Wasserstein perturbations of Markovian transition semigroups

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2023
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Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques = Annales de l'Institut Henri Poincaré (B) Probability and Statistics. Institute of Mathematical Statistics. 2023, 59(2), pp. 904-932. ISSN 0246-0203. eISSN 1778-7017. Available under: doi: 10.1214/22-aihp1270
Zusammenfassung

In this paper, we deal with a class of time-homogeneous continuous-time Markov processes with transition probabilities bearing a nonparametric uncertainty. The uncertainty is modelled by considering perturbations of the transition probabilities within a proximity in Wasserstein distance. As a limit over progressively finer time periods, on which the level of uncertainty scales proportionally, we obtain a convex semigroup satisfying a nonlinear PDE in a viscosity sense. A remarkable observation is that, in standard situations, the nonlinear transition operators arising from nonparametric uncertainty coincide with the ones related to parametric drift uncertainty. On the level of the generator, the uncertainty is reflected as an additive perturbation in terms of a convex functional of first order derivatives. We additionally provide sensitivity bounds for the convex semigroup relative to the reference model. The results are illustrated with Wasserstein perturbations of Lévy processes, infinite-dimensional Ornstein–Uhlenbeck processes, geometric Brownian motions, and Koopman semigroups.

Zusammenfassung in einer weiteren Sprache

Dans cet article, nous traitons d’une classe de processus de Markov à temps continu homogène dans le temps avec des probabilités de transition portant une incertitude non paramétrique. L’incertitude est modélisée en considérant des perturbations de probabilités de transition proches en distance de Wasserstein. Comme limite sur des périodes de temps de plus en plus fines, sur lesquelles le niveau d’incertitude s’étend proportionnellement, nous obtenons un semigroupe convexe satisfaisant une EDP non linéaire dans un sens de viscosité. Une observation remarquable est que, dans des situations standards, les opérateurs de transition non linéaires découlant de l’incertitude non paramétrique coïncident avec ceux liés à l’incertitude paramétrique de dérive. Au niveau du générateur, l’incertitude se traduit par une perturbation additive en termes d’une fonction convexe de dérivées de premier ordre. Nous fournissons en outre des bornes de sensibilité pour le semigroupe convexe par rapport au modèle de référence. Les résultats sont illustrés par les perturbations de Wasserstein des processus de Lévy, les processus d’Ornstein–Uhlenbeck de dimension infinie, les mouvements browniens géométriques et les semigroupes de Koopman.

Fachgebiet (DDC)
510 Mathematik
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Convex semigroup, Markov process, nonlinear PDE, Nonparametric uncertainty, viscosity solution, Wasserstein distance
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Zitieren
ISO 690FUHRMANN, Sven, Michael KUPPER, Max NENDEL, 2023. Wasserstein perturbations of Markovian transition semigroups. In: Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques = Annales de l'Institut Henri Poincaré (B) Probability and Statistics. Institute of Mathematical Statistics. 2023, 59(2), pp. 904-932. ISSN 0246-0203. eISSN 1778-7017. Available under: doi: 10.1214/22-aihp1270
BibTex
@article{Fuhrmann2023Wasse-67068,
  year={2023},
  doi={10.1214/22-aihp1270},
  title={Wasserstein perturbations of Markovian transition semigroups},
  number={2},
  volume={59},
  issn={0246-0203},
  journal={Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques = Annales de l'Institut Henri Poincaré (B) Probability and Statistics},
  pages={904--932},
  author={Fuhrmann, Sven and Kupper, Michael and Nendel, Max}
}
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