Tail behavior of Markov-modulated generalized Ornstein-Uhlenbeck processes

TL;DR

This paper investigates the tail behavior of solutions to Markov-modulated stochastic differential equations, extending renewal theory and Kesten's work to analyze stationary distributions in various applied probability contexts.

Contribution

It introduces a novel approach combining Markov renewal theory with affine function systems to analyze tail behavior of Markov-modulated Ornstein-Uhlenbeck processes.

Findings

Characterizes tail behavior of stationary distributions.

Extends Goldie's implicit renewal theory to Markov-modulated systems.

Provides tools applicable in finance, queueing, and population dynamics.

Abstract

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.