马氏过程与随机分析学术论坛

发布时间:2017年11月24日 作者:唐颖   消息来源:业务办    阅读次数:[]

马氏过程与随机分析学术论坛

报告题目:Estimates of Invariant Probability Measures for Singular SDEs

报告人:王凤雨 天津大学

报告时间: 2017年11月25日上午08:20-08:55

摘要:In terms of a nice reference probability measure, integrability conditions on the path-dependent drift are presented for (infinite-dimensional) degenerate PDEs to have regular positive solutions. To this end, the corresponding stochastic (partial) differential equations are proved to possess the weak existence and uniqueness of solutions, as well as the existence, uniqueness and entropy estimates of invariant probability measures. When the reference measure satisfies the log-Sobolev inequality, Sobolev estimates are derived for the density of invariant probability measures. Some results are new even for non-degenerate SDEs with path-independent drifts. The main results are applied to nonlinear functional SPDEs and degenerate functional SDEs/SPDEs.

报告题目:Scalar BSDEs with weakly

-integrable terminal values

报告人:汤善健 复旦大学

报告时间: 2017年11月25日上午08:55-09:30

摘要:Here I would report a recent joint work with Ying Hu (University o Rennes, France). Consider a scalar linearly growing BSDE with a weakly

-integrable terminal value. We prove that the BSDE admits a solution if the terminal value satisfies some

-integrability condition with the positive parameter

being less than a critical value

, which is weaker than the usual

(p>1) integrability and stronger than

integrability. We show by a counterexample that the conventionally expected

integrability and even the

-integrability (for

) are not sufficient for the existence of solution to a BSDE of a linearly growing generator.

报告题目:Harnack inequality and W-entropy formula on manifolds with super Ricci flows

报告人:李向东 中国科学院

报告时间: 2017年11月25日上午09:30-10:05

摘要:Inspired by G. Perelman's work on entropy formula for the Ricci flow, we prove a Li-Yau-Hamilton-Perelman type Harnack inequality for the heat equation on manifolds with non-negative m-Bakry-Emery Ricci curvature. We establish the relationship between the LYHP quantity and the W-entropy for the heat equation associated with the time dependent Witten Laplacian on manifolds equipped with time dependent metrics. As a consequence, we rederive the W-entropy formula on manifolds with super Ricci flows.

报告题目:Decay Properties of Quadratic Markov Branching Processes

报告人:陈安岳 南方科技大学

报告时间: 2017年11月25日上午10:20-10:55

摘要:In this talk, we address the decay properties of the non-liner Markov branching processes, particularly for the quadratic branching process. We show that the decay parameter of a quadratic branching process is equal to the first eigenvalue of a second order differential operator associated with the PDE to which the generating function of the transition probability of the process must satisfy. Both the upper and lower bounds of the decay parameter are given explicitly by applying the classical Hardy's inequality.

报告题目:Dirichlet Principle for Non-reversible Markov Chains

报告人:毛永华 北京师范大学

报告时间: 2017年11月25日上午10:55-11:30

摘要:In this talk, I will show that how Dirichlet principle works for non-reversible Markov chain, which indicates that the non-reversible Markov chain is better than its reversible one, via mixing time and asymptotic variance.

报告题目:单死过程的遍历性和强遍历性判别准则

报告人:张余晖 北京师范大学

报告时间: 2017年11月25日上午11:30-12:05

摘要:分支过程是单死过程的特殊情形,有很多研究成果,主要基于母函数方法,而对一般单死过程的研究则几乎是空白的.在这个报告里,我们将介绍近期得到的单死过程遍历性和强遍历性的判别准则,指数遍历的一个充分条件,常返性的判别准则和灭绝(回返)概率的表示等等,这些结果是显式的,即基于转移速率,因而是可计算的.

报告题目:Risk-sensitive optimality for continuous-time Markov decision processes

报告人:郭先平 中山大学

报告时间: 2017年11月25日下午13:45-14:20

摘要:In this talk, we consider the risk-sensitive optimality problems for continuous-time Markov decision processes on a finite-horizon, and focuses on the more general case that the transition rates are unbounded, cost rates are allowed to be unbounded from below and from above, the policies can be history-dependent, and the state and action spaces are Polish ones. Under mild conditions, we establish the existence of a solution to the corresponding optimality equation (OE). Using the OE and the extension of E.B. Dynkin formula developed here, we prove the existence of an optimal Markov policy, and verify that the value function is the unique solution to the OE. Finally, we give an example to illustrate the difference between our conditions and those in the previous literature.

报告题目:大数据下社会网络与共享经济中的级联马氏过程:共享、匹配、敏感性与相变

报告人:李泉林 燕山大学

报告时间: 2017年11月25日下午14:20-14:55

摘要:物联网与大数据在全球的快速发展已经极大地推动了当今许多重要实际领域的研究变革与应用创新。这个报告在大数据环境下分析了国际上两个非常重要的应用领域:社会网络及其级联马氏过程;共享经济中的大型马氏过程系统。我们的研究工作包括如下方面:

l 在大数据环境下,如何构造社会网络中的级联马氏过程?这样的马氏过程具有怎样的结构特征与基本性质?

l 大数据如何能够有效地提供给社会网络中级联马氏过程的统计特征与物理行为?从扰动的马氏过程出发,分析这种级联马氏过程的敏感性,并由此观察社会网络中系统参数和物理结构的临界性,从而分析社会网络的柔性与脆弱性。

l 在大数据环境下,讨论共享单车系统。首先,我们介绍在共享自行车系统中已经取得的三类研究成果与数学方法。然后,我们讨论共享单车及其性能评价与价格激励的排队网络方法。研究共享单车系统的图上马氏过程以及变拓扑结构马氏过程。最后,我们讨论由共享单车的信息平台所诱导的双边匹配市场。

l 在大数据下的共享经济系统中,我们将讨论动态的多边匹配市场、共享信息平台之间的合作关系与竞争行为,以及由它们所构建的马氏过程与随机博弈等重要问题。利用社会网络研究共享经济系统的组织行为与结构特征。

报告题目:Quenched weighted moments of a supercritical branching process in a random environment

报告人:李应求 长沙理工大学

报告时间: 2017年11月25日下午14:55-15:30

摘要:We consider a supercritical branching process

in an independent and identically distributed random environment

. Let

be the limit of the natural martingale

, where

denotes the conditional expectation given the environment

. We find a necessary and sufficient condition for the existence of quenched weighted moments of

of the form

, where

and

is a positive function slowly varying at

. The same conclusion is also proved for the maximum of the martingale

instead of the limit variable

. In the proof we first show an extended version of Doob’s inequality about weighted moments for nonnegative sub-martingales, which is of independent interest.

报告题目:Optimization of Markov Decision Processes under the Variance Criterion

报告人:夏俐 清华大学

报告时间: 2017年11月25日下午15:30-16:05

摘要:In this talk, we study a variance minimization problem in an infinite stage discrete time Markov decision process (MDP), regardless of the mean performance. For the Markov chain under the variance criterion, since the value of the cost function at the current stage will be affected by future actions, this problem is not a standard MDP and the traditional MDP theory is not applicable. We convert the variance minimization problem into a standard MDP by introducing a concept called pseudo variance. Then we derive a variance difference formula that quantifies the difference of variances of Markov systems under any two policies. With the difference formula, the correlation of the variance cost function at different stages can be decoupled through a nonnegative term. A necessary condition of the optimal policy is obtained. It is also proved that the optimal policy with the minimal variance can be found in the deterministic policy space. Furthermore, we propose an efficient iterative algorithm to reduce the variance of Markov systems. We prove that this algorithm can converge to a local optimum. Finally, we demonstrate the efficiency of our approach by applying it to the fluctuation reduction problem of renewable energy with storage systems.

报告题目:Stochastic Damping Hamiltonian Systems with State-Dependent Switching

报告人:席福宝 北京理工大学

报告时间: 2017年11月25日下午16:20-16:55

摘要:In this talk, a class of regime-switching stochastic damping Hamiltonian systems with continuous-state-dependent switching will be discussed. First, for a special Markovian switching case, the existence of a globally weak solution is constructed by making using of the martingale approach. Next, for the general continuous-state-dependent switching case, the existence of a globally weak solution is established by virtue of the Radon-Nikodym derivative method. Then, strong Feller property is proved by the killing technique and the Radon-Nikodym derivative method. Moreover, based on the above results, exponential ergodicity is obtained under the Foster-Lyapunov drift condition. Finally, some examples are presented for illustration.

报告题目:Stabilization of regime-switching processes by feedback control based on discrete time observations

报告人:邵井海 天津大学

报告时间: 2017年11月25日下午16:55-17:30

摘要:In this talk, we study the feedback control depending on discrete-time observations of the state process and the switching process. Our criteria depend explicitly on the regular conditions of the coefficients of stochastic differential equations and on the stationary distribution of the switching process. The sharpness of our criteria is shown through studying the stability of linear systems, which also shows explicitly that the stability of hybrid stochastic differential equations depends essentially on the long time behavior of the switching process.

报告题目:Optimal dividend problem for PDMPs risk models

报告人: 刘国欣 石家庄铁道大学

报告时间: 2017年11月25日下午17:30-18:05

摘要:We consider the optimal dividend problem for an insurance company with surplus-dependent premium and surplus-dependent claim-arrival. The objective is to maximize the expected cumulative discounted dividend payments received until the time of ruin. In the absence of dividend payments, such a risk process is a particular case of so-called piecewise deterministic Markov processes (PDMPs). Specially, the inter-arrival time is not limited to be absolutely continuous distributed. Due to this, we identify that the associated dynamic programming equation (DPE) is measure-valued. We present firstly an analytic characterization of a Markov control, which is equivalent to that the dividend strategy is an additive functional of the controlled surplus process. And we show that each Markov control has a band structure. Therefore it is surplus-dependent. A verification theorem is proved. We also demonstrate that the optimal dividend strategy is a Markov control and hence a band strategy. As an illustration of the measure-valued DPE here, we give a short discussion on the optimal dividend problem for a insurance risk model with surplus- dependent premiums, studied by Marciniak and Palmowski (2016).

报告题目:Exponential ergodicity for SDEs driven by $\alpha$-stable processes with Markovian switching in Wasserstein distances

报告人:张振中 东华大学副教授

报告时间:2017年11月26日上午9:00-10:00

报告地点:数学与统计学院一楼142

报告摘要:

In this paper, we consider the ergodicity for stochastic differential equations driven by symmetric $\alpha$-stable processes with Markovian switching in Wasserstein distances. Some sufficient conditions for the exponential ergodicity are presented by using the theory of M-matrix, coupling method and Lyapunov function method. As applications, the Ornstein-Uhlenbeck type processes and some other processes driven by symmetric $\alpha$-stable processes with Markovian switching are presented to illustrate our results. In addition, under some conditions, the explicit expressions of invariant measures for Ornstein-Uhlenbeck processes are given.

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