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

TL;DR

This paper introduces a Markov-modulated generalized Ornstein-Uhlenbeck process, providing explicit solutions, conditions for stationarity, and an application in risk theory to model stochastic investments and calculate ruin probabilities.

Contribution

It derives a new class of Markov-modulated Ornstein-Uhlenbeck processes, explicitly solves their stochastic differential equations, and applies them to risk modeling with stochastic investment environments.

Findings

Explicit solution in terms of Markov-additive processes

Necessary and sufficient conditions for strict stationarity

Ruin probability formula in Markov-modulated risk model

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.