Tail behavior of Markov-modulated generalized Ornstein-Uhlenbeck processes

We study the tail behavior of Markov-modulated generalized Ornstein-Uhlenbeck processes -- that is, solutions to Langevin-type stochastic differential equations driven by a background continuous-time Markov chain. To this end, we consider a sequence of Markov modulated random affine functions $\Psi_{n} : \mathbb{R} \to \mathbb{R}$, $n \in \mathbb{N}$, and the associated iterated function system defined recursively by $X_0^x := x$ and $X_{n}^x := \Psi_{n-1}(X_{n-1}^x)$ for $x \in \mathbb{R}$, $n \in \mathbb{N}$. We analyze the tail behavior of the stationary distribution of such a Markov chain using tools from Markov renewal theory. Our approach extends Goldie's implicit renewal theory~\cite{Goldie:91} and can be seen as an adaptation of Kesten's work on products of random matrices~\cite{Kesten:73} to the one-dimensional setting of random affine function systems. These results have applications in diverse areas of applied probability, including queueing theory, econometrics, mathematical finance, and population dynamics.