A note on convex ordering for stable stochastic integrals

We establish a convex ordering between stochastic integrals driven by strictly α-stable processes with index α ∈ (1,2). Our approach is based on the forward–backward stochastic calculus for martingales together with a suitable decomposition of stable stochastic integrals.     

A test for the bidirectional stochastic order with an application to quality control theory

A test for the bidirectional stochastic ordering is developed in this paper. The main properties of such a test are investigated. The asymptotic distribution of the statistic of the test is obtained under conditions which allow the construction of critical regions with a specific level of significance. It is also proved that the test is consistent on the whole set of alternatives. An application of such a test to quality control theory is developed.     

A concept of negative dependence using stochastic ordering

A concept of negative dependence called negative dependence by stochastic ordering is introduced. This concept satisfies various closure properties. It is shown that three models for negetive dependence satisfy it and that it implies the basic negative orthant inequalities. This concept is also satisfied by the multinomial, multivariate hypergeometric. Dirichlet and Dirichlet compound multinomial distributions. Furthermore, the joint distribution of ranks of a sample and the multivariate normal with nonpositive pairwise correlations also satisfy this condition. The positive dependence analog of this condition is also studied.     

The continuity of the optimum in parametric programming and applications to stochastic programming

It is proved a sufficient condition that the optimal value of a linear program be a continuous function of the coefficients. The condition isessential, in the sense that, if it is not imposed, then examples with discontinuous optimal-value function may be found. It is shown that certain classes of linear programs important in applications satisfy this condition. Using the relation between parametric linear programming and the distribution problem in stochastic programming, a necessary and sufficient condition is given that such a program has optimal value. Stable stochastic linear programs are introduced, and a sufficient condition of such stability, important in computation problems, is established.This note is a slightly modified version of a paper presented at the Institute of Econometrics and Operations Research of the University of Bonn, Bonn, Germany, 1972.The author is grateful to G. B. Dantzig and S. Karamardian for useful comments on an earlier draft of this paper. In particular, S. Karamardian proposed modifications which made clearer the proof of Lemma 2.1.     

Ruin problems with stochastic premium stochastic return on investments

In this paper, ruin problems in the risk model with stochastic premium incomes and stochastic return on investments are studied. The logarithm of the asset price process is assumed to be a Lévy process. An exact expression for expected discounted penalty function is established. Lower bounds and two kinds of upper bounds for expected discounted penalty function are obtained by inductive method and martingale approach. Integro-differential equations for the expected discounted penalty function are obtained when the Lévy process is a Brownian motion with positive drift and a compound Poisson process, respectively. Some analytical examples and numerical examples are given to illustrate the upper bounds and the applications of the integro-differential equations in this paper.      

Monotonicity and bounds for convex stochastic control models

We consider a general convex stochastic control model. Our main interest concerns monotonicity results and bounds for the value functions and for optimal policies. In particular, we show how the value functions depend on the transition kernels and we present conditions for a lower bound of an optimal policy. Our approach is based on convex stochastic orderings of probability measures. We derive several interesting sufficient conditions of these ordering concepts, where we make also use of the Blackwell ordering. The structural results are illustrated by partially observed control models and Bayesian information models.     

On extremal solutions to stochastic control problems

We identify two solutions of a controlled diffusion if the corresponding one-dimensional marginals of the state and control process agree. The extreme points of the set of such equivalence classes are shown to correspond to Markov controls.     

A note on quasi-equilibrium problems

The purpose of this paper is to prove the existence of solutions of quasi-equilibrium problems without any generalized monotonicity assumption. Additionally, we give some applications.     

Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems

This paper is concerned with Kalman-Bucy filtering problems of a forward and backward stochastic system which is a Hamiltonian system arising from a stochastic optimal control problem. There are two main contributions worthy pointing out. One is that we obtain the Kalman-Bucy filtering equation of a forward and backward stochastic system and study a kind of stability of the aforementioned filtering equation. The other is that we develop a backward separation technique, which is different to Wonham's separation theorem, to study a partially observed recursive optimal control problem. This new technique can also cover some more general situation such as a partially observed linear quadratic non-zero sum differential game problem is solved by it. We also give a simple formula to estimate the information value which is the difference of the optimal cost functionals between the partial and the full observable information cases.     

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.     

具有非线性采购成本库存控制问题的研究现状与挑战

库存管理是基于运筹学而发展起来的一门学科,并成为近几十年来运筹学和管理科学重要的研究领域之一。在库存系统中,采购成本是必不可少的成本之一,主要包含产品成本、运输成本、装卸成本等。现实中,采购成本依赖于采购量,且往往是采购量的非线性函数。介绍了几类常见的采购成本函数：依赖于采购量的固定成本、增量折扣、全量折扣、车载容量折扣和凸采购成本等。基于周期盘点库存模型和连续盘点库存模型,综述了带有这些非线性采购成本函数的库存模型研究进展。虽然经过了几十年的研究,但很多带有非线性采购成本的库存模型的最优采购策略因为其复杂性至今未能被完整刻画。通过综述来简单讨论该类问题的挑战和机会。     

About an unsolved stability problem for a stochastic difference equation with continuous time

An unsolved problem of stability for stochastic difference equation with continuous time is proposed for consideration.     

Conditions for the existence of decision horizons for discounted problems in a stochastic environment: A note

In this note, we extend the sufficiency conditions obtained by Bean and Smith for the existence of decision and forecast horizons in discounted deterministic problems to a stochastic environment. Also developed are some examples for illustration of the results and the need for the development of necessary and sufficient conditions for undiscounted problems.     

Uniform estimate for maximum of randomly weighted sums with applications to insurance risk theory

This paper obtains the uniform estimate for maximum of sums of independent and heavy-tailed random variables with nonnegative random weights,which can be arbi- trarily dependent of each other.Then the applications to ruin probabilities in a discrete time risk model with dependent stochastic returns are considered.     

A note on deterministic equivalents to stochastic linear programming problems

Summary Generalized inverse matrices are used as a tool for a study of two-stage linear program under uncertainty. For a special choice of M which represents an extension of the so-called complete problem, a deterministic equivalent is given in the explicite form.     

On extremal solutions to stochastic control problems,II

The set of attainable laws of the joint state-control process of a controlled diffusion is analyzed from a convex analytic viewpoint. Various equivalence relations depending on one-dimensional marginals thereof are defined on this set and the corresponding equivalence classes are studied.