By embedding a Markov-modulated random recurrence equation in continuous time, we derive the Markov-modulated generalized Ornstein-Uhlenbeck process. This process turns out to be the unique solution of a 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 certain exponential functional of Markov-additive processes. Finally, we propose a Markov-modulated risk model with investment that generalizes Paulsen’s risk process to a Markov-switching environment, and derive a formula for the ruin probability in this risk model.
The authors thank Paolo Di Tella for comments on an earlier draft of this paper that lead to improvements of the manuscript. They also thank two anonymous referees for their helpful comments.
"Markov-modulated generalized Ornstein-Uhlenbeck processes and an application in risk theory." Bernoulli 28 (2) 1309 - 1339, May 2022. https://doi.org/10.3150/21-BEJ1389