Tail of a linear diffusion with Markov switching

Benoîte de Saporta; Jian-Feng Yao

doi:10.1016/j.crma.2004.09.022

Probability Theory

Tail of a linear diffusion with Markov switching

Presented by: Marc Yor

Benoîte de Saporta ¹ ; Jian-Feng Yao ¹

¹ IRMAR, université de Rennes I, campus de Beaulieu, 35042 Rennes cedex, France

Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 643-646

Abstract (Original language)
Résumé

Let Y be a Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic Markov jump process X, i.e. $d Y_{t} = a (X_{t}) Y_{t} d t + σ (X_{t}) d W_{t}$ , $Y_{0} = y_{0}$ . Ergodicity conditions for Y have been obtained. Here we investigate the tail property of the stationary distribution of this model. A characterization of the only two possible cases is established: light tail or polynomial tail. Our method is based on discretizations and renewal theory.

Soit Y une diffusion de Ornstein–Uhlenbeck dirigée par un processus Markovien de saut X stationnaire et ergodique : $d Y_{t} = a (X_{t}) Y_{t} d t + σ (X_{t}) d W_{t}$ , $Y_{0} = y_{0}$ . On connaît des conditions d'ergodicité pour Y. Ici on s'intéresse à la queue de la loi stationnaire de ce modèle. Par des méthodes de discrétisation et de renouvellement, on donne une caractérisation complète des deux seuls cas possibles : queue polynômiale ou existence de moment à tout ordre.

Received: 2004-01-08
Accepted: 2004-09-21
Published online: 2004-10-27

DOI: 10.1016/j.crma.2004.09.022

Author's affiliations:

Benoîte de Saporta ¹ ; Jian-Feng Yao ¹

¹ IRMAR, université de Rennes I, campus de Beaulieu, 35042 Rennes cedex, France

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     title = {Tail of a linear diffusion with {Markov} switching},
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Benoîte de Saporta; Jian-Feng Yao. Tail of a linear diffusion with Markov switching. Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 643-646. doi: 10.1016/j.crma.2004.09.022

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Cited by

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