Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Deep factorisation of the stable process II: Potentials and applications

Andreas E. Kyprianou, Victor Rivero, and Batı Şengül

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Abstract

Here, we propose a different perspective of the deep factorisation in (Electron. J. Probab. 21 (2016) Paper No. 23, 28) based on determining potentials. Indeed, we factorise the inverse of the MAP-exponent associated to a stable process via the Lamperti–Kiu transform. Here our factorisation is completely independent from the derivation in (Electron. J. Probab. 21 (2016) Paper No. 23, 28), moreover there is no clear way to invert the factors in (Electron. J. Probab. 21 (2016) Paper No. 23, 28) to derive our results. Our method gives direct access to the potential densities of the ascending and descending ladder MAP of the Lamperti-stable MAP in closed form.

In the spirit of the interplay between the classical Wiener–Hopf factorisation and the fluctuation theory of the underlying Lévy process, our analysis will produce a collection of new results for stable processes. We give an identity for the law of the point of closest reach to the origin for a stable process with index $\alpha\in(0,1)$ as well as an identity for the the law of the point of furthest reach before absorption at the origin for a stable process with index $\alpha\in(1,2)$. Moreover, we show how the deep factorisation allows us to compute explicitly the limiting distribution of stable processes multiplicatively reflected in such a way that it remains in the strip $[-1,1]$.

Résumé

On propose une perspective différente à la factorisation du type Wiener–Hopf, dite deep factorisation en anglais, obtenue dans (Electron. J. Probab. 21 (2016) Paper No. 23, 28), qui consiste en une factorisation de la matrice exposant caractéristique du processus de Markov additif (MAP) associé à un processus stable via la transformation de Lamperti–Kiu. Ici on décrit les mesures potentiel, au lieu de la mesure de Lévy, la dérive et le terme de mort. Les méthodes utilisés ici sont complètement différentes de celles de (Electron. J. Probab. 21 (2016) Paper No. 23, 28), ceci est dû, d’un part, au fait qu’il n’y a pas de méthode claire pour inverser les facteurs apparaissant dans cette référence, et, d’autre part, nos méthodes nous permettent d’obtenir explicitement les mesures potentiel des processus d’échelle croissant et décroissant.

D’une manière analogue à la conjonction entre la factorisation de Wiener–Hopf et la théorie des fluctuations des processus de Lévy, notre analyse nous permet de produire une collection de résultats nouveaux pour les processus stables. On donne une identité pour la loi du point le plus proche de l’origine pour un processus stable d’indice $\alpha\in(0,1)$, ainsi qu’une identité pour la loi du point le plus lointain avant le premier temps d’atteinte de zéro pour un processus stable d’indice $\alpha\in(1,2)$. De plus, nos résultats nous permettent de calculer explicitement la limite en loi du processus stable réfléchi multiplicativement de telle sorte à rester dans l’intervalle $[-1,1]$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 1 (2018), 343-362.

Dates
Received: 23 November 2015
Revised: 18 August 2016
Accepted: 14 November 2016
First available in Project Euclid: 19 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1519030831

Digital Object Identifier
doi:10.1214/16-AIHP806

Mathematical Reviews number (MathSciNet)
MR3765892

Zentralblatt MATH identifier
06880057

Subjects
Primary: 60G18: Self-similar processes 60G52: Stable processes 60G51: Processes with independent increments; Lévy processes

Keywords
Stable processes Self-similar Markov processes Wiener–Hopf factorisation Radial reflection

Citation

Kyprianou, Andreas E.; Rivero, Victor; Şengül, Batı. Deep factorisation of the stable process II: Potentials and applications. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 1, 343--362. doi:10.1214/16-AIHP806. https://projecteuclid.org/euclid.aihp/1519030831


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