Conditioned real self-similar Markov processes

Kyprianou, Andreas E.; Rivero, Víctor; Satitkanitkul, Weerapat

doi:10.48550/arxiv.1510.01781

Cited by 8 publications

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“…The reader again notes that, for d ≥ 2, the change of measure (59) rewards paths that remain close to the origin and penalises those that stray far from the origin. Just as in Kyprianou et al (2015) it can be shown that P •…”

Section: I3 Stable Processes As Self-similar Markov Processesmentioning

confidence: 87%

“…Stable processes killed on entering (−∞, 0). This first example was introduced in detail in Caballero and Chaumont (2006a); see also Kyprianou et al (2015). To some extent, the former of these two references marks the beginning of the modern treatment of stable processes through the theory of self-similar Markov processes.…”

Section: I2 Positive Self-similar Markov Processes and Stable Processesmentioning

confidence: 98%

“…This comes about by a change of measure, which corresponds to a Doob h-transform to the semigroup of a two-sided jumping stable process killed on first hitting the origin if α ∈ (1, 2). As it is instructive for future discussion, we give a proof of the following result which is originally from Chaumont et al (2013), for α ∈ (1, 2), and Kyprianou et al (2015), for α ∈ (1, 2). Our proof differs slightly from its original setting.…”

Section: I3 Stable Processes As Self-similar Markov Processesmentioning

confidence: 99%

“…It penalises those paths that approach the origin and rewards those that stray away from the origin. In fact, it has been shown in Kyprianou et al (2015) that, for α ∈ (0, 1), in the appropriate sense, the change of measure is equivalent to conditioning the stable process to continuously absorb at the origin, and when α ∈ (1, 2), in Chaumont et al (2013); Kuznetsov et al (2014) it is shown that the change of measure is equivalent to conditioning the stable process to avoid the origin. I.3.4.…”

Section: I3 Stable Processes As Self-similar Markov Processesmentioning

confidence: 99%

“…With only the sporadic works of e.g. ; Vuolle-Apiala and ; Kiu (1980) in the 1980s, only very recently has the bigger picture begun to emerge some 30 or more years later in Xiao (1998); Nane et al (2010); Chaumont et al (2013); Kuznetsov et al (2014); Alili et al (2017); Kyprianou et al (2015); Kyprianou (2016); Kyprianou et al (2016a,b).…”

Section: Onward Researchmentioning

confidence: 99%

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Stable Lévy processes, self-similarity and the unit ball

Kyprianou¹

2018

ALEA

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Section: I3 Stable Processes As Self-similar Markov Processesmentioning

confidence: 87%

Section: I2 Positive Self-similar Markov Processes and Stable Processesmentioning

confidence: 98%

Section: I3 Stable Processes As Self-similar Markov Processesmentioning

confidence: 99%

Section: I3 Stable Processes As Self-similar Markov Processesmentioning

confidence: 99%

Section: Onward Researchmentioning

confidence: 99%

See 3 more Smart Citations

Stable Lévy processes, self-similarity and the unit ball

Kyprianou¹

2018

ALEA

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

2019

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In this paper, we continue our understanding of the stable process from the perspective of the theory of self-similar Markov processes in the spirit of [10,13]. In particular, we turn our attention to the case of d-dimensional isotropic stable process, for d ≥ 2. Using a completely new approach we consider the distribution of the point of closest reach. This leads us to a number of other substantial new results for this class of stable processes. We engage with a new radial excursion theory, never before used, from which we develop the classical Blumenthal-Getoor-Ray identities for first entry/exit into a ball, cf. [3], to the setting of n-tuple laws. We identify explicitly the stationary distribution of the stable process when reflected in its running radial supremum. Moreover, we provide a representation of the Wiener-Hopf factorisation of the MAP that underlies the stable process through the Lamperti-Kiu transform.where the two terms either side of the multiplication sign constitute the two Wiener-Hopf factors. See e.g. Chapter VI in [2] for background. Recall that if Ψ is the characteristic exponent of any Lévy process, then there exist two Bernstein functions κ andκ (see [18] for a definition) such that, up to a multiplicative constant,Identity (1.7) is what we refer to as the Wiener-Hopf factorisation. The left-hand factor codes the range of the running maximum and the right-hand factor codes the range of the running 2 infimum of ξ. It can be checked that both belong to the class of so-called beta subordinators (see [8], as well as some of the discussion later in this paper) and, in particular, have infinite activity. This implies that ξ is regular for both the upper and lower half-lines, which in turn, means that any sphere of radius r > 0 is regular for both its interior and exterior for X. This and the fact that X has càdlàg paths ensures that, denotingthe quantity X G(t) is well defined as the point of closest reach to the origin up to time t in the sense that X G(t)− = X G(t) andThe process (G(t), t ≥ 0) is monotone increasing and hence there is no problem defining G(∞) = lim t→∞ G(t) almost surely. Moreover, as X is transient in the sense of (1.3), it is also clear that, almost surely, G(∞) = G(t) for all t sufficiently large and thatOur first main result provides explicitly the law of X G(∞) .Theorem 1.1 (Point of Closest Reach to the origin). The law of the point of closest reach to the origin is given by P x (X G(∞) ∈ dy) = π −d/2 Γ (d/2) 2 Γ ((d − α)/2) Γ (α/2) (|x| 2 − |y| 2 ) α/2 |x − y| d |y| α dy, 0 < |y| < |x|.

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Stable windings at the origin

Kyprianou

Vakeroudis

2018

Stochastic Processes and their Applications

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Conditioned real self-similar Markov processes

Cited by 8 publications

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Stable Lévy processes, self-similarity and the unit ball

Stable Lévy processes, self-similarity and the unit ball

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

Stable windings at the origin

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