-integrable terminal values

-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.

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。

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

. 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.

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.