停止过程与Doob选择抽样定理
Stopped processes and Doob's optional sampling theorem
摘要：
使用停止时间$\mathbb{S},$的谱测度$μ_\mathbb{S}$，我们将停止元素$X_\mathbb{S}$定义为自适应随机过程$(X_t)_{t\in J}$的Daniell积分$\int X_t\,dμ_\mathbb{S}$，该自适应随机过程是Daniell可和向量值函数。这是一个扩展的定义，以前给出的右序连续子鞅的Doob-Meyer分解性质。$X_\mathbb{S}$的更一般的定义需要Doob的可选抽样定理的新证明，因为之前给出的下鞅的定义隐含地使用了Doob定理应用于鞅。我们提供了这样一个证明，从而消除了迄今为止必要的假设Doob-Meyer分解性质的结果。本文提出的另一个进步是我们使用的无界阶收敛，这适当的特点的概念，几乎处处收敛的经典理论。利用序投影代替传统的指示函数，推广了一致可积序列的概念。在我们的主要定理的一个重要组成部分，我们证明了一致可积序列收敛于无界阶收敛也收敛于同一元素在$\mathcal{L}^1$。
英文摘要：
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 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 $\mathcal{L}^1$.
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