Stopped processes and Doob's optional sampling theorem

摘要

Using the spectral measure mu(S) of the stopping time S, we define the stopping element X-S as a Daniell integral integral X(t)d(mu S) for an adapted stochastic process (X-t) (t is an element of 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-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 continuity of a stochastic process, which properly characterizes the notion of continuity of sample paths almost everywhere, found in the classical theory. (C) 2020 Elsevier Inc. All rights reserved.