Deep Factorisation of the Stable Process III: the View from Radial Excursion Theory and the Point of Closest Reach
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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.
KeywordsStable processes Lévy processes Excursion theory Riesz–Bogdan–Żak transform Lamperti–Kiu transform
Mathematics Subject Classification (2010)Primary: 60G18 60G52 Secondary: 60G51
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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