An Increment-Type Set-Indexed Markov Property

Abstract

We present and study a Markov property, named C-Markov, adapted to processes indexed by a general collection of sets. This new definition fulfils one important expectation for a set-indexed Markov property: there exists a natural generalization of the concept of transition operator which leads to characterization and construction theorems of C-Markov processes. Several usual Markovian notions, including Feller and strong Markov properties, are also developed in this framework. Actually, the C-Markov property turns out to be a natural extension of the two-parameter * -Markov property to the multiparameter and the set-indexed settings. Moreover, extending a classic result of the real-parameter Markov theory, sample paths of multiparameter C-Feller processes are proved to be almost surely right-continuous. Concepts and results presented in this study are illustrated with various examples. AMS 2000 subject classifications: 60G10, 60G15, 60G60, 60J25.

A set-indexed Ornstein-Uhlenbeck process

“…We note that these filtrations are not necessarily outer-continuous. In the following result, we prove that the ssiOU process satisfies the C-Markov property introduced in [6].…”

“…, X U n C , and similarly x C is used for a vector of variables. Thereby, according to [6], the extension ∆X of X on the class C satisfies…”

“…Among these, the most commonly studied ones are sharp-Markov [27,13], germ-Markov [29,30,25] and * -Markov [9,24] properties. We refer to [6] for a more complete description of these concepts. The study multiparameter processes is still a very active area of research, particularly the analysis of sample paths and geometric properties (see e.g.…”

“…Set-indexed processes constitute a natural generalization of multiparameter stochastic processes and their local regularity have been considered in the Gaussian case since the early work of Dudley [14] (see also [1,3,8]). Extending the literature on random fields, several different subjects have been recently investigated, including set-indexed martingales [21], setindexed Markov [20,5,6] and Lévy processes [1,7,19], and set-indexed fractional Brownian motion [17,18]. Although the set-indexed formalism appears to be more abstract, it usually offers a simpler and more condensed way to express technical concepts of multiparameter processes.…”

“…Although the set-indexed formalism appears to be more abstract, it usually offers a simpler and more condensed way to express technical concepts of multiparameter processes. For instance, the present work intensively uses the C-Markov property introduced and developed in [6]. In the latter, the Chapman-Kolmogorov equation related to transition probabilities turns out to be more easily expressed using the set-indexed formalism than the two-parameter framework.…”