Formulation and well-posedness of the Cauchy problem for a diffusion equation with discontinuous degenerating coefficients

The choice of a differential diffusion operator with discontinuous coefficients that corresponds to a finite flow velocity and a finite concentration is substantiated. For the equation with a uniformly elliptic operator and a nonzero diffusion coefficient, conditions are established for the existence and uniqueness of a solution to the corresponding Cauchy problem. For the diffusion equation with degeneration on a half-line, it is proved that the Cauchy problem with an arbitrary initial condition has a unique solution if and only if there is no flux from the degeneration domain to the ellipticity domain of the operator. Under this condition, a sequence of solutions to regularized problems is proved to converge uniformly to the solution of the degenerate problem in L ₁(R) on each interval.     

随机单调的离散时间马氏链强遍历速度的估计I

研究了离散时间马氏链的强遍历性,对随机单调的离散时间马氏链,给出了最大强遍历收敛速度的下界估计。     

Bayesian Copulae Distributions, with Application to Operational Risk Management

The aim of this paper is to introduce a new methodology for operational risk management, based on Bayesian copulae. One of the main problems related to operational risk management is understanding the complex dependence structure of the associated variables. In order to model this structure in a flexible way, we construct a method based on copulae. This allows us to split the joint multivariate probability distribution of a random vector of losses into individual components characterized by univariate marginals. Thus, copula functions embody all the information about the correlation between variables and provide a useful technique for modelling the dependency of a high number of marginals. Another important problem in operational risk modelling is the lack of loss data. This suggests the use of Bayesian models, computed via simulation methods and, in particular, Markov chain Monte Carlo. We propose a new methodology for modelling operational risk and for estimating the required capital. This methodology combines the use of copulae and Bayesian models.      

Modeling and Analysis of Modal Switching in Networked Transport Systems

We consider networked transport systems defined on directed graphs: the dynamics on the edges correspond to solutions of transport equations with space dimension one. In addition to the graph setting, a major consideration is the introduction and propagation of discontinuities in the solutions when the system may discontinuously switch modes, naturally or as a hybrid control. This kind of switching has been extensively studied for ordinary differential equations, but not much so far for systems governed by partial differential equations. In particular, we give well-posedness results for switching as a control, both in finite horizon open loop operation and as feedback based on sensor measurements in the system.     

Optimal server resource reservation policies for priority classes of users under cyclic non-homogeneous markov modeling

Resource availability optimization is studied on a server–client system where different users are partitioned into priority classes. The aim is to provide higher resource availability according to the priority of each class. For this purpose, resource reservation is modeled by a homogeneous continuous time Markov chain (CTMC), but also by a cyclic non-homogeneous Markov chain (CNHMC) as there is a cyclic behavior of the users’ requests for resources. The contribution of the work presented consists in the formulation of a multiobjective optimization problem for both the above cases that aims to determine the optimal resource reservation policy providing higher levels of resource availability for all classes. The optimization problem is solved either with known methods or with a proposed kind of heuristic algorithm. Finally, explicit generalized approximate inverse preconditioning methods are adopted for solving efficiently sparse linear systems that are derived, in order to compute resource availability.     

A stabilization law for two semi-infinite interacting strings of characters

We consider a Markov chain that describes the evolution of two interacting strings of symbols. The transitions probalitities of this Markov chain depend only on the rightmost symbols of both strings. The main goal of the present paper is to prove a limit theorem (stabilization law): the distribution of the rightmost symbols converges to some limit correlation function.1 Partially supported by FAPESP (2002/01501-9) and RFBR (02-01-00415)2 Partially supported by RFBR (02-01-00415)     

On Certain Extensions of a Rotation Invariant Markov Process

Let (X _t ,P ^x ) be a rotation invariant (RI) Feller process on R ^d {0}, d2. We study certain type of strong Markov extensions of (X _t ,P ^x ) to R ^d . It turns out that that the unique (RI) extension always exists and is of that type. We give a kind of quasi-ergodicy condition (E) under which the (RI) extension is the only extension of that type. A class of processes fulfilling (E) is also characterized.     

Geometric Convergence Rates for Time-Sampled Markov Chains

We consider time-sampled Markov chain kernels, of the form P = _n n P ⁿ . We prove bounds on the total variation distance to stationarity of such chains. We are motivated by the analysis of near-periodic MCMC algorithms.     

一类混杂动态系统的能控性（I）——基本结果

首次将时滞现象引入到线性切换系统的模型中，研究含有时滞线性切换系统的能控性及其判定条件．全部工作由三部分组成，第I部分首先，提出含时滞的线性切换系统的数学模型，并介绍切换系统的基本概念——切换序列．其次，引入列空间、循环不变子空间和广义循环不变子空间等基本几何概念，给出一些有关概念的基本性质，特别是分离引理．然后以一个基本引理的形式揭式某一积分方程的解集与广义循环不变子空间之间的联系，这个引理将在能控性的判定中起关键作用．这些概念和引理都将作为以后展开能控性分析所必需的研究工具．