Markov-modulated Ornstein–Uhlenbeck processes

Abstract

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 = t 1 , . . ., 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.