Deep Factorisation of the Stable Process III: the View from Radial Excursion Theory and the Point of Closest Reach

Abstract

We compute explicitly the distribution of the point of closest reach to the origin in the path of any d-dimensional isotropic stable process, with d ≥ 2. Moreover, we develop a new radial excursion theory, from which we push the classical Blumenthal–Getoor–Ray identities for first entry/exit into a ball (cf. Blumenthal et al. Trans. Amer. Math. Soc., 99, 540–554 1961) into the more complex setting of n-tuple laws for overshoots and undershoots. We identify explicitly the stationary distribution of any d-dimensional isotropic stable process when reflected in its running radial supremum. Finally, for such processes, and as consequence of some of the analysis of the aforesaid, we provide a representation of the Wiener–Hopf factorisation of the MAP that underlies the stable process through the Lamperti–Kiu transform. Our analysis continues in the spirit of Kyprianou (Ann. Appl. Probab., 20(2), 522–564 2010) and Kyprianou et al. (2015) in that our methodology is largely based around treating stable processes as self-similar Markov processes and, accordingly, taking advantage of their Lamperti-Kiu decomposition.

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Acknowledgements

The authors would like to thank Ron Doney who pointed out the distributional interpretations in Remarks 1.1 and 1.5. We would also like to thank two anonymous referees who provided two extremely helpful and thorough reports on an earlier version of this paper.

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Correspondence to Andreas E. Kyprianou.

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Victor Rivero was supported by EPSRC grant EP/M001784/1. Andreas E. Kyprianou was supported by EPSRC grant EP/L002442/1 and EP/M001784/1

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Kyprianou, A.E., Rivero, V. & Satitkanitkul, W. Deep Factorisation of the Stable Process III: the View from Radial Excursion Theory and the Point of Closest Reach. Potential Anal 53, 1347–1375 (2020). https://doi.org/10.1007/s11118-019-09809-4

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Keywords

  • Stable processes
  • Lévy processes
  • Excursion theory
  • Riesz–Bogdan–Żak transform
  • Lamperti–Kiu transform

Mathematics Subject Classification (2010)

  • Primary: 60G18
  • 60G52
  • Secondary: 60G51