Markov-modulated generalized Ornstein-Uhlenbeck processes and an application in risk theory

Abstract: We derive the Markov-modulated generalized Ornstein-Uhlenbeck process by embedding a Markov-modulated random recurrence equation in continuous time. The obtained process turns out to be the unique solution of a certain stochastic differential equation driven by a bivariate Markov-additive process. We present this stochastic differential equation as well as its solution explicitely in terms of the driving Markov-additive process. Moreover, we give necessary and sufficient conditions for strict stationarity of the Markov-modulated generalized Ornstein-Uhlenbeck process, and prove that its stationary distribution is given by the distribution of a specific exponential functional of Markov-additive processes. Finally we propose an application of the Markov-modulated generalized Ornstein-Uhlenbeck process as Markov-modulated risk model with stochastic investment. This generalizes Paulsen's risk process to a Markov-switching environment. We derive a formula in this risk model that expresses the ruin probability in terms of the distribution of an exponential functional of a Markov-additive process.