Advances in Applied Probability

Markov-modulated Ornstein-Uhlenbeck processes

G. Huang, H. M. Jansen, M. Mandjes, P. Spreij, and K. De Turck

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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.

Article information

Source
Adv. in Appl. Probab., Volume 48, Number 1 (2016), 235-254.

Dates
First available in Project Euclid: 8 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.aap/1457466164

Mathematical Reviews number (MathSciNet)
MR3473576

Zentralblatt MATH identifier
1337.60191

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 60G44: Martingales with continuous parameter 60G15: Gaussian processes

Keywords
Ornstein-Uhlenbeck process Markov modulation regime switching central limit theorems martingale techniques

Citation

Huang, G.; Jansen, H. M.; Mandjes, M.; Spreij, P.; De Turck, K. Markov-modulated Ornstein-Uhlenbeck processes. Adv. in Appl. Probab. 48 (2016), no. 1, 235--254. https://projecteuclid.org/euclid.aap/1457466164


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