Stopped processes and Doob's optional sampling theorem

Abstract

Using the spectral measure μS of the stopping time S, we define the stopping element XS as a Daniell integral ∫XtdμS for an adapted stochastic process (Xt)t∈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 XS 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 continuity of a stochastic process, which properly characterizes the notion of continuity of sample paths almost everywhere, found in the classical theory.