By embedding a Markov-modulated random recurrence equation in continuous time, we derive the Markovmodulated 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 Markovadditive 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.
|Number of pages||31|
|Journal||Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability|
|Publication status||Published - May 2022|
DFG Classification of Subject Areas according to Review Boards
ASJC Scopus subject areas
- Exponential functional, Generalized Ornstein-Uhlenbeck process, Lévy process, Markov additive process, Markov-modulated random recurrence equation, Markov-switching model, Risk theory, Ruin probability, Stationary process