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

Markov-modulated Ornstein-Uhlenbeck processes. (English) Zbl 1337.60191

Adv. Appl. Probab. 48, No. 1, 235-254 (2016).
Summary: In this paper, we consider an Ornstein-Uhlenbeck (OU) process \((M(t))_{t \geq 0}\) whose parameters are determined by an external Markov process \((X(t))_{t \geq 0}\) on a finite state space \(\{1,\ldots, d\}\); 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 \(t = t_{1},\ldots, t_{K}\). 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.

Cited in 11 Documents

MSC:

60J60 Diffusion processes
60J27 Continuous-time Markov processes on discrete state spaces
60F17 Functional limit theorems; invariance principles
60F05 Central limit and other weak theorems
60G44 Martingales with continuous parameter
60G15 Gaussian processes

Keywords:

Ornstein-Uhlenbeck processes; Markov modulation; regime switching; functional central limit theorem; martingale techniques
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\textit{G. Huang} et al., Adv. Appl. Probab. 48, No. 1, 235--254 (2016; Zbl 1337.60191)
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