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March 2016 Markov-modulated Ornstein-Uhlenbeck processes
G. Huang, H. M. Jansen, M. Mandjes, P. Spreij, K. De Turck
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Adv. in Appl. Probab. 48(1): 235-254 (March 2016).

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.

Citation

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G. Huang. H. M. Jansen. M. Mandjes. P. Spreij. K. De Turck. "Markov-modulated Ornstein-Uhlenbeck processes." Adv. in Appl. Probab. 48 (1) 235 - 254, March 2016.

Information

Published: March 2016
First available in Project Euclid: 8 March 2016

zbMATH: 1337.60191
MathSciNet: MR3473576

Subjects:
Primary: 60K25
Secondary: 60G15, 60G44

Rights: Copyright © 2016 Applied Probability Trust

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Vol.48 • No. 1 • March 2016
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