Stable processes conditioned to hit an interval continuously from the outside

Döring, Leif; Weissmann, Philip

doi:10.3150/19-bej1134

Cited by 5 publications

(3 citation statements)

References 18 publications

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“…There are multiple examples of conditioning a univariate Lévy process in some limiting sense, which alternatively can be described by Doob h-transforms, see [4,12,11] and references therein. Most often the focus is on establishing properties directly related to these conditional processes.…”

Section: Introductionmentioning

confidence: 99%

Lévy processes conditioned to stay in a half-space with applications to directional extremes

Ivanovs¹,

Thøstesen²

2022

Modern Stochastics: Theory and Applications

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This paper provides a multivariate extension of Bertoin’s pathwise construction of a Lévy process conditioned to stay positive or negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged.

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Section: Introductionmentioning

confidence: 99%

Lévy processes conditioned to stay in a half-space with applications to directional extremes

Ivanovs¹,

Thøstesen²

2022

Modern Stochastics: Theory and Applications

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show abstract

“…There are multiple examples of conditioning a univariate Lévy process in some limiting sense, which alternatively can be described by Doob h-transforms, see [4,11,12] and references therein. Most often the focus is on establishing properties directly related to these conditional processes.…”

Section: Introductionmentioning

confidence: 99%

Lévy processes conditioned to stay in a half-space with applications to directional extremes

Ivanovs¹,

Thøstesen²

2021

Preprint

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This paper provides a multivariate extension of Bertoin's pathwise construction of a Lévy process conditioned to stay positive/negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged.

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“…Our results naturally complement those of the recent paper [16], which considers a similar type of conditioning, albeit requiring the stable process to additionally remain either inside or outside of the unit ball. Other related works include [13], who considered a real valued Lévy process conditioned to continuously approach the boundary of the interval [−1, 1] from the outside.…”

Section: Introductionmentioning

confidence: 99%

Oscillatory attraction and repulsion from a subset of the unit sphere or hyperplane for isotropic stable Lévy processes

Mateusz¹,

Kyprianou²,

Palau³

et al. 2020

Preprint

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Suppose that S is a closed set of the unit sphere S d−1 = {x ∈ R d : |x| = 1} in dimension d ≥ 2, which has positive surface measure. We construct the law of absorption of an isotropic stable Lévy process in dimension d ≥ 2 conditioned to approach S continuously, allowing for the interior and exterior of S d−1 to be visited infinitely often. Additionally, we show that this process is in duality with the underlying stable Lévy process. We can replicate the aforementioned results by similar ones in the setting that S is replaced by D, a closed bounded subset of the hyperplane {x ∈ R d : (x, v) = 0} with positive surface measure, where v is the unit orthogonal vector and where (•, •) is the usual Euclidean inner product. Our results complement similar results of the authors [16] in which the stable process was further constrained to attract to and repel from S from either the exterior or the interior of the unit sphere.

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Stable processes conditioned to hit an interval continuously from the outside

Cited by 5 publications

References 18 publications

Lévy processes conditioned to stay in a half-space with applications to directional extremes

Lévy processes conditioned to stay in a half-space with applications to directional extremes

Lévy processes conditioned to stay in a half-space with applications to directional extremes

Oscillatory attraction and repulsion from a subset of the unit sphere or hyperplane for isotropic stable Lévy processes

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