# 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.
Document type :
Journal articles
Domain :

https://hal-ecp.archives-ouvertes.fr/hal-00734421
Contributor : Erick Herbin <>
Submitted on : Friday, September 21, 2012 - 5:23:30 PM
Last modification on : Wednesday, April 8, 2020 - 3:58:10 PM

### Citation

Paul Balança, Erick Herbin. A set-indexed Ornstein-Uhlenbeck process. Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2012, 17 (39), pp.1-14. ⟨10.1214/ECP.v17-1903⟩. ⟨hal-00734421⟩

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