{
  "annotation_id": "97a354d1-3270-436e-bc5e-b7664e50d307",
  "date_created": "2026-02-17T05:53:20.106000Z",
  "date_modified": "2026-02-17T05:53:20.106000Z",
  "file_hash": "f778d53598442d79ca11de0d6b0c1605f6183fbec858dd94a8bc4cda7a743240",
  "private": false,
  "record": {
    "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\u0027s implicit renewal theory~\\cite{Goldie:91} and can be seen as an adaptation of Kesten\u0027s 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.",
    "arxiv_id": "2601.09314",
    "authors": [
      "Gerold Alsmeyer",
      "Anita Behme"
    ],
    "categories": [
      "math.PR"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Tail behavior of Markov-modulated generalized Ornstein-Uhlenbeck processes",
    "url": "https://arxiv.org/abs/2601.09314",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "81b879a2-e984-49b4-bb4d-8d6b6b4b035f",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}