论文标题
停止过程和DOOB的可选抽样定理
Stopped processes and Doob's optional sampling theorem
论文作者
论文摘要
Using the spectral measure $μ_\mathbb{S}$ of the stopping time $\mathbb{S},$ we define the stopping element $X_\mathbb{S}$ as a Daniell integral $\int X_t\,dμ_\mathbb{S}$ for an adapted stochastic process $(X_t)_{t\in J}$ that是Daniell可总结的矢量值函数。这是先前针对具有DOOB-MEYER分解属性的右顺序子赛的定义的扩展。 $ x_ \ mathbb {s} $的更一般的定义需要一个新的Doob的可选采样定理证明，因为Sub-Martingales先前给出的定义隐式使用了doob的定理，应用于martingales。我们提供了这样的证据，从而消除了结果中迄今为止对Doob-Meyer分解属性的必要假设。 本文提出的另一个进步是我们使用了无限制的顺序融合，这正确地表征了经典理论中几乎所有地方的融合的概念。使用顺序预测代替传统指标函数，我们还推广了统一的可集成序列的概念。在上面提到的主要定理的基本要素中，我们证明，相对于无界顺序收敛的均匀集成序列也将$ \ MATHCAL {l}^1 $收敛到同一元素。
Using the spectral measure $μ_\mathbb{S}$ of the stopping time $\mathbb{S},$ we define the stopping element $X_\mathbb{S}$ as a Daniell integral $\int X_t\,dμ_\mathbb{S}$ for an adapted stochastic process $(X_t)_{t\in J}$ that is a Daniell summable vector-valued function. This is an extension of the definition previously given for right-order-continuous sub-martingales with the Doob-Meyer decomposition property. The more general definition of $X_\mathbb{S}$ necessitates a new proof of Doob's optional sampling theorem, because the definition given earlier for sub-martingales implicitly used Doob's theorem applied to martingales. We provide such a proof, thus removing the heretofore necessary assumption of the Doob-Meyer decomposition property in the result. Another advancement presented in this paper is our use of unbounded order convergence, which properly characterizes the notion of almost everywhere convergence found in the classical theory. Using order projections in place of the traditional indicator functions, we also generalize the notion of uniformly integrable sequences. In an essential ingredient to our main theorem mentioned above, we prove that uniformly integrable sequences that converge with respect to unbounded order convergence also converge to the same element in $\mathcal{L}^1$.
