An increment type set-indexed Markov property

A set-indexed Ornstein-Uhlenbeck process

“…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…”

“…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.…”