Hitting distributions of $\alpha$-stable processes via path censoring and self-similarity

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

doi:10.1214/12-aop790

Cited by 37 publications

(64 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…for α ∈ (0, 1). The first of these probabilities is taken from Corollary 1 of [21] and the second from Corollary 1.2 of [20]. For the general case, no such detailed formulae are available and a different approach is needed.…”

Section: Remarks On the Case Of Stable Processesmentioning

confidence: 99%

Conditioned real self-similar Markov processes

Kyprianou¹,

Rivero²,

Satitkanitkul³

2019

Stochastic Processes and their Applications

Self Cite

View full text Add to dashboard Cite

Section: Remarks On the Case Of Stable Processesmentioning

confidence: 99%

Conditioned real self-similar Markov processes

Kyprianou¹,

Rivero²,

Satitkanitkul³

2019

Stochastic Processes and their Applications

Self Cite

View full text Add to dashboard Cite

“…Note that this process is different from (actually simpler than) Pistorius' doubly reflected process of [34], since here the boundary behaviour at 0 is not reflection, and is also different from Lambert's [28] process confined in a finite interval (conditioned not to exit (0, b)). The discussion of these three cases is connected to the discussion in [10] and [27,Remark 3.3], which distinguish three types of exit at a boundary (A) continuously, (B) by a jump or (C) not at all (while preserving self-similarity of the process). In our context (two boundaries, self-similarity being meaningless on an interval, but re-entry being allowed), Lambert studies (C,C) exits from (0, b), no entry needed, Pistorius studies the (A,B) exit (A,A) entrance, and we find (A,B) exit (B,A) entrance.…”

Section: Uniform Hölder Continuity Of Local Timesmentioning

confidence: 99%

Uniform control of local times of spectrally positive stable processes

Forman¹,

Pal²,

Rizzolo³

et al. 2018

Ann. Appl. Probab.

View full text Add to dashboard Cite

We establish two results about local times of spectrally positive stable processes. The first is a general approximation result, uniform in space and on compact time intervals, in a model where each jump of the stable process may be marked by a random path. The second gives moment control on the Hölder constant of the local times, uniformly across a compact spatial interval and in certain random time intervals. For the latter, we introduce the notion of a Lévy process restricted to a compact interval, which is a variation of Lambert's Lévy process confined in a finite interval and of Pistorius' doubly reflected process. We use the results of this paper to exhibit a class of path-continuous branching processes of Crump-Mode-Jagers type with continuum genealogical structure. A further motivation for this study lies in the construction of diffusion processes in spaces of interval partitions and R-trees, which we explore in forthcoming articles.

show abstract

“…where the last asymptotic relation was obtained by taking the leading term in s. For the denominator, note that where the prefactors are expressed in a form suitable for comparison with the prefactors obtained from other methods [34,45,26,28]. Even though the derivation here is rough, with the key asymptotic result (C.9) obtained using computer algebra software [37], it is important to note that all previous methods have ultimately derived, to the knowledge of the author, from techniques due to Lamperti, either using the theorem due to Skorokhod [25], the formalised Lamperti transform [7], or directly [34,45]; all of which make use of similar intermediate expressions in order to arrive at the prefactor. However, the approach here used elementary techniques on the first-hit distribution along with the free propagator.…”

Section: Discussionmentioning

confidence: 99%

First passage leapovers of Lévy flights and the proper formulation of absorbing boundary conditions

Wardak

2020

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

An important open problem in the theory of Lévy flights concerns the analytically tractable formulation of absorbing boundary conditions. Although numerical studies using the correctly defined nonlocal approach have yielded substantial insights regarding the statistics of first passage, the resultant modifications to the dynamical equations hinder the detailed analysis possible in the absence of these conditions. In this study it is demonstrated that using the first-hit distribution, related to the first passage leapover, as the absorbing sink preserves the tractability of the dynamical equations for a particle undergoing Lévy flight. In particular, knowledge of the first-hit distribution is sufficient to fully determine the first passage time and position density of the particle, without requiring integral truncation or numerical simulations. In addition, we report on the first-hit and leapover properties of first passages and arrivals for Lévy flights of arbitrary skew parameter, and extend these results to Lévy flights in a certain ubiquitous class of potentials satisfying an integral condition.

show abstract

Hitting distributions of $\alpha$-stable processes via path censoring and self-similarity

Cited by 37 publications

References 33 publications

Conditioned real self-similar Markov processes

Conditioned real self-similar Markov processes

Uniform control of local times of spectrally positive stable processes

First passage leapovers of Lévy flights and the proper formulation of absorbing boundary conditions

Contact Info

Product

Resources

About