Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-05T04:21:39.313Z Has data issue: false hasContentIssue false
  • English
  • Français

The Stationary Distributions of Two Classes of Reflected Ornstein–Uhlenbeck Processes

Part of: Mathematical economics

Published online by Cambridge University Press:  14 July 2016

Xiaoyu Xing ,
Wei Zhang  and
Yongjin Wang
Xiaoyu Xing*
Affiliation:
Hebei University of Technology
Wei Zhang*
Affiliation:
Central South University
Yongjin Wang*
Affiliation:
Nankai University
*
∗Postal address: School of Sciences, Hebei University of Technology, Beichen District, Tianjin, 300401, P. R. China. Email address: [email protected]
∗∗Postal address: School of Mathematical Sciences, Central South University, Changsha, 410075, P. R. China.
∗∗∗Postal address: School of Business, Nankai University, Tianjin, 300071, P. R. China.
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.

In this paper we consider two classes of reflected Ornstein–Uhlenbeck (OU) processes: the reflected OU process with jumps and the Markov-modulated reflected OU process. We prove that their stationary distributions exist. Furthermore, for the jump reflected OU process, the Laplace transform (LT) of the stationary distribution is given. As for the Markov-modulated reflected OU process, we derive an equation satisfied by the LT of the stationary distribution.

Keywords

Ornstein–Uhlenbeck processreflected processMarkov-modulated processstationary distribution

MSC classification

Secondary: 91B70: Stochastic models
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 3 , September 2009 , pp. 709 - 720
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Supported by NSF 10671052.

References

[1] Applebaum, D. (2004). Lévy Processes and Stochastic Calculus. Cambridge University Press.CrossRefGoogle Scholar
[2] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
[3] De Saporta, B. and Yao, J.-F. (2005). Tail of a linear diffusion with Markov switching. Ann. Appl. Prob. 15, 992–1018.CrossRefGoogle Scholar
[4] Golub, G. H. and Van Loan, C. F. (1983). Matrix Computations. Johns Hopkins University Press, Baltimore, MD.Google Scholar
[5] Guyon, X., Iovleff, S. and Yao, J.-F. (2004). Linear diffusion with stationary switching regime. ESAIM Prob. Statist. 8, 25–35.CrossRefGoogle Scholar
[6] Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. John Wiley, New York.Google Scholar
[7] Novikov, A. A. (2004). Martingales and first-exit times for the Ornstein–Uhlenbeck process with Jumps. Theory Prob. Appl. 48, 288–303.CrossRefGoogle Scholar
[8] Oberhettinger, F. and Badii, L. (1973). Tables of Laplace Transforms. Springer, New York.CrossRefGoogle Scholar
[9] Patie, P. (2005). On a martingale associated to generalized Ornstein–Uhlenbeck processes and an application to finance. Stoch. Process. Appl. 115, 593–607.CrossRefGoogle Scholar
[10] Srikant, R. and Whitt, W. (1996). Simulation run lengths to estimate blocking probabilities. ACM Trans. Model. Comput. Simul. 6, 7–52.CrossRefGoogle Scholar
[11] Ward, A. R. and Glynn, P. W. (2003). A diffusion approximation for a Markovian queue with reneging. Queueing Systems 43, 103–128.CrossRefGoogle Scholar
[12] Ward, A. R. and Glynn, P. W. (2003). Properties of the reflected Ornstein–Uhlenbeck process. Queueing Systems 44, 109–123.CrossRefGoogle Scholar
[13] Ward, A. R. and Glynn, P. W. (2005). A diffusion approximation for a GI/GI/1 queue with balking or reneging. Queueing Systems 50, 371–400.CrossRefGoogle Scholar