Concrete representation of atomic $(F_4)$ filtrations

Atomic blocks for noncommutative martingales

Given a probability space $(Ω,Σ,μ)$, the Hardy space $\mathrm{H}_1(Ω)$ which is associated to the martingale square function does not admit a classical atomic decomposition when the underlying filtration is not regular. In this paper we construct a decomposition of $\mathrm{H}_1(Ω)$ into "atomic blocks"${}$ in the spirit of Tolsa, which we will introduce for martingales. We provide three proofs of this result. Only the first one also applies to noncommutative martingales, the main target of this paper. The other proofs emphasize alternative approaches for commutative martingales. One might be …

On independent sets in purely atomic probability spaces with geometric distribution

We are interested in constructing concrete independent events in purely atomic probability spaces with geometric distribution. Among other facts we prove that there are uncountable many sequences of independent events.

熵与时间递增披露的离散信息的附加效用

Entropy and additional utility of a discrete information disclosed progressively in time

本文研究了时间递增披露信息的熵及其效用，分析了其在渐进设定下的martingale空间稳定性，并计算了离散信息的附加对数效用。

Gundy-Stein分解与显式常数

The Gundy-Stein decomposition with explicit constants

本文研究了鞅的Gundy-Stein分解，证明了显式常数下的分解形式，并给出了各部分的明确界及应用。

Asymmetric Doob inequalities in continuous time

The present paper is devoted to the second part of our project on asymmetric maximal inequalities, where we consider martingales in continuous time. Let $(\mathcal M,τ)$ be a noncommutative probability space equipped with a continuous filtration of von Neumann subalgebras $(\mathcal M_t)_{0\leq t\leq1}$ whose union is weak-$*$ dense in $\mathcal{M}$. Let $\mathcal E_t$ denote the corresponding family of conditional expectations. As for discrete filtrations, we shall prove that for $1 < p < 2$ and $x \in L_p(\mathcal M,τ)$ one can find $a, b \in L_p(\mathcal M,τ)$ and contractions $u_t, v_t \in…

The Chaotic Representation Property of Compensated-Covariation Stable Families of Martingales

In the present paper, we study the chaotic representation property for certain families of square integrable martingales. For this purpose, we introduce the notion of compensated-covariation stability of such families. The chaotic representation property will be defined using iterated integrals with respect to a given family of square integrable martingales having deterministic mutual predictable covariation. The main result of the present paper is: If $\mathscr{X}$ is a compensated-covariation stable family of square integrable martingales such that $\langle{X},{Y}\rangle$ is deterministic fo…