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

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