Singular control problems in bounded intervals

We consider optimal control problems for one-dimensional diffusion processes y _x (t) = y ^v _x (t), solutions dy _x (t) = g(y _x (t) dt + σ(y _x (t)(dw) + dv _t with y _x(0) = x& isinv;[a,b], the control processes v _t are increasing, positive, and adapted. Several types of expected cost structures associated with each policy v(.) are adopted, e.g. discounted cost, long term average cost and time average cost. Our work is related to [2,6,12,14,16 and 21], where diffusions are allowed to evolve in the whole space, and to [13] and [20], where diffusions evolve only in bounded regions. We shall present some analytic results about value functions, mainly their characterizations, by simple dynamic programming arguments. Several simple examples are explicitly solved to illustrate the singular behaviour of our problems     

Asymptotic expansion of stochastic flows

Summary We study the asymptotic expansion in small time of the solution of a stochastic differential equation. We obtain a universal and explicit formula in terms of Lie brackets and iterated stochastic Stratonovich integrals. This formula contains the results of Doss [6], Sussmann [15], Fliess and Normand-Cyrot [7], Krener and Lobry [10], Yamato [17] and Kunita [11] in the nilpotent case, and extends to general diffusions the representation given by Ben Arous [3] for invariant diffusions on a Lie group. The main tool is an asymptotic expansion for deterministic ordinary differential equations, given by Strichartz [14].     

Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability

In this paper we study the existence of the optimal (minimizing) control for a tracking problem, as well as a quadratic cost problem subject to linear stochastic evolution equations with unbounded coefficients in the drift. The backward differential Riccati equation (BDRE) associated with these problems (see [2], for finite dimensional stochastic equations or [21], for infinite dimensional equations with bounded coefficients) is in general different from the conventional BDRE (see [10], [18]). Under stabilizability and uniform observability conditions and assuming that the control weight-costs are uniformly positive, we establish that BDRE has a unique, uniformly positive, bounded on ℝ ₊ and stabilizing solution. Using this result we find the optimal control and the optimal cost. It is known [18] that uniform observability does not imply detectability and consequently our results are different from those obtained under detectability conditions (see [10]).      

Convergence to diffusions with regular boundaries

Earlier results on weak convergence to diffusion processes [8] are generalized to cases where the limiting diffusions may have regular boundaries. The boundaries may be adhesive or reflecting, and in each case we give two different sets of conditions for convergence. It is shown that these conditions are necessary and sufficient for convergence in the same sense as the conditions in [8]. We also extend our results to cases where the coefficients of the diffusions have simple discontinuities, in particular we thereby answer an open question by Keilson and Wellner [9]. Finally we formulate alternative sets of conditions for convergence, with these new sets being more convenient for instance when the sequence under investigation consists of pure jump Markov processes in continuous time.     

A maximum a posteriori estimator for trajectories of diffusion processes

Let x _t be a diffusion process observed via a noisy sensor, whose output is y_t We consider the problem of evaluating the maximum a posteriori trajectory {x_s0≤ s ≤ t Based on results of Stratonovich [1] and Ikeda-Watanabe [2], we show that this estimator is given by the solution of an appropriate variational problem which is a slight modification of the "minimum energy" estimator. We compare our results to the non-linear filtering theory and show that for problems which possess a finite dimensional solution, our approach yields also explicit filters. For linear diffusions observed via linear sensors, these filters are identical to the Kalman-filter     

Representations of general stochastic processes

In this paper we carry on our study [4] of the algebraic representations of general stochastic processes. We give methods for constructing the algebraic representation of a stochastic process from the distribution of the process at a fixed finite number of times, we develope some techniques of integration, and we introduce the notion of a fibre bundle representation of a stochastic process. We then use this fibre bundle representation to study existence, methods of computation and the geometry of Markov process representations of the general stochastic process; thus extending [4] where existence was only discussed for discrete time or simple stochastic processes.     

Mayer and optimal stopping stochastic control problems with discontinuous cost

We study two classes of stochastic control problems with semicontinuous cost: the Mayer problem and optimal stopping for controlled diffusions. The value functions are introduced via linear optimization problems on appropriate sets of probability measures. These sets of constraints are described deterministically with respect to the coefficient functions. Both the lower and upper semicontinuous cases are considered. The value function is shown to be a generalized viscosity solution of the associated HJB system, respectively, of some variational inequality. Dual formulations are given, as well as the relations between the primal and dual value functions. Under classical convexity assumptions, we prove the equivalence between the linearized Mayer problem and the standard weak control formulation. Counter-examples are given for the general framework.     

Adaptive control of stochastic systems with unknown disturbance distribution: discounted criteria

We consider a class of discrete-time stochastic control systems, with Borel state and action spaces, and possibly unbounded costs. The processes evolve according to the equation x _{t +1}=F(x _t , a _t , ξ _t ), t=0, 1, ..., where the ξ _t are i.i.d. random vectors whose common distribution is unknown. Assuming observability of {ξ _t }, we use the empirical estimator of its distribution to construct adaptive policies which are asymptotically discounted cost optimal .AMS Subject Classification (2000) 93E10, 90C40     

An invariance principle for non-symmetric Markov processes and reflecting diffusions in random domains

Summary We study an invariance principle for additive functionals of nonsymmetric Markov processes with singular mean forward velocities. We generalize results of Kipnis and Varadhan [KV] and De Masi et al. [De] in two directions: Markov processes are non-symmetric, and mean forward velocities are distributions. We study continuous time Markov processes. We use our result to homogenize non-symmetric reflecting diffusions in random domains.     

A primer on perfect simulation for spatial point processes

This primer provides a self-contained exposition of the case where spatial birth-and-death processes are used for perfect simulation of locally stable point processes. Particularly, a simple dominating coupling from the past (CFTP) algorithm and the CFTP algorithms introduced in [13], [14], and [5] are studied. Some empirical results for the algorithms are discussed. Received: 30 June 2002     

Mean stochastic comparison of diffusions

Summary Stochastic bounds are derived for one dimensional diffusions (and somewhat more general random processes) by dominating one process pathwise by a convex combination of other processes. The method permits comparison of diffusions with different diffusion coefficients. One interpretation of the bounds is that an optimal control is identified for certain diffusions with controlled drift and diffusion coefficients, when the reward function is convex. An example is given to show how the bounds and the Liapunov function technique can be applied to yield bounds for multidimensional diffusions.This work was supported by the Office of Naval Research under Contract N00014-82-K-0359 and the U.S. Army Research Office under Contract DAAG29-82-K-0091 (administered through the University of California at Berkeley).     

Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise

We consider the controlled stochastic Navier–Stokes equations in a bounded multidimensional domain, where the noise term allows jumps. In order to prove existence and uniqueness of an optimal control w.r.t. a given control problem, we first need to show the existence and uniqueness of a local mild solution of the considered controlled stochastic Navier–Stokes equations. We then discuss the control problem, where the related cost functional includes stopping times dependent on controls. Based on the continuity of the cost functional, we can apply existence and uniqueness results provided in [4], which enables us to show that a unique optimal control exists.     

Estimation of spectral gap for Markov chains

The study of the convergent rate (spectral gap) in theL ²-sense is motivated from several different fields: probability, statistics, mathematical physics, computer science and so on and it is now an active research topic. Based on a new approach (the coupling technique) introduced in [7] for the estimate of the convergent rate and as a continuation of [4], [5], [7–9], [23] and [24], this paper studies the estimate of the rate for time-continuous Markov chains. Two variational formulas for the rate are presented here for the first time for birth-death processes. For diffusions, similar results are presented in an accompany paper [10]. The new formulas enable us to recover or improve the main known results. The connection between the sharp estimate and the corresponding eigenfunction is explored and illustrated by various examples. A previous result on optimal Markovian couplings^[4] is also extended in the paper.Research supported in part by NSFC, Qin Shi Sci & Tech. Foundation and the State Education Commission of China.     

Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions

The classical existence-and-uniqueness theorem of the solution to a stochastic differential delay equation (SDDE) requires the local Lipschitz condition and the linear growth condition (see e.g. [11], [12] and [20]). The numerical solutions under these conditions have also been discussed intensively (see e.g. [4], [10], [13], [16], [17], [18], [21], [22] and [24]). Recently, Mao and Rassias [14] and [15] established the generalized Khasminskii-type existence-and-uniqueness theorems for SDDEs, where the linear growth condition is no longer imposed. These generalized Khasminskii-type theorems cover a wide class of highly nonlinear SDDEs but these nonlinear SDDEs do not have explicit solutions, whence numerical solutions are required in practice. However, there is so far little numerical theory on SDDEs under these generalized Khasminskii-type conditions. The key aim of this paper is to close this gap.     

Handling forecasting problems using fuzzy time series

In [6–9], Song et al. proposed fuzzy time-series models to deal with forecasting problems. In [10], Sullivan and Woodall reviewed the first-order time-invariant fuzzy time series model and the first-order time-variant model proposed by Song and Chissom [6–8], where the models are compared with each other and with a time-invariant Markov model using linguistic labels with probability distributions. In this paper, we propose a new method to forecast university enrollments, where the historical enrollments of the University of Alabama shown in [7,8] are used to illustrate the forecasting process. The average forecasting errors and the time complexity of these methods are compared. The proposed method is more efficient than the ones presented in [7, 8, 10] due to the fact that the proposed method simplifies the arithmetic operation process. Furthermore, the average forecasting error of the proposed method is smaller than the ones presented in [2, 7, 8].     

Adaptive policies for time-varying stochastic systems under discounted criterion

We consider a class of time-varying stochastic control systems, with Borel state and action spaces, and possibly unbounded costs. The processes evolve according to a discrete-time equation x _{n + 1}=G _n (x _n , a _n , ξ_n), n=0, 1, … , where the ξ_n are i.i.d. ℜ^k-valued random vectors whose common density is unknown, and the G _n are given functions converging, in a restricted way, to some function G _∞ as n→∞. Assuming observability of ξ_n, we construct an adaptive policy which is asymptotically discounted cost optimal for the limiting control system x _n+1=G _∞ (x _n , a _n , ξ_n).     

Construction of steiner quadruple systems having a prescribed number of blocks in common

Let q_υ=υ(υ–1)(υ–2)/24 and let I_υ={0, 1, 2, …, q_{υ–14}∪{qυ}–12, q_υ–8, q_υ}, for υ?8 Further, let J[υ] denote the set of all k such that there exists a pair of Steiner quadruple systems of order υ having exactly k blocks in common. We determine J[υ] for all υ=2ⁿ, n?2, with the possible exception of 7 cases for υ=16 and of 5 cases for each υ?32. In particular we show: J[υ]?I_υ for all υ≡2 or 4 (mod 6) and υ?8, J[4]={1}, J[8]=I₈={0, 2, 6, 14}, I₁₆?{103, 111, 115, 119, 121, 122, 123}?J[16], and I_υ? {q_υ–h:h=17, 18, 19, 21, 25}?J[υ] for all υ=2ⁿ, n?5.     

Stationary min-stable stochastic processes

Summary We consider the class of stationary stochastic processes whose margins are jointly min-stable. We show how the scalar elements can be generated by a single realization of a standard homogeneous Poisson process on the upper half-strip [0,1]×R ₊ and a group of L ₁-isometries. We include a Dobrushin-like result for the realizations in continuous time.     

ON PRIME RINGS AND THE RADICALS ASSOCIATED WITH THEIR DEGREES OF PRIMENESS

Abstract

Prime ringsMaybe classified by the sizes of the sets that ‘insulate’ their elements from annihilation. For a cardinal m > 0, the class [Pbar]_r,(m) of all rings that are right prime of ‘bound at most m’ is studied, with particular reference to its closure under constructions such as matrix rings, semigoup rings, orders and extensions. The classes [Pbar]_r,(m) are special in the sense of radical theory for each m > 0. The attendant upper radicals υ[Pbar]_r,(m) are right (and not left) strong; their compatibility with certain ring constructions is examined. In the lattice of radicals (where they form a strictly descending chain), their positions are described, relative to various familiar radicals.     

一类具有扩散的奇异型随机控制的非对称平稳模型

该文讨论了一类奇异型随机控制的平稳模型,其费用结构中的函数不限于偶函数,其状态过程为扩散型且具有“非对称的”(关于原点)漂移及扩散系数.因此,奇异型随机控制中的平稳问题被实质性地推广到更一般的形式。该文求得了与此类问题有关的一个变分方程组的解,并且证明了最佳控制的存在性.