The extended hypergeometric class of Lévy processes

Kyprianou, Andreas E.; Pardo, Juan Carlos; Watson, A. R.

doi:10.1017/s0021900200021409

Cited by 5 publications

(13 citation statements)

References 30 publications

(42 reference statements)

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“…Indeed, following the introduction of the Lamperti-stable Lévy process in [7], for which an explicit Wiener-Hopf factorisation are available, it was quickly discovered that many other explicit Wiener-Hopf factorisations could be found by studying related positive selfsimilar path functionals of stable processes. In part, this stimulated the definition of the class of hypergeometric Lévy processes for which the Wiener-Hopf factorisation is explicit; see [19,21,20]. One might therefore also expect a general class of MAPs to exist, analogous to the class of hypergeometric Lévy processes, for which a matrix factorisation such as the one presented above, is explicitly available.…”

Section: Results On the Deep Factorisation Of Stable Processesmentioning

confidence: 99%

Deep factorisation of the stable process II: Potentials and applications

Kyprianou

Rivero²,

Sengul³

2018

Ann. Inst. H. Poincaré Probab. Statist.

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Here, we propose a different perspective of the deep factorisation in [22] based on determining potentials. Indeed, we factorise the inverse of the MAP-exponent associated to a stable process via the Lamperti-Kiu transform. Here our factorisation is completely independent from the derivation in [22], moreover there is no clear way to invert the factors in [22] to derive our results. Our method gives direct access to the potential densities of the ascending and descending ladder MAP of the Lamperti-stable MAP in closed form.In the spirit of the interplay between the classical Wiener-Hopf factorisation and the fluctuation theory of the underlying Lévy process, our analysis will produce a collection of new results for stable processes. We give an identity for the law of the point of closest reach to the origin for a stable process with index α ∈ (0, 1) as well as an identity for the the law of the point of furthest reach before absorption at the origin for a stable process with index α ∈ (1, 2). Moreover, we show how the deep factorisation allows us to compute explicitly the limiting distribution of stable processes multiplicatively reflected in such a way that it remains in the strip [−1, 1].height processes respectively. The ascending and descending ladder height processes, say (h t : t ≥ 0) and (ĥ t : t ≥ 0), are subordinators that correspond to a time change of X t := sup s≤t X s , t ≥ 0 and −X t := − inf s≤t X s , t ≥ 0, and therefore have the same range, respectively. Additional information comes from the exponents in that they also provide information about the potential measures associated to their respective ladder height processes. So for example, U (dx) := ∞ 0 P(h t ∈ dx)dt, x ≥ 0, has Laplace transform given by 1/κ. A similar identity hold for U , the potential ofĥ.These potential measures appear in a wide variety of fluctuation identities. Perhaps the most classical example concerns the stationary distribution of the process reflected in its maximum, X t − X t , t ≥ 0 in the case that lim t→∞ X t = ∞. In that case, we may take lim t→∞ P x (X t − X t ∈ dx) = κ(0) U (dx); c.f. [29]. The ladder height potential measures also feature heavily in first passage identities that describe the joint law of the triple (X τ +x , X τ + x − , X τ + x − ), where τ + x = inf{t > 0 : X t > x} and x > 0; cf. [14]. Specifically, one has, for u > 0, v ≥ y and y ∈ [0, x],

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Section: Results On the Deep Factorisation Of Stable Processesmentioning

confidence: 99%

Deep factorisation of the stable process II: Potentials and applications

Kyprianou

Rivero²,

Sengul³

2018

Ann. Inst. H. Poincaré Probab. Statist.

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“…For the proof of (i), we refer the reader to [5]. Also, the proof of (ii) can be found in [7]. We will now prove (iii).…”

Section: Proof Of Theorem 21mentioning

confidence: 90%

“…To show ξ is a meromorphic Lévy process, apply [3, Theorem 1(v)] in the killed case and [3, Corollary 2] in the unkilled case. For Lévy processes in the meromorphic class, it is known that their Lévy measure has a density, π, of the form given in (7). We will now compute the coefficients a k ρ k andâ kρk in the representation (7) and accordingly prove that π is equivalent to the expression given in the statement of the theorem.…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

“…We will now review the theory that has been developed for the hypergeometric Lévy processes in [5] and [7] respectively. Note that the former reference refers to their generalisation of the Lévy processes and in the latter reference as the extended hypergeometric class.…”

Section: Introductionmentioning

confidence: 99%

“…Remark 2.2. Note that the parameter ranges Aβ EHG and A β EHG defined in [7,Remark 3] are included in the parameter ranges A 3 and A 4 respectively. Remark 2.3.…”

Section: Introductionmentioning

confidence: 99%

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More on hypergeometric Lévy processes

Horton

Kyprianou

2016

Adv. Appl. Probab.

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Kuznetsov and co-authors in 2011-14 introduced the family of hypergeometric Lévy processes. They appear naturally in the study of fluctuations of stable processes when one analyses stable processes through the theory of positive self-similar Markov processes. Hypergeometric Lévy processes are defined through their characteristic exponent, which, as a complex-valued function, has four independent parameters. In 2014 it was shown that the definition of a hypergeometric Lévy process could be taken to include a greater range of the aforesaid parameters than originally specified. In this short article, we push the parameter range even further.

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