报告题目：Estimates of Invariant Probability Measures for Singular SDEs
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
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
) 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
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
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
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：Risk-sensitive optimality for continuous-time Markov decision processes
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：Quenched weighted moments of a supercritical branching process in a random environment
摘要：We consider a supercritical branching process
in an independent and identically distributed random environment
be the limit of the natural martingale
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
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
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
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
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
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
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.