Markov-modulated Ornstein-Uhlenbeck processes

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, . . ., 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.