Stopped processes and Doob's optional sampling theorem

TL;DR

This paper extends the theory of stopped processes by defining a new spectral measure-based stopping element, generalizes Doob's optional sampling theorem without the need for the Doob-Meyer property, and introduces unbounded order convergence for a more comprehensive understanding of convergence in stochastic processes.

Contribution

It introduces a spectral measure approach to defining stopped processes, removes the Doob-Meyer assumption in the optional sampling theorem, and employs unbounded order convergence for a broader convergence framework.

Findings

Defined the stopping element $X_S$ via spectral measure integration.

Provided a new proof of Doob's optional sampling theorem without Doob-Meyer assumption.

Demonstrated convergence properties of uniformly integrable sequences under unbounded order convergence.

Abstract

Using the spectral measure $\mu_\mathbb{S}$ of the stopping time $\mathbb{S},$ we define the stopping element $X_\mathbb{S}$ as a Daniell integral $\int X_t\,d\mu_\mathbb{S}$ 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 $\mathcal{L}^1$.