Markov-modulated Ornstein-Uhlenbeck processes
Résumé
In this paper we consider an Ornstein-Uhlenbeck () process (M (t)) t0 whose parameters are determined by an external Markov process (X(t)) t0 on a nite state space {1,. .. , d}; this process is usually referred to as Markov-modulated Ornstein-Uhlenbeck (or:). We use stochastic integration theory to determine explicit expressions for the mean and variance of M (t). Then we establish a system of partial diierential equations (s) for the Laplace transform of M (t) and the state X(t) of the background process, jointly for time epochs t = t 1 ,. .. , t K. Then we use this to set up a recursion that yields all moments of M (t) and its stationary counterpart; we also nd an expression for the covariance between M (t) and M (t + u). We then establish a central limit theorem for M (t) for the situation that certain parameters of the underlying processes are scaled, in combination with the modulating Markov process being accelerated; interestingly, speciic scalings lead to drastically diierent limiting processes. We conclude the paper by considering the situation of a single Markov process modulating multiple processes.
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