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May 2022 Markov-modulated generalized Ornstein-Uhlenbeck processes and an application in risk theory
Anita Behme, Apostolos Sideris
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Bernoulli 28(2): 1309-1339 (May 2022). DOI: 10.3150/21-BEJ1389

Abstract

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.

Acknowledgements

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.

Citation

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Anita Behme. Apostolos Sideris. "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

Information

Received: 1 December 2020; Revised: 1 July 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

Digital Object Identifier: 10.3150/21-BEJ1389

Keywords: 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

Rights: Copyright © 2022 ISI/BS

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Vol.28 • No. 2 • May 2022
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