A stochastic optimal control approach to a class of production and inventory problems

A comprehensible and unified system control approach is presented to solve a class of production/inventory smoothing problems. A nonstationary, non-Gaussian, finite-time linear optimal solution with an attractive computation scheme is obtained for a general quadratic and linear cost structure. A complete solution to a classical production/inventory control problem is given as an example. A general solution to the discrete-time optimal regulator with arbitrary but known disturbance is provided and discussed in detail. A computationally attractive closed-loop suboptimal scheme is presented for problems with constraints or nonquadratic costs. Implementation and interpretation of the results are discussed.     

A martingale method for the convergence of a sequence of processes to a jump-diffusion process

Summary A convenient method for proving weak convergence of a sequence of non-Markovian processesx (·) to a jump-diffusion process is proved. Basically, it is shown that the limit solves the martingale problem of Strook and Varadhan. The proofs are relatively simple, and the conditions apparently weaker than required by other current methods (in particular, for limit theorems for a sequence of ordinary differential equations with random right hand sides). In order to illustrate the relative ease of applicability in many cases, a simpler proof of a known result on averaging is given.Brown University, Divisions of Applied Mathematics and Engineering and Lefschetz Center for Dynamical Systems. Research supported in part by the Air Force Office of Scientific Research under AFOSR AF-76-3063, by the National Science Foundation under NSF Eng. 73-03846A03 and in part by the Office of Naval Research ONR N00014-76-C-0279 P003     

A new approach to optimum state estimation in optimum system control

We consider optimum system control, in which the controller has imperfect information on the availability of the state of the plant to control. The system is controlled via an estimation of the state. We propose a general estimation criterion, which should permit determination of the optimal estimator for a given dynamics and a given performance cost. This approach includes the separation theorem as a special case.     

On a Lie-theoretic approach to generalized doubly stochastic matrices and applications

In this article, we study generalized doubly stochastic matrices using the theory of Lie groups and Lie algebras. Applications to the inverse eigenvalue problem for symmetric doubly stochastic matrices are presented.     

On a Lie-theoretic approach to generalized doubly stochastic matrices and applications

In this article, we study generalized doubly stochastic matrices using the theory of Lie groups and Lie algebras. Applications to the inverse eigenvalue problem for symmetric doubly stochastic matrices are presented.     

Fractional programming approach to two stochastic inventory problems

This paper considers two popular inventory models: the continuous review and periodic review reorder-point, order-quantity, control systems. Specifically we present two procedures which determine optimal values for the two control parameters (i.e., reorder-point and order-quantity) when the holding-and-shortage costs are non-quasi-convex. This cost structure may arise when non-linear cost rate is considered, for instance when the shortage cost is the shadow cost of a service-level constraint. The algorithms based on a fractional programming method are intuitive and efficient, and as the holding-and-shortage cost functions become quasi-convex, they are compatible to existing algorithms.     

A martingale approach to scan statistics

Scan statistics are commonly used in biology, medicine, engineering and other fields where interest is in the probability of observing clusters of events in a window at an unknown location. Due to the dependent nature of the number of events in a large number of overlapping window locations, even approximate solutions for the simplest scan statistics may require elaborate calculations. We propose a new martingale method which allows one to approximate the distribution for a wide variety of scan statistics, including some for which analytical results are computationally infeasible.     

A martingale approach to minimal surfaces

We provide a probabilistic approach to studying minimal surfaces in R³. After a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used to study two classes of results in minimal surface theory, maximum principle-type results, such as weak and strong halfspace theorems and the maximum principle at infinity, and Liouville theorems.     

A martingale approach to directed polymers in a random environment

The diffusive behavior for a system of directed polymers in a random environment was first rigorously discussed by Imbrie and Spencer, and then by Bolthausen. By means of some basic properties of martingales we extend some results due to Imbrie and Spencer concerning the asymptotic behaviour of the mean square displacement. We also obtain a Wiener process behaviour with probability one for this system. Bolthausen already used some martingale limit theorems to prove a central limit theorem for this system.Partly supported by AvH Foundation.     

A stochastic control approach to reciprocal diffusion processes

The problem of forcing a nondegenerate diffusion process to a given final configuration is considered. Using the logarithmic transformation approach developed by Fleming, it is shown that the perturbation of the drift suggested by Jamison solves an optimal stochastic control problem. Such perturbation happens to have minimum energy between all controls that bring the diffusion to the desired final distribution. A special property of the change of measure on the path-space that corresponds to the aforesaid perturbation of the drift is also shown.     

A Lagrangian relaxation approach for network inventory control of stochastic revenue management with perishable commodities

Airline seat inventory control is the allocation of seats in the same cabin to different fare classes such that the total revenue is maximized. Seat allocation can be modelled as dynamic stochastic programs, which are computationally intractable in network settings. Deterministic and probabilistic mathematical programming models are therefore used to approximate dynamic stochastic programs. The probabilistic model, which is the focus of this paper, has a nonlinear objective function, which makes the solution of large-scale practical instances with off-the-shelf solvers prohibitively time consuming. In this paper, we propose a Lagrangian relaxation (LR) method for solving the probabilistic model by exploring the fact that LR problems are decomposable. We show that the solutions of the LR problems admit a simple analytical expression which can be resolved directly. Both the booking limit policy and the bid-price policy can be implemented using this method. Numerical simulations demonstrate the effectiveness of the proposed method.     

Asymptotic distribution of the nonparametric distribution estimator based on a martingale approach in doubly censored data

For analysis of time-to-event data with incomplete information beyond right-censoring, many generalizations of the inference of the distribution and regression model have been proposed. However, the development of martingale approaches in this area has not progressed greatly, while for right-censored data such an approach has spread widely to study the asymptotic properties of estimators and to derive regression diagnosis methods. In this paper, focusing on doubly censored data, we discuss a martingale approach for inference of the nonparametric maximum likelihood estimator (NPMLE). We formulate a martingale structure of the NPMLE using a score function of the semiparametric profile likelihood. Finally, an expression of the asymptotic distribution of the NPMLE is derived more conveniently without depending on an infinite matrix expression as in previous research. A further useful point is that a variance-covariance formula of the NPMLE computable in a larger sample is obtained as an empirical version of the limit form presented here.     

A stochastic approach to the flood control problem

We propose a stochastic model in conjunction with reliability analysis concepts to improve estimates for the protection volume that should be allocated in a reservoir to control a flood wave. In this approach, the inflow that reaches the reservoir during a flood is considered to be a load, and the reservoir capacity to control this flood is considered to be the resistance that the reservoir offers against the propagation of the flood. Here, the load and the resistance are modeled as a diffusion stochastic process, and the protection volume is determined via Itô's formula. In this scenario, an explicit formula for the failure risk is derived. The parameter inference is carried out by a Bayesian approach for a time discrete version of the load, and the estimates are obtained by using Monte Carlo Markov Chain Algorithms (MCMC). The maximum likelihood estimators are used in the comparison. The record utilized comprises nine years of daily inflow rates during flood periods that come to the Chavantes hydroelectric power plant (CHPP) in Southeast Brazil. The protection volumes estimated through the proposed model are compared to the volumes obtained by other existing methods.     

A market-based computational approach to collaborative organizational learning

We draw upon the concepts of knowledge market, organizational tacit knowledge, credit assignment, and single-loop learning in proposing a market-based conceptual model for collaborative organizational learning. Our proposed model is characterized by the local competition among seller agents and the global collaboration among winner agents in forming a plan, through a chain of ‘upstream–downstream’ working relationship, for task accomplishment. This feature is achieved through three closely coupled processes: the expert selection process, the capital reallocation process, and the plan formation process. Our model is intended for multiple-step learning environment in which each task consists of a sequence of single-step learning tasks. Learning at the global level is the result of a sequence of nested single-loop learning at the local level.     

A geometric programming approach to the Van der Waerden conjecture on doubly stochastic matrices

LetD _p be the set of all doubly stochastic square matrices of orderp i.e. the set of allp × p matrices with non-negative entries with row and column sums equal to unity. The permanent of ap × p matrixA = (a _ij ) is defined byP(A) = _Sp II _i=1 ^p a _i(i) whereS _p is the symmetric group of orderp. Van der Waerden conjectured thatP(A) p !/p ^p for all A AD _p with equality occurring if and only ifA = J _p , whereJ _p is the matrix all of whose entries are equal to 1/p.The validity of this conjecture has been shown for a few values ofp and for generalp under certain assumptions. In this paper the problem of finding the minimum of the permanent of a doubly stochastic matrix has been formulated as a reversed geometric program with a single constraint and an equivalent dual program is given. A related problem of reversed homogeneous posynomial programming problem is also studied.     

A martingale approach to state estimation in delay-differential systems

A rigorous derivation of filtering arid smoothing equations for linear stochastic systems with time delay is presented. The estimation equations are obtained in term of the innovation process of the problem under consideration. The method used is based on a representation theorem on Gaussian martingales.     

A Bayesian approach to equity valuation

Quantitative approaches to equity valuation in common use includemethods based upon the dividend discount model, which equatesthe value of an enterprise to the present value of the dividendsthat it pays to its owners, and methods which begin from therelation between the current share price and the expected earningsfor the next 12 months. A short coming of each of these approachesis that they are forced to take near-term earnings and dividendforecasts as their basic data, when in fact the values theyare trying to estimate are dominated by longer-term considerations.As a result, model estimates are often too sensitive to changesin near-term earnings which have little effect on a company'slong-term prospects. In relatively efficient markets, however, the price of a company'sshares at any time reflects information which is not readilyquantifiable and which may represent a longer-term view. Thatinformation can be used to create a stable quantitative valuationmodel in which the information content of near-term changesis balanced against longer-term considerations via a Bayesianupdating procedure. The structure of the Bayesian valuation model is essentiallythat of a truncated dividend discount model, where the fairvalue today is expressed as the present value of a terminaldividend (i.e. the fair value at a future date, discounted tothe present), plus the present value of the interim dividends.In the Bayesian model, the terminal dividend is a variable whoseprobability distribution depends upon the sequence of earningsbetween the valua tion date and the terminal date.     

A generalization of doubly stochastic matrices over a field

Let X and Y be m×n matrices over a field F such that Y^TX is nonsingular, and let Λ and Λ′ be sets of n-square matrices over F. Solutions A to the simultaneous equations AX = XK and $\text{Y}{}^{\mathrm{T}}\text{A =}\text{K}\text{?}\text{Y}{}^{\mathrm{T}}$ where K?Λ and $\text{K}\text{?}\text{? Λ′}$ are considered. It is shown that many properties of doubly stochastic matrices over a field have a natural generalization in terms of the set Δ(Λ,Λ′) of all such solutions.