A set-indexed Ornstein-Uhlenbeck process

Paul Balança,Erick Herbin

doi:10.1214/ecp.v17-1903

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 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.

Highlights

The study of multiparameter processes goes back to the 70’ and the theory developed for years covers multiple properties of random fields
Using the Markov kernel obtained, we are able to introduce in Definition 3.1 a general set-indexed Ornstein-Uhlenbeck process, whose law is consistent with the covariance structure (1.3) in particular case of initial Dirac distributions
We prove that the stationary set-indexed Ornstein-Uhlenbeck process (ssiOU) process satisfies the C-Markov property introduced in [6]

Summary

Introduction

The study of multiparameter processes goes back to the 70’ and the theory developed for years covers multiple properties of random fields (we refer to the recent books [22] and [2] for a modern review). To the case of martingales, different interesting Markov properties can be formalized for multiparameter processes. Extending the literature on random fields, several different subjects have been recently investigated, including set-indexed martingales [21], setindexed Markov [20, 5, 6] and Levy processes [1, 7, 19], and set-indexed fractional Brownian motion [17, 18]. Using the Markov kernel obtained, we are able to introduce in Definition 3.1 a general set-indexed Ornstein-Uhlenbeck process, whose law is consistent with the covariance structure (1.3) in particular case of initial Dirac distributions.

Definition and first properties

Definition of a general set-indexed Ornstein-Uhlenbeck process

Multiparameter Ornstein-Uhlenbeck process