Markov-modulated Ornstein-Uhlenbeck processes

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 nite state space {1, . . . , d}; this process is usually referred to as Markov-modulated Ornstein-Uhlenbeck (or: mmou). We use stochastic integration theory to determine explicit expressions for the mean and variance of M(t). Then we establish a system of partial di erential equations (pde s) 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 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 ou processes are scaled, in combination with the modulating Markov process being accelerated; interestingly, speci c scalings lead to drastically di erent limiting processes. We conclude the paper by considering the situation of a single Markov process modulating multiple ou processes.