Hostname: page-component-7dd5485656-gs9qr Total loading time: 0 Render date: 2025-10-31T08:40:35.223Z Has data issue: false hasContentIssue false
  • English
  • Français

Markov-modulated Ornstein–Uhlenbeck processes

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  24 March 2016

G. Huang ,
H. M. Jansen ,
M. Mandjes ,
P. Spreij  and
K. De Turck
G. Huang*
Affiliation:
University of Amsterdam
H. M. Jansen*
Affiliation:
University of Amsterdam and Ghent University
M. Mandjes*
Affiliation:
University of Amsterdam, CWI and EURANDOM
P. Spreij*
Affiliation:
University of Amsterdam
K. De Turck*
Affiliation:
Ghent University
*
* Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands.
* Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands.
* Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands.
* Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands.
****** Postal address: TELIN, Ghent University, St.-Pietersnieuwstraat 41, B9000 Gent, Belgium. Email address: kdeturck@telin.ugent.be
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider an Ornstein–Uhlenbeck (OU) process (M(t))t≥0 whose parameters are determined by an external Markov process (X(t))t≥0 on a finite state space {1, . . ., d}; this process is usually referred to as Markov-modulated Ornstein–Uhlenbeck. We use stochastic integration theory to determine explicit expressions for the mean and variance of M(t). Then we establish a system of partial differential equations (PDEs) for the Laplace transform of M(t) and the state X(t) of the background process, jointly for time epochs t = t1, . . ., tK. Then we use this PDE to set up a recursion that yields all moments of M(t) and its stationary counterpart; we also find an expression for the covariance between M(t) and M(t + u). We then establish a functional central limit theorem for M(t) for the situation that certain parameters of the underlying OU processes are scaled, in combination with the modulating Markov process being accelerated; interestingly, specific scalings lead to drastically different limiting processes. We conclude the paper by considering the situation of a single Markov process modulating multiple OU processes.

Keywords

Ornstein–Uhlenbeck processMarkov modulationregime switchingcentral limit theoremsmartingale techniques

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60G44: Martingales with continuous parameter 60G15: Gaussian processes

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 1 , March 2016 , pp. 235 - 254
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Alili, L., Patie, P. and Pedersen, J. L. (2005). Representations of the first hitting time density of an Ornstein–Uhlenbeck process. Stoch. Models 21, 967–980. Google Scholar
[2]Anderson, D., Blom, J., Mandjes, M., Thorsdottir, H. and De Turck, K. (2014). A functional central limit theorem for a Markov-modulated infinite-server queue. To appear in Methodology Comput. Appl. Prob.Google Scholar
[3]Ang, A. and Bekaert, G. (2002). Regime switches in interest rates. J. Business Econom. Statist. 20, 163–182. CrossRefGoogle Scholar
[4]Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York. Google Scholar
[5]Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. World Scientific, Hackensack, NJ. CrossRefGoogle Scholar
[6]Banachewicz, K., Lucas, A. and van der Vaart, A. (2008). Modelling portfolio defaults using hidden Markov models with covariates. Econometrics J. 11, 155–171. Google Scholar
[7]Blom, J. and Mandjes, M. (2013). A large-deviations analysis of Markov-modulated inifinite-server queues. Operat. Res. Lett. 41, 220–225. Google Scholar
[8]Blom, J., De Turck, K. and Mandjes, M. (2013). Rare event analysis of Markov-modulated infinite-server queues: a Poisson limit. Stoch. Models 29, 463–474. Google Scholar
[9]Blom, J., De Turck, K. and Mandjes, M. (2015). Analysis of Markov-modulated infinite-server queues in the central-limit regime. Prob. Eng. Inf. Sci. 29, 433–459. Google Scholar
[10]Blom, J., Mandjes, M. and Thorsdottir, H. (2013). Time-scaling limits for Markov-modulated infinite-server queues. Stoch. Models 29, 112–127. CrossRefGoogle Scholar
[11]Blom, J., De Turck, K., Kella, O. and Mandjes, M. (2014). Tail asymptotics of a Markov-modulated infinite-server queue. Queueing Systems 78, 337–357. CrossRefGoogle Scholar
[12]Blom, J., Kella, O., Mandjes, M. and Thorsdottir, H. (2014). Markov-modulated infinite-server queues with general service times. Queueing Systems 76, 403–424. CrossRefGoogle Scholar
[13]Coolen-Schrijner, P. and van Doorn, E. A. (2002). The deviation matrix of a continuous-time Markov chain. Prob. Eng. Inf. Sci. 16, 351–366. Google Scholar
[14]D'Auria, B. (2008). M/M/∞ queues in semi-Markovian random environment. Queueing Systems 58, 221–237. Google Scholar
[15]Elliott, R. J. and Mamon, R. S. (2002). An interest rate model with a Markovian mean reverting level. Quant. Finance 2, 454–458. CrossRefGoogle Scholar
[16]Elliott, R. J. and Siu, T. K. (2009). On Markov-modulated exponential-affine bond price formulae. Appl. Math. Finance 16, 1–15. Google Scholar
[17]Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. John Wiley, New York. CrossRefGoogle Scholar
[18]Fralix, B. H. and Adan, I. J. B. F. (2009). An infinite-server queue influenced by a semi-Markovian environment. Queueing Systems 61, 65–84. CrossRefGoogle Scholar
[19]Giampieri, G., Davis, M. and Crowder, M. (2005). Analysis of default data using hidden Markov models. Quant. Finance 5, 27–34. Google Scholar
[20]Guyon, X., Iovleff, S. and Yao, J.-F. (2004). Linear diffusion with stationary switching regime. ESAIM Prob. Statist. 8, 25–35. CrossRefGoogle Scholar
[21]Hale, J. K. (1980). Ordinary Differential Equations, 2nd edn. Krieger, Huntington, NY. Google Scholar
[22]Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384. CrossRefGoogle Scholar
[23]Horn, R. A. and Johnson, C. R. (1985). Matrix Analysis. Cambridge University Press. CrossRefGoogle Scholar
[24]Huang, G., Mandjes, M. and Spreij, P. (2014). Weak convergence of Markov-modulated diffusion processes with rapid switching. Statist. Prob. Lett. 86, 74–79. CrossRefGoogle Scholar
[25]Huang, G.et al. (2014). Markov-modulated Ornstein–Uhlenbeck processes. Preprint. Available at http://arxiv.org/abs/1412.7952v1. Google Scholar
[26]Jacobsen, M. (1996). Laplace and the origin of the Ornstein–Uhlenbeck process. Bernoulli 2, 271–286. Google Scholar
[27]Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin. Google Scholar
[28]Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York. Google Scholar
[29]Keilson, J. (1979). Markov Chain Models—Rarity and Exponentiality. Springer, New York. Google Scholar
[30]Kemeny, J. G. and Snell, J. L. (1960). Finite Markov Chains. Van Nostrand, Princeton, NJ. Google Scholar
[31]Neuts, M. F. (1981). Matrix–Geometric Solutions in Stochastic Models: An Algorithmic Approach. John Hopkins University Press, Baltimore, MD. Google Scholar
[32]O'Cinneide, C. A. and Purdue, P. (1986). The M/M/∞ queue in a random environment. J. Appl. Prob. 23, 175–184. Google Scholar
[33]Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin. Google Scholar
[34]Robert, P. (2003). Stochastic Networks and Queues. Springer, Berlin. CrossRefGoogle Scholar
[35]Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales, Vol. 2, Itô Calculus, 2nd edn. Cambridge University Press. Google Scholar
[36]Syski, R. (1978). Ergodic potential. Stoch. Process. Appl. 7, 311–336. Google Scholar
[37]Uhlenbeck, G. E. and Ornstein, L. S. (1930). On the theory of Brownian motion. Phys. Rev. 36, 823–841. Google Scholar
[38]Williams, D. (1991). Probability with Martingales. Cambridge University Press. Google Scholar
[39]Xing, X., Zhang, W. and Wang, Y. (2009). The stationary distributions of two classes of reflected Ornstein–Uhlenbeck processes. J. Appl. Prob. 46, 709–720. Google Scholar
[40]Yuan, C. and Mao, X. (2003). Asymptotic stability in distribution of stochastic differential equations with Markovian switching. Stoch. Process. Appl. 103, 277–291. Google Scholar