A set-indexed Ornstein-Uhlenbeck process

Abstract

The purpose of this article is a set-indexed extension of the well-known Ornstein-Uhlenbeck process. The first part is devoted to a stationary definition of the random field and ends up with the proof of a complete characterization by its $L^2$-continuity, stationarity and set-indexed Markov properties. This specific Markov transition system allows to define a general \emph{set-indexed Ornstein-Uhlenbeck (SIOU) process} with any initial probability measure. Finally, in the multiparameter case, the SIOU process is proved to admit a natural integral representation.Comment: 13 page

An Increment-Type Set-Indexed Markov Property

“…As proved by Wang [46], the two-parameter process Y is * -Markov. A calculus similar to Proposition 2.5 from [6] shows that it is also a multiparameter C-Markov process. Moreover, suppose P and P denote the C-transition systems of X and Y , respectively.…”

“…The Gaussian particular case α = 2 is studied in more details in [6]. Its kernel P is characterized by the following transition densities…”

“…Even though the form of the transition probabilities exhibited in Equations (2.23) and (2.24) may not seem very intuitive, Balança and Herbin [6] have adopted a more constructive presentation of the Gaussian Ornstein-Uhlenbeck process which helps to understand our present C-Markov definition. Finally, similarly to the set-indexed Lévy processes, we observe that P is homogeneous when the measure m is compatible with the family of operators (θ U ) U∈A and Feller under the mild condition (2.20).…”

“…Multiparameter Ornstein-Uhlenbeck processes). The set-indexed Gaussian Ornstein-Uhlenbeck process presented in[6] has an integral representation in the multipa-rameter setting. More precisely,∀t ∈ R N + ; X t = e − α,t X 0 + σ (−∞,t]\(−∞,0]e α,u dW u ,…”