizon with a continuous debt repayment until bankruptcy in this paper we consider the optimization problem of an agent who wants to maximize the total expected discounted utility from consumption over an infinite horizon the agent is under obligation to pay a debt at a fixed rate until he she declares bankruptcy at that point after paying a fixed cost the agent will be able to keep a certain fraction of the present wealth and the debt will be forgiven the selection of the bankruptcy time is taken to be at the discretion of the agent the novelty of this paper is that at the time of bankruptcy the wealth process has a discontinuity and that the agent continues to invest and consume after bankruptcy we show that the solution of a free boundary problem satisfying some additional conditions is the value function of the above optimization problem particular examples such as the logarithmic and the power utility functions will be provided and in these cases explicit forms will be given for the optimal bankruptcy time investment and consumption processes

optimal investment consumption and timing of annuity purchase under a preference change abstract in this paper we study the optimal investment and consumption strategies for a retired individual who has the opportunity of choosing a discretionary stopping time to purchase an annuity we assume that the individual receives a fixed annuity income and changes his her preference after paying a fixed cost for annuitization by using the martingale method and the variational inequality method we tackle this problem and obtain the optimal strategies and the value function explicitly for the case of constant force of mortality and constant relative risk aversion crra utility function

on a representation of the limit occupational measures set of a control system with applications to singularly perturbed control systems a representation of the limit occupational measures set of a control system in terms of the vector function defining the system s dynamics is established applications in averaging of singularly perturbed control systems are demonstrated

singular perturbation of a finite horizon problem with state space constraints we study the singular perturbation of optimal control problems for nonlinear systems with constraints on the fast state variables and a cost functional either of bolza type or involving the exit time of the system from a given domain under a controllability assumption on the fast variables we show that these variables become controls in the limit problem our method consists of passing to the limit in the associated hamilton jacobi bellman hjb equations by means of some tools in the theory of viscosity solutions

multiscale singularly perturbed control systems limit occupational measures sets and averaging an averaging technique for nonlinear multiscale singularly perturbed control systems is developed issues concerning the existence and structure of limit occupational measures sets generated by such systems are discussed general results are illustrated with special cases

an optimal consumption investment and voluntary retirement choice problem with disutility and subsistence consumption constraints a dynamic programming approach abstract in this paper we investigate an optimal consumption portfolio problem of an agent with voluntary retirement and subsistence consumption constraints before retirement we assume that the agent s utility function of consumption is of crra type and the agent suffers a utility loss from labor before retirement we use the dynamic programming method to obtain explicit forms of the optimal consumption portfolio and retirement time

singular perturbations of nonlinear degenerate parabolic pdes a general convergence result the main result of the paper is a general convergence theorem for the viscosity solutions of singular perturbation problems for fully nonlinear degenerate parabolic pdes partial differential equations with highly oscillating initial data it substantially generalizes some results obtained previously in 2 under the only assumptions that the hamiltonian is ergodic and stabilizing in a suitable sense the solutions are proved to converge in a relaxed sense to the solution of a limit cauchy problem with appropriate effective hamiltonian and initial data in its formulation our convergence result is analogous to the stability property of barles and perthame it should thus reveal a useful tool for studying general singular perturbation problems by viscosity solutions techniques a detailed exposition of ergodicity and stabilization is given with many examples applications to homogenization and averaging are also discussed

consistent sets and contrary inferences reply to griffiths and hartle it was pointed out recently a kent phys rev lett 78 1997 2874 that the consistent histories approach allows contrary inferences to be made from the same data these inferences correspond to projections p and q belonging to different consistent sets with the properties that pq qp 0 and p neq 1 q to many this seems undesirable in a theory of physical inferences it also raises a specific problem for the consistent histories formalism since that formalism is set up so as to eliminate contradictory inferences i e inferences p and q where p 1 q yet there seems to be no sensible physical distinction between contradictory and contrary inferences it seems particularly hard to defend the asymmetry since i there is a well defined quantum histories formalisms which admits both contradictory and contrary inferences and ii there is also a well defined formalism based on ordered consistent sets of histories which excludes both in a recent comment griffiths and hartle while accepting the validity of the examples given in the above paper restate their own preference for the consistent histories formalism as this brief reply explains in so doing they fail to address the arguments made against their approach to quantum theory

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

uniqueness of constrained viscosity solutions in hybrid control systems we study constrained viscosity solutions with an unbounded growth for a class of first order hamilton jacobi bellman equations arising in hybrid control systems to deal with the boundary constraint and rapid growth of the solutions we construct a particular set of test functions and under very mild conditions establish a comparison theorem which gives the estimate of distance between the subsolution and the supersolution the comparison theorem implies uniqueness of the constrained viscosity solution if its existence is ensured and under some additional assumptions we give an existence result by showing that the value function is a constrained viscosity solution we then apply the obtained uniqueness results to an optimal scheduling problem and finally to stochastic manufacturing systems

hamilton jacobi equations with state constraints in the present paper we consider hamilton jacobi equations of the form h x u vu 0 x e q where q is a bounded open subset of rn h is a given continuous real valued function of x s p e q x r x rn and vu is the gradient of the unknown function u we are interested in particular solutions of the above equation which are required to be supersolutions in a suitable weak sense of the same equation up to the boundary of q this requirement plays the role of a boundary condition the main motivation for this kind of solution comes from deterministic optimal control and differential games problems with constraints on the state of the system as well from related questions in constrained geodesics

singular perturbation of optimal control problems on multidomains the goal of this paper is to study a singular perturbation problem in the framework of optimal control on multi domains we consider an optimal control problem in which the controlled system contains a fast and a slow variables this problem is reformulated as an hamilton jacobi bellman hjb equation the main difficulty comes from the fact that the fast variable lives in a multi domain the geometric singularity of the multi domains leads to the discontinuity of the hamiltonian under a controllability assumption on the fast variables the limit equation as the velocity of the fast variable goes to infinity is obtained via a pde approache and by means of the tools of the control theory

comparison of optimal portfolios with and without subsistence consumption constraints abstract we present the effects of the subsistence consumption constraints on a portfolio selection problem for an agent who is free to choose when to retire with a constant relative risk aversion crra utility function by comparing the previous studies with and without the constraints expressed by the minimum consumption requirement the changes of a retirement wealth level and the amount of money invested in the risky asset are derived explicitly as a result the subsistence constraints always lead to lower retirement wealth level but do not always induce less investment in the risky asset this implies that even though the agent who has a restriction on consumption retires with lower wealth level she invests more money near the retirement when her risk aversion lies inside a certain range

viscosity solutions methods for singular perturbations in deterministic and stochastic control viscosity solutions methods are used to pass to the limit in some penalization problems for first order and second order degenerate parabolic hamilton jacobi bellman equations this characterizes the limit of the value functions of singularly perturbed optimal control problems for deterministic systems and for controlled degenerate diffusions the results apply to cases where the usual order reduction method does not give the correct limit and to systems with fast state variables depending nonlinearly on the control some connections with ergodic control and periodic homogenization are discussed

linearly positive histories probabilities for a robust family of sequences of quantum events non negative probabilities that obey the sum rules may be assigned to a much wider family of sets of histories than decohering histories the resulting linearly positive histories avoid the highly restrictive decoherence conditions and yet give the same probabilities when those conditions apply thus linearly positive histories are a broad extension of decohering histories moreover the resulting theory is manifestly time reversal invariant

optimal consumption and portfolio selection problem with downside consumption constraints abstract we study a general optimal consumption and portfolio selection problem of an infinitely lived investor whose consumption rate process is subjected to downside constraint that is her consumption rate is greater than or equals to some positive constant we obtain the general optimal policies in an explicit form using martingale method and feynman kac formula we derive some numerical results of optimal consumption and portfolio in the special case of a constant relative risk aversion crra utility function

singular perturbations in manufacturing an asymptotic analysis for a large class of stochastic optimization problems arising in manufacturing is presented a typical example of the problems considered in this paper is a production planning problem with random capacity and demand in this example it is assumed that the capacity of the system fluctuates faster than the other quantities the general model considered here also has a fast controlled markov process in its state description by using the difference in the time scales of different quantities the problem is simplified by averaging out the fast process then asymptotically optimal strategies are constructed from the optimal solutions of the limiting problems the proofs of these results use the theory of viscosity solutions to dynamic programming equations however the formal construction of the asymptotically optimal strategies does not require knowledge of this theory

stochastic portfolio optimization with default risk abstract a stochastic portfolio optimization problem with default risk on an infinite time horizon is investigated the default risk premium and the default intensity corresponding to the defaultable bond are assumed to rely on a stochastic factor formulated by a diffusion process we study the optimal allocation and consumption policies to maximize the infinite horizon expected discounted non log hara utility of the consumption and we use the dynamic programming principle to derive the hamilton jacobi bellman hjb equation then we explore the hjb equation by employing a so called sub super solution approach the optimal allocation and consumption policies are finally presented in a verification theorem and also a numerical simulation is given at the end of the paper

decoherent histories and realism we reconsider the decoherent histories approach to quantum mechanics and analyze some problems related to its interpretation which we believe have not been adequately clarified by its proponents we put forward some assumptions which in our opinion are necessary for a realistic interpretation of the probabilities that the formalism attaches to decoherent histories we prove that such assumptions unless one limits the set of the decoherent families which can be taken into account lead to a logical contradiction the line of reasoning we follow is conceptually different from other arguments which have been presented and which have been rejected by the supporters of the decoherent histories approach the conclusion is that the decoherent histories approach to be considered as an interesting realistic alternative to the orthodox interpretation of quantum mechanics requires the identification of a mathematically precise criterion to characterize an appropriate set of decoherent families which does not give rise to any problem

optimal control with state space constraint i optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain finally the uniform continuity of the value function is obtained under a condition on the deterministic drift

optimal portfolio consumption and retirement decision under a preference change we investigate an optimal portfolio consumption and retirement decision problem in which an economic agent can determine the discretionary stopping time as a retirement time with constant labor wage and disutility we allow the preference of the agent to be changed before and after retirement it is assumed that the agent s coefficient of relative risk aversion becomes higher after retirement under a constant relative risk aversion crra utility function we obtain the optimal policies in closed forms using martingale methods and variational inequality methods we give some numerical results of the optimal policies we also consider the relation between the level of disutility and the labor wage with the optimal retirement wealth level

optimal investment consumption and retirement choice problem with disutility and subsistence consumption constraints in this paper we consider a general optimal consumption portfolio selection problem of an infinitely lived agent whose consumption rate process is subject to subsistence constraints before retirement that is her consumption rate should be greater than or equal to some positive constant before retirement we integrate three optimal decisions which are the optimal consumption the optimal investment choice and the optimal stopping problem in which the agent chooses her retirement time in one model we obtain the explicit forms of optimal policies using a martingale method and a variational inequality arising from the dual function of the optimal stopping problem we treat the optimal retirement time as the first hitting time when her wealth exceeds a certain wealth level which will be determined by a free boundary value problem and duality approaches we also derive closed forms of the optimal wealth processes before and after retirement some numerical examples are presented for the case of constant relative risk aversion crra utility class

applications of singular perturbation techniques to control problems this paper discusses typical applications of singular perturbation techniques to control problems in the last fifteen years the first three sections are devoted to the standard model and its time scale stability and controllability properties the next two sections deal with linear quadratic optimal control and one with cheap near singular control then the composite control and trajectory optimization are considered in two sections and stochastic control in one section the last section returns to the problem of modeling this time in the context of large scale systems the bibliography contains more than 250 titles

exit time problems in optimal control and vanishing viscosity method the authors study the connections between deterministic exit time control problems and possibly discontinuous viscosity solutions of a first order hamilton jacobi hj equation up to the boundary this equation admits a maximum and a minimum solution that are the value functions associated to stopping time problems on the boundary when these solutions are equal they can be obtained through the vanishing viscosity method finally when the hj equation has a continuous solution it is proved to be the value function for the first exit time of the domain it is also the vanishing viscosity limit arising in particular in some large deviations problems

logical reformulation of quantum mechanics i foundations the basic rules of quantum mechanics are reformulated they deal primarily with individual systems and do not assume that every ket may represent a physical state the customary kinematic and dynamic rules then allow to construct consistent boolean logics describing the history of a system following essentially griffiths proposal logical implication is defined within these logics the multiplicity of which reflects the complementary principle only one interpretative rule of quantum mechanics is necessary in such a framework it states that these logics providebona fide foundations for the description of a quantum system and for reasoning about it one attempts to build up classical physics including classical logic on these quantum foundations the resulting theory of measurement needs not to statea priori that the eigenvalues of an observable have to be the results of individual measurements nor to assume wave packet reduction both these properties can be obtained as consequences of the basic rules one also needs not to postulate that every observable is measurable even in principle a proposition calculus is obtained allowing in principle the replacement of the discussion of problems concerned with the practical interpretation of experiments by due calculations

consistent histories and the interpretation of quantum mechanics the usual formula for transition probabilities in nonrelativistic quantum mechanics is generalized to yield conditional probabilities for selected sequences of events at several different times called consistent histories through a criterion which ensures that within limits which are explicitly defined within the formalism classical rules for probabilities are satisfied the interpretive scheme which results is applicable to closed isolated quantum systems is explicitly independent of the sense of time i e past and future can be interchanged has no need for wave function collapse makes no reference to processes of measurement though it can be used to analyze such processes and can be applied to sequences of microscopic or macroscopic events or both as long as the mathematical condition of consistency is satisfied when applied to appropriate macroscopic events it appears to yield the same answers as other interpretative schemes for standard quantum mechanics though from a different point of view which avoids the conceptual difficulties which are sometimes thought to require reference to conscious observers or classical apparatus

consistent quantum realism a reply to griffiths we call attention on the fact that we have already replied to the criticisms put forward by r b griffiths to our position about decoherent histories

periodic near optimal control we consider the infinite time optimal control problem of minimizing an average cost functional for nonlinear singularly perturbed systems in particular we are interested in the existence of near optimal trajectories which are periodic for this purpose we investigate nonlinear control systems with three time scales and construct a limiting system for the slowest motion by an iterated averaging procedure we furthermore show that global tools invariant control sets and local tools lie algebra ranks of vector fields are preserved under singular perturbations the existence of periodic near optimal trajectories is verified under the condition that the fast subsystem and an unperturbed slow subsystem are controllable and that a standard lie algebra rank condition for local accessibility holds

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

complete description of a quantum system at a given time a generalization of the description of a quantum system in the time interval between two measurements is presented a new concept of a generalized state is introduced generalized states yield a complete description of a quantum system when information about the system is available both from the past snd from the future the formalism of generalized states provides a natural language for describing many peculiar situations in particular situations in which one can ascertain the result of a measurement of any one of several non commuting variables are analysed weak measurements on quantum systems described by generalized states are discussed the relation belween weak and strong measurements is investigated