We give new proofs of certain equivalent conditions for the existence of generalized moments of a Lévy process $(X_t)_{t\geq 0}$; in particular, the existence of a generalized $g$-moment is equivalent to the uniform integrability of $(g(X_t))_{t\in [0,1]}$. As a consequence, certain functions of a Lévy process which are integrable and local martingales are already true martingales. Our methods extend to moments of stochastically continuous additive processes, and we give new, short proofs for the characterization of lattice distributions and the transience of Lévy processes.
This note establishes convergence in mean of order $p$, $0<p\le 1$ for $d$-dimensional arrays of random vectors in Hilbert spaces under the Cesàro uniform integrability conditions. In the case where $0<p<1$, our $L_p$ convergence is valid irrespective of any dependence structure. In the case where $p=1$, the underlying random vectors are supposed to be pairwise independent. The mean convergence results are established for maximal partial sums while previous contributions were so far considered partial sums only. Some results in the literature are extended. Various properties of the Cesàro unif…
Let $L^0$ be the vector space of all (equivalence classes of) real-valued random variables built over a probability space $(Ω, \mathcal{F}, P)$, equipped with a metric topology compatible with convergence in probability. In this work, we provide a necessary and sufficient structural condition that a set $X \subseteq L^0$ should satisfy in order to infer the existence of a probability $Q$ that is equivalent to $P$ and such that $X$ is uniformly $Q$-integrable. Furthermore, we connect the previous essentially measure-free version of uniform integrability with local convexity of the $L^0$-topolog…
This paper is devoted to the existence, uniqueness and comparison theorem on unbounded solutions of a scalar backward stochastic differential equation (BSDE) whose generator grows (with respect to both unknown variables $y$ and $z$) in a super-linear way like $|y||\ln |y||^{(λ+1/2)\wedge 1}+|z||\ln |z||^λ$ for some $λ\geq 0$. For the following four different ranges of the growth power parameter $λ$: $λ=0$, $λ\in (0,1/2)$, $λ=1/2$ and $λ>1/2$, we give reasonably weakest possible different integrability conditions of the terminal value for the existence of an unbounded solution to the BSDE. In t…
A Note on Uniform Nonintegrability of Random Variables
在最近的一份文件\cite{CHR16}，钱德拉，胡和Rosalsky介绍的概念序列的随机变量是一致不可积的，并提出了一系列有趣的结果，这种一致不可积性。本文引入了随机变量一致不可积的一个较弱定义，给出了一致不可积的一个充要条件，并给出了一致不可积的两个等价刻画，其中之一是一致不可积的de La Vallée Poussin准则的一个W-UNI类比.此外，我们给出了一些评论，其中之一给出了一个否定的答案\cite{CHR16}中提出的开放问题。
