NCD系统的数学建模与稳态分析

本文以有限齐次马尔科夫链对NCD（无索赔折扣）系统建模，严谨地证明了，任一NCD系统皆存在唯一的平稳分布。此外，给出了求解这一平稳分布的一般算法，并揭示了此平稳分布的结构。特别地，对两类给定的折扣类转移法则，还给出了平稳分布的显式。     

"NCD"系统中保险双方的最优决策

在"NCD"系统中,利用Markov决策过程,获得了投保双方博弈行为的最优结果.对被保险人来说,确定了其最优临界损失值;对保险人来说,确定了最优保费与折扣值.     

非平衡态随机动力学

本文主要评述随机过程的统计力学研究方面的基本概念,包括随机过程中的熵产生率、流、位势、平衡态、温度及其数学定义,以及由此得到的一些随机系统的基本性质,如涨落定理、Green-Kubo公式、功率谱密度的单调性等深刻的数学性质.本文要特别提出,利用Boltzman原理,在随机过程中引入温度参数的思想,并指出由此可以得到马氏过程表达的系统的分层集结聚类,这个分层聚类,反映了状态之间的相互作用网络结构.本文最后还例举了利用随机过程在应用问题中建模,进而得到对自然科学中关心的问题的解答、解释及深入认识.     

动力学系统参数辨识问题最优控制解的理论与方法(Ⅱ)——随机系统参数辨识及应用实例

基于文(Ⅰ)(《应用数学和力学》,1998,20(2))的内容和随机最优控制理论,本文首先介绍了随机动力学系统参数辨识问题最优控制解的概念.然后讨论了建立参数辨识问题HJB方程的过程以及参数辨识的算法.最后给出了一个应用实例:解决动力学系统局部非线性参数辨识问题的方法.     

Markov切换随机双线性系统依概率渐进稳定性

本文研究随机双线性系统大范围渐进稳定性。利用数学期望不等式给出了随机双线性系统渐进稳定的新标准。设计了一种非线性状态反馈控制器,利用类Riccati不等式推出了Markov切换随机双线性系统大范围依概率渐进稳定的充分条件。数值算例表明本文提出的方法是可行的。     

奖惩系统在随机优序下的极小与极大平稳分布

奖惩系统(Bonus-M a lus System)是世界各国机动车辆险中广泛采用的一种经验费率厘定机制.文[1]在最一般的框架下,给出了奖惩系统的数学建模与稳态分析.本文将进一步证明,文[1]中给出的奖惩系统两种特定的平稳分布恰分别是奖惩系统在随机优序下的极小与极大平稳分布.特别地,基于这一事实严格证明了文[1]提出的有关封闭型奖惩系统年度总保费的一个猜想.     

随机动力学:内源与外源噪声的数学模型及其应用

复杂系统与过程的数学建模需要用随机动力学(stochastic dynamics)的思想和方法.随机动力学的理论有着两种不同的数学表述:随机过程(stochastic processes)和随机动力系统(random dynamical systems).后者是比前者更为精细的数学模型,它不但给出对应于每一个初值的随机过程,还全面地描述不同初值的多条随机轨道如何同时随时间变化.前者恰恰表述了有内在随机性的个体的运动,而后者则反映了多个相同的确定性个体同时经历同一个随机环境.本文称这两种情形为内源噪声和外源噪声.两者都在化学和生物学中有广泛的应用.近年来兴起的以图G(V,E)为基础的概率布尔网络正是一类以{0,1}~V为状态空间的随机动力系统(RDS).本文介绍有关离散时间离散空间的RDS,同时也给出一个它在统计推断隐Markov模型的收敛速率估算中的应用.     

随机延迟微分方程的Milstein方法的非线性均方稳定性

本文针对一般的非线性随机延迟微分方程,证明了当系统理论解满足均方稳定性条件时,则当方程的漂移和扩散项满足一定的条件时,Milstein方法也是均方稳定的.数学实验进一步验证了我们的结论.     

非平衡态统计物理的随机数学理论

非平衡态热力学和统计物理理论从20世纪70至80年代开始进入了蓬勃发展的阶段,特别是其中的随机模型可以用来描述亚宏观尺度的生物物理过程,恰好契合了现代物理化学和生物化学高精度实验技术的发展潮流.本文将回顾过去三十多年里非平衡态统计物理随机数学理论的发展,包括非平衡态热力学理论的起源、非平衡定态的随机数学理论、非平衡暂态的随机数学理论以及非平衡定态的初步相变理论等.在这套随机数学理论中,随机过程的时间可逆性、熵产生率和环流是最核心的概念,而寻求在随机轨道层面热力学第一定律和第二定律的对应形式是最重要的目标.这套数学物理理论能够很好地被应用于细胞层面重要的生物化学过程,通过严格的数学论证探寻其中重要的规律,以期对处于非平衡态的生命现象得到更深刻的认识和理解.     

随机交通均衡配流模型及其等价的变分不等式问题

本文讨论了交通网络系统的随机用户均衡原理的数学表述问题.在路段出行成本是流量的单调函数的较弱条件下,对具有固定需求和弹性需求的模式,首次证明了随机均衡配流模型可表示为一个变分不等式问题,同时也说明了该变分不等式问题与相应的互补问题以及一个凸规划问题之间的等价关系.     

Infection Time in Multistable Gene Networks. A Backward Stochastic Variational Inequality with Nonconvex Switch-Dependent Reflection Approach

We investigate a mathematical model associated to the infection time in multistable gene networks. The mathematical processes are of hybrid switch type. The switch is governed by pure jump modes and linked to DNA bindings. The differential component follows backward stochastic dynamics reflected in some mode-dependent nonconvex domains. First, we study the existence of solutions to the resulting stochastic variational inclusions, by reducing the model to a family of ordinary variational inclusions with generalized reflection in semiconvex domains. Second, by considering control-dependent drivers, we hint to some model-selection approach by embedding the controlled backward stochastic variational inclusion in a family of regular measures. Regularity and structural properties of these sets are given.     

Quantitative stability analysis of stochastic quasi-variational inequality problems and applications

We consider a parametric stochastic quasi-variational inequality problem (SQVIP for short) where the underlying normal cone is defined over the solution set of a parametric stochastic cone system. We investigate the impact of variation of the probability measure and the parameter on the solution of the SQVIP. By reformulating the SQVIP as a natural equation and treating the orthogonal projection over the solution set of the parametric stochastic cone system as an optimization problem, we effectively convert stability of the SQVIP into that of a one stage stochastic program with stochastic cone constraints. Under some moderate conditions, we derive Hölder outer semicontinuity and continuity of the solution set against the variation of the probability measure and the parameter. The stability results are applied to a mathematical program with stochastic semidefinite constraints and a mathematical program with SQVIP constraints.     

State constrained reachability for stochastic hybrid systems

Many control problems can be formulated as driving a system to reach some target states while avoiding some unwanted states. We study this problem for systems with regime change operating in uncertain environments. Nowadays, it is a common practice to model such systems in the framework of stochastic hybrid system models. In this casting, the problem is formalized as a mathematical problem named state constrained stochastic reachability analysis. In the state constrained stochastic reachability analysis, this probability is computed by imposing a constraint on the system to avoid the unwanted states. The scope of this paper is twofold. First we define and investigate the state constrained reachability analysis in an abstract mathematical setting. We define the problem for a general model of stochastic hybrid systems, and we show that the reach probabilities can be computed as solutions of an elliptic integro-differential equation. Moreover, we extend the problem by considering randomized targets. We approach this extension using stochastic dynamic programming. The second scope is to define a developmental setting in which the state constrained reachability analysis becomes more tractable. This framework is based on multilayer modelling of a stochastic system using hierarchical viewpoints. Viewpoints represent a method originated from software engineering, where a system is described by multiple models created from different perspectives. Using viewpoints, the reach probabilities can be easily computed, or even symbolically calculated. The reach probabilities computed in one viewpoint can be used in another viewpoint for improving the system control. We illustrate this technique for trajectory design.     

正倒向随机微分方程组的数值解法

1990年,Pardoux和Peng(彭实戈)解决了非线性倒向随机微分方程(backward stochastic differential equation,BSDE)解的存在唯一性问题,从而建立了正倒向随机微分方程组(forward backward stochastic differential equations,FBSDEs)的理论基础;之后,正倒向随机微分方程组得到了广泛研究,并被应用于众多研究领域中,如随机最优控制、偏微分方程、金融数学、风险度量、非线性期望等.近年来,正倒向随机微分方程组的数值求解研究获得了越来越多的关注,本文旨在基于正倒向随机微分方程组的特性,介绍正倒向随机微分方程组的主要数值求解方法.我们将重点介绍讨论求解FBSDEs的积分离散法和微分近似法,包括一步法和多步法,以及相应的数值分析和理论分析结果.微分近似法能构造出求解全耦合FBSDEs的高效高精度并行数值方法,并且该方法采用最简单的Euler方法求解正向随机微分方程,极大地简化了问题求解的复杂度.文章最后,我们尝试提出关于FBSDEs数值求解研究面临的一些亟待解决和具有挑战性的问题.     

Global asymptotic stability of a Lotka–Volterra competition model with stochasticity in inter-specific competition

In the paper we first propose a two-species Lotka–Volterra competition model with the stochastic terms related to the inter-specific competition rates and the coexistence equilibrium of the deterministic model. Then we establish the global asymptotic stability of the coexistence equilibrium. Finally, we provide some discussions and numerical examples to illustrate our mathematical results.     

Properties of minimal mathematical expectations

The aim of this paper is to discuss how to compute a class of minimal mathematical expectations by using backward stochastic differential equations. We also prove that the minimal mathematical expectation operator still preserves some properties of the mathematical expectation operator.     

The Cauchy problem for fractional conservation laws driven by Lévy noise

In this article, we explore some of the main mathematical problems connected to multidimensional fractional conservation laws driven by Lévy processes. Making use of an adapted entropy formulation, a result of existence and uniqueness of a solution is established. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that the Lévy noise depends only on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. Furthermore, we establish a result on vanishing non-local regularization of scalar stochastic conservation laws.     

PROMISE: A DSS for multiple objective stochastic linear programming problems

Most of the multiple objective linear programming (MOLP) methods which have been proposed in the last fifteen years suppose deterministic contexts, but because many real problems imply uncertainty, some methods have been recently developed to deal with MOLP problems in stochastic contexts. In order to help the decision maker (DM) who is placed before such stochastic MOLP problems, we have built a Decision Support System called PROMISE. On the one hand, our DSS enables the DM to identify many current stochastic contexts: risky situations and also situations of partial uncertainty. On the other hand, according to the nature of the uncertainty, our DSS enables the DM to choose the most appropriate interactive stochastic MOLP method among the available methods, if such a method exists, and to solve his problem via the chosen method.     

Long time behaviour of a stochastic nanoparticle

In this article, we are interested in the behaviour of a single ferromagnetic mono-domain particle submitted to an external field with a stochastic perturbation. This model is the first step toward the mathematical understanding of thermal effects on a ferromagnet. In a first part, we present the stochastic model and prove that the associated stochastic differential equation is well defined. The second part is dedicated to the study of the long time behaviour of the magnetic moment and in the third part we prove that the stochastic perturbation induces a non-reversibility phenomenon. Last, we illustrate these results through numerical simulations of our stochastic model.     

The Analysis of Stochastic Fiber Lay-Down Models: Geometry and Convergence to Equilibrium of the Basic Model

The so-called fiber lay-down models arise in the production process of nonwovens. We introduce the generalized version of the basic fiber lay-down model which can precisely be formulated in abstract form as some manifold-valued stochastic differential equation. An important criterion for the quality of the nonwoven material is how the solution to the associated Fokker-Planck equation converges towards its stationary state. Especially, one is interested in determining the speed of convergence. Here we present some results concerning the long-time behavior by using classical stochastic methods as well as modern analytic methods from the theory of hypocoercivity. Demanding mathematical difficulties arising since the equation is degenerate. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)