Deep Factorisation of the Stable Process III: the View from Radial Excursion Theory and the Point of Closest Reach
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Deep Factorisation of the Stable Process III: the View from Radial Excursion Theory and the Point of Closest Reach Andreas E. Kyprianou1
· Victor Rivero2 · Weerapat Satitkanitkul1,3
Received: 7 May 2018 / Accepted: 18 October 2019 / © The Author(s) 2019
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. ˙ Keywords Stable processes · L´evy processes · Excursion theory · Riesz–Bogdan–Zak transform · Lamperti–Kiu transform Mathematics Subject Classification (2010) Primary: 60G18 · 60G52 ; Secondary: 60G51 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 Andreas E. Kyprianou
[email protected] Victor Rivero [email protected] Weerapat Satitkanitkul [email protected] 1
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK
2
CIMAT A. C., Calle Jalisco s/n, Col. Valenciana, A. P. 402, C.P. 36000, Guanajuato, Gto., Mexico
3
Facult´e des Sciences Bˆatiment I, D´epartement de math´ematiques, Universit´e d’Angers, 2 Boulevard Lavoisier, 49045 Angers cedex 01, France
A.E. Kyprianou et al.
1 Introduction and Main Results For d ≥ 1, let X := (Xt : t ≥ 0), with probabilities Px , x ∈ Rd , be a d-dimensional isotropic stable process of index α ∈ (0, 2). That is to say that X is a Rd -valued L´evy process having characteristic triplet (0, 0, ), where 2α ((d + α)/2) 1 (B) = d/2 dy, B ∈ B (R). (1.1) π |(−α/2)| B |y|α+d Equivalently, this means X is a d-dimensional L´evy process with characteristic exponent (θ ) = − log E0 (eiθX1 ) which satisfies (θ) = |θ |α ,
θ ∈ R.
Stable processes are also self-similar in the sense that they satisfy a scaling property. More precisely, for c > 0 and x ∈ Rd \ {0}, under Px , the law of (cXc−α t , t ≥ 0) is equal to Pcx .
(1.2)
As such, stable processes are useful for the study of the class of general L´evy processes and, more recently, for the study of the class of self-similar Marko
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