Discrete-time stochastic processes on Riesz spaces.
type: Item
instance of: scholarly article
title: Discrete-time stochastic processes on Riesz spaces.
zbMATH Open document ID: 1057.60041
DOI: 10.1016/S0019-3577(04)80010-7
author: Wen-Chi Kuo
author: Coenraad C. A. Labuschagne
author: Bruce A. Watson
published in: Indagationes Mathematicae. New Series
publication date: 2005-01-03T00:00:00Z
review text: Measure and integration theory has been as inspiring source for the development of the theory of Riesz spaces and positive operators. For example, the space of all measurable functions on some measurable space is a Riesz space and the integral is a positive operator on the space of all integrable functions. Moreover, even the construction of the Lebesgue integral is based on the notions of positivity and order convergence. The authors translate several notions of discrete martingale theory into the framework of Riesz spaces having a weak order unit; weak order units are needed as a substitute for the constants in spaces of measurable functions. The basic notion is that of a conditional expectation which is defined as an order continuous positive projection having a Dedekind complete range and preserving weak order units. Conditional expectations are then used to define filtrations and stopping times: A filtration is defined as a sequence of conditional expectations \( (T_i) \) satisfying \( T_jT_i = T_i = T_iT_j \) for all \( i \leq j \), and a stopping time with respect to the filtration \( (T_i) \) is defined as an increasing sequence of conditional expectations \( (P_i) \) satisfying \( P_i \leq I \) for all \( i \) and \( T_jP_i = P_iT_j \) for all \( i \leq j \). The definition of a (sub-/super-)martingale is then straightforward. In this abstract setting, the authors prove, among other results, the Doob-Meyer decomposition and the optional stopping theorem. The paper is pleasant to read since the authors take care to prepare the abstract definitions in the Riesz space setting by a thorough discussion of the corresponding traditional notions from martingale theory. It is likely that the general theory developed here can also be applied to asymptotic martingales.
Mathematics Subject Classification ID: 60G42
Mathematics Subject Classification ID: 60G48
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 47N30
zbMATH DE Number: 2124781
zbMATH Keywords: conditional expectation, martingales, stopping times
zbMATH Keywords: positive operators
reviewed by: Klaus D. Schmidt
MaRDI profile type: Publication
cites work: Q4781239
cites work: Conditional expectation and ordering
cites work: Q4148368
cites work: Characterizations of conditional expectation-type operators
cites work: Q4422013
cites work: Q3157507
cites work: Conditional measures and operators
cites work: Two Characterizations of Conditional Probability
cites work: Q4049667
cites work: Q3742442
cites work: Q5690471
full work available at URL: s0019-3577(04)80010-7
OpenAlex ID: W2077014107
description: scientific article; zbMATH DE number 2124781