Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T05:58:31.142Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Exponential Ergodicity of Lévy-Driven Ornstein–Uhlenbeck Processes

Part of: Stochastic analysis Stochastic processes Markov processes

Published online by Cambridge University Press:  30 January 2018

Jian Wang
Jian Wang*
Affiliation:
Fujian Normal University
*
∗ Postal address: School of Mathematics and Computer Science, Fujian Normal University, 350007, Fuzhou, P. R. China. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Based on the explicit coupling property, the ergodicity and the exponential ergodicity of Lévy-driven Ornstein–Uhlenbeck processes are established.

Keywords

Lévy processOrnstein–Uhlenbeck processcoupling propertyexponential ergodicity

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 60J75: Jump processes 60G51: Processes with independent increments; L'evy processes
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 4 , December 2012 , pp. 990 - 1004
Copyright
© Applied Probability Trust 

References

Applebaum, D. (2005). Lévy Processes and Stochastic Calculus. Cambridge University Press.Google Scholar
Bodnarchuk, S. V. and Kulik, O. M. (2009). Conditions for the existence and smoothness of the distribution density for an Ornstein-Uhlenbeck process with Lévy noise. Theoret. Prob. Math. Statist. 79, 23–38.Google Scholar
Böttcher, B., Schilling, R. L. and Wang, J. (2011). Constructions of coupling processes for Lévy processes. Stoch. Process. Appl. 121, 1201–1216.Google Scholar
Cerrai, S. (2001). Second Order PDE's in Finite and Infinite Dimension (Lecture Notes Math. 1762). Springer, Berlin.Google Scholar
Masuda, H. (2004). On multidimensional Ornstein-Uhlenbeck processes driven by a general Lévy process. Bernoulli 10, 97–120.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
Priola, E. and Zabczyk, J. (2009). Densities for Ornstein-Uhlenbeck processes with Jumps. Bull. London Math. Soc. 41, 41–50.Google Scholar
Priola, E. and Zabczyk, J. (2009). On linear evolution equations with cylindrical Lévy noise. Preprint. Available at http://arxiv.org/abs/0908.0356v1.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Sato, K.-I. and Yamazato, M. (1984). Operator-self-decomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type. Stoch. Process. Appl. 17, 73–100.Google Scholar
Schilling, R. L. and Wang, J. (2011). On the coupling property of Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 47, 1147–1159.Google Scholar
Schilling, R. L. and Wang, J. (2012). On the coupling property and the Liouville theorem for Ornstein-Uhlenbeck processes. J. Evol. Equat. 12, 119–140.CrossRefGoogle Scholar
Schilling, R. L., Sztonyk, P. and Wang, J. (2012). Coupling property and gradient estimates of Lévy processes via symbol. Bernoulli 18, 1128–1149.CrossRefGoogle Scholar
Simon, T. (2011). On the absolute continuity of multidimensional Ornstein-Uhlenbeck processes. Prob. Theory Relat. Fields 151, 173–190.CrossRefGoogle Scholar
Stettner, Ł. (1994). Remarks on ergodic conditions for Markov processes on Polish spaces. Bull. Polish Acad. Sci. Math. 42, 103–114.Google Scholar
Wang, F.-Y. (2011). Coupling for Ornstein-Uhlenbeck Jump processes with Jumps. Bernoulli 17, 1136–1158.Google Scholar
Wang, F.-Y. (2011). Gradient estimate for Ornstein-Uhlenbeck Jump processes. Stoch. Process. Appl. 121, 466–478.CrossRefGoogle Scholar
Wang, F.-Y. and Wang, J. (2012). Coupling and strong Feller for Jump processes on Banach spaces. Preprint. Available at http://arxiv.org/abs/1111.3795v2.Google Scholar