Entrance and exit at infinity for stable jump diffusions

Leif Doering, Andreas Kyprianou

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8 Citations (SciVal)

Abstract

In his seminal work from the 1950s, William Feller classified all onedimensional diffusions on -∞ ≤ a < b≤∞ in terms of their ability to access the boundary (Feller's test for explosions) and to enter the interior from the boundary. Feller's technique is restricted to diffusion processes as the corresponding differential generators allow explicit computations and the use of Hille-Yosida theory. In the present article, we study exit and entrance from infinity for the most natural generalization, that is, jump diffusions of the form dZt = σ(Z t-)dX t, driven by stable Lévy processes for α ε (0, 2). Many results have been proved for jump diffusions, employing a variety of techniques developed after Feller's work but exit and entrance from infinite boundaries has long remained open. We show that the presence of jumps implies features not seen in the diffusive setting without drift. Finite time explosion is possible for α ε (0, 1), whereas entrance from different kinds of infinity is possible for α ε [1, 2). Accordingly, we derive necessary and sufficient conditions on s. Our proofs are based on very recent developments for path transformations of stable processes via the Lamperti-Kiu representation and new Wiener-Hopf factorisations for Lévy processes that lie therein. The arguments draw together original and intricate applications of results using the Riesz-Bogdan- Z˙ak transformation, entrance laws for self-similar Markov processes, perpetual integrals of Lévy processes and fluctuation theory, which have not been used before in the SDE setting, thereby allowing us to employ classical theory such as Hunt-Nagasawa duality and Getoor's characterisation of transience and recurrence.

Original languageEnglish
Pages (from-to)1220-1265
Number of pages46
JournalAnnals of Probability
Volume48
Issue number3
Early online date17 Jun 2020
DOIs
Publication statusPublished - 30 Jun 2020

Bibliographical note

Funding Information:
Acknowledgments. Both authors would like to thank Jean Bertoin for introducing them to the problem. Part of this work was carried out when the AEK was on sabbatical at ETH Zürich and he would like thank the Forschungsinstitut für Mathematik for its hospitality. Both authors are especially grateful to an anonymous referee who read an initial version of this manuscript with great care, offering very helpful comments that led to its improvement. We would also like to thank Cyril Labbé for discussions. The first author was supported by Ambizione Grant of the Swiss Science Foundation.

Funding Information:
The second author was supported by EPSRC Grants EP/L002442/1 and EP/M001784/1.

Publisher Copyright:
© Institute of Mathematical Statistics, 2020.

Keywords

  • Duality
  • Entrance
  • Explosion
  • Kelvin transform
  • SDEs
  • Stable lévy processes

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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