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Tail behavior of Markov-modulated generalized Ornstein-Uhlenbeck processes

Gerold Alsmeyer, Anita Behme

TL;DR

This work extends Goldie’s implicit renewal framework to Markov-modulated affine recursions, deriving Pareto-type tails for the stationary distribution of MMGOU processes driven by a finite-state MAP. By constructing a measure-change and employing MRW renewal theory, the authors characterize the tail exponent $\kappa$ as the unique positive solution to $\rho(\kappa)=1$ for the Cramér-transformed map and provide explicit tail constants $C_i^{\pm}$ in terms of MAP characteristics and eigenstructure. The results are obtained via a two-pronged strategy: (i) an implicit renewal theorem in finite Markovian environments for MMLIFS, and (ii) a discretization/aggregation approach that links continuous-time MMGOU tails to discrete-time perpetuity-type recursions. The findings unify and generalize previous unmodulated cases, with broad implications for queueing, econometrics, finance, and population dynamics where regime-switching stochastic dynamics are relevant.

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 $ Ψ_{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 := Ψ_{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.

Tail behavior of Markov-modulated generalized Ornstein-Uhlenbeck processes

TL;DR

This work extends Goldie’s implicit renewal framework to Markov-modulated affine recursions, deriving Pareto-type tails for the stationary distribution of MMGOU processes driven by a finite-state MAP. By constructing a measure-change and employing MRW renewal theory, the authors characterize the tail exponent as the unique positive solution to for the Cramér-transformed map and provide explicit tail constants in terms of MAP characteristics and eigenstructure. The results are obtained via a two-pronged strategy: (i) an implicit renewal theorem in finite Markovian environments for MMLIFS, and (ii) a discretization/aggregation approach that links continuous-time MMGOU tails to discrete-time perpetuity-type recursions. The findings unify and generalize previous unmodulated cases, with broad implications for queueing, econometrics, finance, and population dynamics where regime-switching stochastic dynamics are relevant.

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 , , and the associated iterated function system defined recursively by and for , . 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.
Paper Structure (14 sections, 13 theorems, 212 equations)

This paper contains 14 sections, 13 theorems, 212 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. An implicit renewal theorem in finite Markovian environment
  4. Prerequisites
  5. Time-reversal and duality
  6. Induced Markov random walk and its dual
  7. The Cramér transforms of $P$ and its dual $\widehat{P}$
  8. Measure change
  9. Lattice-type of $(\xi_{n},S_{n})_{n\geqslant 0}$ and its dual
  10. Condition (B1): Making life easier by geometric sampling
  11. Subsampling at hitting times
  12. Exponential smoothing
  13. Proof of Theorem \ref{['thm:main_1']}
  14. Proof of Theorem \ref{['thm:main_2']}

Key Result

Theorem 2.1

Let $(J,(\zeta,\eta))$ be a bivariate MAP such that the modulating process $J$ has finite state space $\mathscr{S}$ and is ergodic with stationary distribution $\pi$. Assume that $\eta\not\equiv 0$ and there exists no sequence $\{c_j,\,j\in\mathscr{S}\}$ such that eq-MMGOUdegenerate holds. If all Le Then there exists a nondegenerate random variable $V_0$, which is conditionally independent of $(J,

Theorems & Definitions (27)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 4.1
  • Proposition 4.2
  • ...and 17 more