Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Affine processes on $\mathbb{R}_{+}^{m}\times\mathbb{R}^{n}$ and multiparameter time changes

M. Emilia Caballero, José Luis Pérez Garmendia, and Gerónimo Uribe Bravo

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We present a time change construction of affine processes with state-space $\mathbb{R}_{+}^{m}\times\mathbb{R}^{n}$. These processes were systematically studied in (Ann. Appl. Probab. 13 (2003) 984–1053) since they gather interesting classes of processes such as Lévy processes, continuous-state branching processes with immigration, and of the Ornstein–Uhlenbeck type. The construction is based on a (basically) continuous functional of a multidimensional Lévy process which implies that limit theorems for Lévy processes (both almost surely and in distribution) can be inherited to affine processes. The construction can be interpreted as a multiparameter time change scheme or as a (random) ordinary differential equation driven by discontinuous functions. In particular, we propose approximation schemes for affine processes based on the Euler method for solving the associated discontinuous ODEs, which are shown to converge.


Nous présentons une construction des processus affines à valeurs dans $\mathbb{R}_{+}^{m}\times\mathbb{R}^{n}$ à partir de changement de temps. Ces processus ont été systématiquement étudiés dans (Ann. Appl. Probab. 13 (2003) 984–1053) car ils regroupent certaines classes intéressantes de processus tels que les processus de Lévy, les processus de branchement continu avec immigration et du type Ornstein–Uhlenbeck. La construction se base sur une fonctionnelle (presque) continue d’un processus de Lévy multidimensionnel, ce qui implique que les théorèmes limites pour les processus de Lévy (que ce soit presque sûrement ou en loi) peuvent être transmis aux processus affines. La construction peut être interprétée comme un changement de temps à plusieurs paramètres ou comme une équation différentielle ordinaire aléatoire dirigée par des fonctions discontinues. En particulier, on propose des schémas d’approximation pour les processus affines basés sur la méthode d’Euler pour résoudre les EDO discontinues associées, dont la convergence est démontrée.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1280-1304.

Received: 6 March 2015
Revised: 16 March 2016
Accepted: 20 March 2016
First available in Project Euclid: 21 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F17: Functional limit theorems; invariance principles

Lévy processes Continuous-state branching processes with immigration Ornstein–Uhlenbeck processes Multiparameter time change


Caballero, M. Emilia; Pérez Garmendia, José Luis; Uribe Bravo, Gerónimo. Affine processes on $\mathbb{R}_{+}^{m}\times\mathbb{R}^{n}$ and multiparameter time changes. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1280--1304. doi:10.1214/16-AIHP755.

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