Comptes Rendus
Probability Theory
Tail of a linear diffusion with Markov switching
Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 643-646.

Let Y be a Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic Markov jump process X, i.e. dYt=a(Xt)Ytdt+σ(Xt)dWt, Y0=y0. 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 : dYt=a(Xt)Ytdt+σ(Xt)dWt, Y0=y0. 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:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.09.022
Benoîte de Saporta 1; Jian-Feng Yao 1

1 IRMAR, université de Rennes I, campus de Beaulieu, 35042 Rennes cedex, France
@article{CRMATH_2004__339_9_643_0,
     author = {Beno{\^\i}te de Saporta and Jian-Feng Yao},
     title = {Tail of a linear diffusion with {Markov} switching},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {643--646},
     publisher = {Elsevier},
     volume = {339},
     number = {9},
     year = {2004},
     doi = {10.1016/j.crma.2004.09.022},
     language = {en},
}
TY  - JOUR
AU  - Benoîte de Saporta
AU  - Jian-Feng Yao
TI  - Tail of a linear diffusion with Markov switching
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 643
EP  - 646
VL  - 339
IS  - 9
PB  - Elsevier
DO  - 10.1016/j.crma.2004.09.022
LA  - en
ID  - CRMATH_2004__339_9_643_0
ER  - 
%0 Journal Article
%A Benoîte de Saporta
%A Jian-Feng Yao
%T Tail of a linear diffusion with Markov switching
%J Comptes Rendus. Mathématique
%D 2004
%P 643-646
%V 339
%N 9
%I Elsevier
%R 10.1016/j.crma.2004.09.022
%G en
%F CRMATH_2004__339_9_643_0
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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.09.022/

[1] G.K. Basak; A. Bisi; M.K. Ghosh Stability of random diffusion with linear drift, J. Math. Anal. Appl., Volume 202 (1996), pp. 604-622

[2] C.M. Goldie Implicit renewal theory and tails of solutions of random equations, Ann. Appl. Probab., Volume 1 (1991), pp. 26-166

[3] X. Guyon; S. Iovleff; J.-F. Yao Linear diffusion with stationary switching regime, ESAIM Probability and Statistics, Volume 8 (2004), pp. 25-35

[4] J.D. Hamilton Estimation, inference and forecasting of time series subject to change in regime (G. Maddala; C.R. Rao; D.H. Vinod, eds.), Handbook of Statistics, vol. 11, 1993, pp. 230-260

[5] H. Kesten Random difference equations and renewal theory for products of random matrices, Acta Math., Volume 131 (1973), pp. 207-248

[6] E. Le Page, Théorèmes de renouvellement pour les produits de matrices aléatoires. Equations aux différences aléatoires, Séminaires de probabilités de Rennes, 1983

[7] B. de Saporta Renewal theorem for a system of renewal equations, Ann. Inst. H. Poincaré Probab. Statist., Volume 39 (2003), pp. 823-838

Cited by Sources:

Comments - Policy


Articles of potential interest

Tail of the stationary solution of the stochastic equation Yn+1=anYn+bn with Markovian coefficients

Benoîte de Saporta

C. R. Math (2005)


On the multidimensional stochastic equation Yn+1=AnYn+Bn

Benoîte de Saporta; Yves Guivarc'h; Emile Le Page

C. R. Math (2004)