带小随机扰动的中偏差原理(英文)

本文研究了带小随机扰动的中偏差原理.运用收缩原理和指数逼近方法,Freidlin-Wentzell定理给出了Xε的大偏差原理,从而得到了Xε的中偏差原理.     

Déviations modérées pour la moyennisation d'une EDS avec petite diffusion

We prove in this Note the moderate deviation principle (MDP) for the averaging principle of a stochastic differential equation (SDE) in a fast random environment, modelized by an exponentially ergodic Markov process independent of the Wiener process driving the SDE. The main tools will be the method of Puhalskii for exponential tightness and a MDP for inhomogeneous functionals of Markov processes established in [5].     

Large deviations for the 3D stochastic Navier-Stokes-Voight equations

In this paper, using the weak convergence method, a large deviation principle for 3D stochastic Navier–Stokes–Voight equations is proved.     

带移民超布朗运动的占位时中偏差

本文证明了当底空间维数d≥3时,一类带移民超布朗运动占位时过程的中偏差,其移民由Lebesgue 测度控制.可以清楚地看出,中偏差的规范化因子和速度函数恰好介于中心极限定理和大偏差之间,在 这个意义下,中偏差填补了中心极限定理和大偏差之间的空白.     

Suboptimal policy determination for large-scale Markov decision processes,Part 1: Description and bounds

This paper is the first of two papers that present and evaluate an approach for determining suboptimal policies for large-scale Markov decision processes (MDP). Part 1 is devoted to the determination of bounds that motivate the development and indicate the quality of the suboptimal design approach; Part 2 is concerned with the implementation and evaluation of the suboptimal design approach. The specific MDP considered is the infinite-horizon, expected total discounted cost MDP with finite state and action spaces. The approach can be described as follows. First, the original MDP is approximated by a specially structured MDP. The special structure suggests how to construct associated smaller, more computationally tractable MDP's. The suboptimal policy for the original MDP is then constructed from the solutions of these smaller MDP's. The key feature of this approach is that the state and action space cardinalities of the smaller MDP's are exponential reductions of the state and action space cardinalities of the original MDP.     

面向多检查的门诊患者调度研究

及时的检查对于患者病情诊断和治疗非常重要。然而,患者不同的紧急程度、检查项目的多样性、以及患者的行为因素如失约等,使门诊患者调度问题难以求解。为解决该问题,本文考虑患者对两种检查项目的不同需求,两类不同的紧急程度,以及患者失约和医生加班,建立了有限时域马尔可夫决策过程(MDP)模型,目标是使得患者检查所得的期望收益最大化以及期望加班时间惩罚成本最小化。由于MDP模型复杂,难以用解析方法来分析最优控制策略,因此本文基于MDP模型进行数值实验,观察最优解的结构特征,进一步构造了两种参数化启发式调度策略,并采用遗传算法对调度策略的参数进行优化。数值实验比较了最优控制策略、两种启发式调度策略以及先到先服务规则,实验结果表明本文所提的调度策略性能偏离最优解不超过10%;当工作负荷非常大时,启发式调度策略远远优于先到先服务规则。     

Well-posedness of Hamilton–Jacobi equations in population dynamics and applications to large deviations

We prove Freidlin–Wentzell type large deviation principles for various rescaled models in populations dynamics that have immigration and possibly harvesting: birth–death processes, Galton–Watson trees, epidemic SI models, and prey–predator models.The proofs are carried out using a general analytic approach based on the well-posedness of a class of associated Hamilton–Jacobi equations. The notable feature for these Hamilton–Jacobi equations is that the Hamiltonian can be discontinuous at the boundary. We prove a well-posedness result for a large class of Hamilton–Jacobi equations corresponding to one-dimensional models, and give partial results for the multi-dimensional setting.     

Large deviations for perturbed reflected diffusion processes

In this article, we establish a large deviation principle for the solutions of perturbed reflected diffusion processes. The key is to prove a uniform Freidlin–Ventzell estimate of perturbed diffusion processes.     

Asymptotic approximation for the probability density function of an arbitrary sequence of random variables

We establish a large deviation approximation for the density function of an arbitrary sequence of random variables. The results are analogous to those obtained by Chaganty and Sethuraman (1985). We apply our theorems to the sample variance and the Mann–Whitney two-sample statistic.     

Asymptotics for stochastic reaction–diffusion equation driven by subordinate Brownian motion

We study the ergodicity of stochastic reaction–diffusion equation driven by subordinate Brownian motion. After establishing the strong Feller property and irreducibility of the system, we prove the tightness of the solution’s law. These properties imply that this stochastic system admits a unique invariant measure according to Doob’s and Krylov–Bogolyubov’s theories. Furthermore, we establish a large deviation principle for the occupation measure of this system by a hyper-exponential recurrence criterion. It is well known that S(P)DEs driven by $\alpha$-stable type noises do not satisfy Freidlin–Wentzell type large deviation, our result gives an example that strong dissipation overcomes heavy tailed noises to produce a Donsker–Varadhan type large deviation as time tends to infinity.     

Sharp trace Gagliardo–Nirenberg–Sobolev inequalities for convex cones,and convex domains

We find a new sharp trace Gagliardo–Nirenberg–Sobolev inequality on convex cones, as well as a sharp weighted trace Sobolev inequality on epigraphs of convex functions. This is done by using a generalized Borell–Brascamp–Lieb inequality, coming from the Brunn–Minkowski theory.     

Karhunen–Loève expansion for additive Slepian processes

For Gaussian random fields defined as additive processes based on Slepian processes, we study their Karhunen–Loève expansions and obtain the Pythagorean type distribution identities. As applications, the corresponding small deviation estimates are given.     

Robust empirical optimization is almost the same as mean–variance optimization

We formulate a distributionally robust optimization problem where the deviation of the alternative distribution is controlled by a $?$-divergence penalty in the objective, and show that a large class of these problems are essentially equivalent to a mean–variance problem. We also show that while a “small amount of robustness” always reduces the in-sample expected reward, the reduction in the variance, which is a measure of sensitivity to model misspecification, is an order of magnitude larger.     

Maximal mean/standard deviation ratio in an undiscounted MDP

A stationary policy in an MDP (Markov decision process) induces a stationary probability distribution of the reward from each initial state. The problem analyzed here is maximization of the mean/standard deviation ratio of the stationary distribution. In the unichain case, a solution is obtained via parametric analysis of a linear program having the same number of variables and one more constraint than the formulation for gain-rate optimization. The same linear program suffices in the multichain case if the initial state is an element of choice. The easier problem of maximizing the mean/variance ratio is mentioned at the end of the paper.     

Asymptotic behaviours for the trajectory fitting estimator in Ornstein–Uhlenbeck process with linear drift

We study the asymptotic behaviours for the trajectory fitting estimator in the Ornstein–Uhlenbeck process with linear drift. Deviation inequality, moderate deviations, Berry–Esseen bound and the law of iterated logarithm (LIL) of this estimator can be obtained. Moreover, as an application of the Berry–Esseen bound, we can get the precise rate in LIL. The main method of this paper is the deviation inequality for multiple Wiener-Itô integrals [P. Major, On a multivariate version of Bernsteins inequality. Electron. J. Prob. 12 (2007), pp. 966–988; P. Major, Tail behavior of multiple integrals and U-statistics. Probab. Surv. 2 (2005), pp. 448–505].     

A note on heuristic approach based on UBQP formulation of the maximum diversity problem

The maximum diversity problem (MDP) is a challenging NP-hard problem with a wide range of real applications. Several researchers have pointed out close relationship between the MDP and unconstrained binary quadratic program (UBQP). In this paper, we provide procedures to solve MDP ideas from the UBQP formulation of the problem. We first give some local optimality results for r-flip improvement procedures on MDP. Then, a set of highly effective diversification approaches based on sequential improvement steps for MDP are presented. Four versions of the approaches are used within a simple tabu search and applied to 140 benchmark MDP problems available on the Internet. The procedures solve all 80 small- to medium-sized problems instantly to the best known solutions. For 22 of the 60 large problems, the procedures improved by significant amounts the best known solutions in reasonably short CPU time.     

A decomposition algorithm for limiting average Markov decision problems

We consider a Markov decision process (MDP) under average reward criterion. We investigate the decomposition of such MDP into smaller MDPs by using the strongly connected classes in the associated graph. Then, by introducing the associated levels, we construct an aggregation-disaggregation algorithm for the computation of an optimal strategy for the original MDP.     

Suboptimal policy determination for large-scale Markov decision processes,part 2: Implementation and numerical evaluation

We present an implementation of the procedure for determining a suboptimal policy for a large-scale Markov decision process (MDP) presented in Part 1. An operation count analysis illuminates the significant computational benefits of this procedure for determining an optimal policy relative to a procedure for determining a suboptimal policy based on state and action space aggregation. Results of a preliminary numerical study indicate that the quality of the suboptimal policy produced by the 3MDP approach shows promise.This research has been supported by NSF Grants Nos. ECS-80-18266 and ECS-83-19355.     

Modelling silicosis: Existence,uniqueness and basic properties of solutions

We present a model for the silicosis disease mechanism following the original proposal by Tran et al. (1995), as modified recently by da Costa et al. (2020). The model consists in an infinite ordinary differential equation system of coagulation–fragmentation–death type. Results of existence, uniqueness, continuous dependence on the initial data and differentiability of solutions are proved for the initial value problem.     

Estimating Mixture of Dirichlet Process Models

Abstract

Current Gibbs sampling schemes in mixture of Dirichlet process (MDP) models are restricted to using “conjugate” base measures that allow analytic evaluation of the transition probabilities when resampling configurations, or alternatively need to rely on approximate numeric evaluations of some transition probabilities. Implementation of Gibbs sampling in more general MDP models is an open and important problem because most applications call for the use of nonconjugate base measures. In this article we propose a conceptual framework for computational strategies. This framework provides a perspective on current methods, facilitates comparisons between them, and leads to several new methods that expand the scope of MDP models to nonconjugate situations. We discuss one in detail. The basic strategy is based on expanding the parameter vector, and is applicable for MDP models with arbitrary base measure and likelihood. Strategies are also presented for the important class of normal-normal MDP models and for problems with fixed or few hyperparameters. The proposed algorithms are easily implemented and illustrated with an application.