Entrance and exit at infinity for stable jump diffusions

Doering, Leif; Kyprianou, Andreas E.

doi:10.48550/arxiv.1802.01672

Cited by 5 publications

(7 citation statements)

References 43 publications

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“…Actually, the above condition implies that the associated CB-process with competition in a Lévy random environment comes down from infinity. This phenomenon has been observed and studied by several authors in branching processes with interactions, see for instance González-Casanova et al [12], Lambert [18], Li [23], Li et al [24] and Pardoux [28] and also for stable jump diffusions by Döring and Kyprianuo [9] and some jump diffusions by Bansaye [4]. Formally, we define the property of coming down from infinity in the sense that ∞ is a continuous entrance point, i.e.…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

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Extinction and coming down from infinity of CB-processes with competition in a Lévy environment

Leman¹,

Pardo²

2018

Preprint

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In this note, we are interested on the event of extinction and the property of coming down from infinity of continuous state branching (or CB for short) processes with competition in a Lévy environment whose branching mechanism satisfies the so-called Grey's condition. In particular, we deduce, under the assumption that the Lévy environment does not drift towards infinity, that for any starting point the process becomes extinct in finite time a.s. Moreover if we impose an integrability condition on the competition mechanism, then the process comes down from infinity regardless the long term behaviour of the environment.

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Section: Introduction and Main Resultsmentioning

confidence: 97%

“…where T M = inf{t ≥ 0 : Z t ≤ M } and the original process can be extended into a Feller process on [0, ∞] (see for instance Theorem 20.13 in Kallenberg [15] for the diffusion case or Definition 2.2 for Feller processes in [9]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Extinction and coming down from infinity of CB-processes with competition in a Lévy environment

Leman¹,

Pardo²

2018

Preprint

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“…and the original process can be extended into a Feller process on [0, ∞] (see for instance Theorem 20.13 in Kallenberg [11] for the diffusion case or Definition 2.2 for Feller processes in [4]).…”

Section: General Casementioning

confidence: 99%

Extinction time of logistic branching processes in a Brownian environment

Leman¹,

Pardo²

2019

Preprint

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In this paper, we study the extinction time of logistic branching processes which are perturbed by an independent random environment driven by a Brownian motion. Our arguments use a Lamperti-type representation which is interesting on its own right and provides a one to one correspondence between the latter family of processes and the family of Feller diffusions which are perturbed by an independent spectrally positive Lévy process. When the independent random perturbation (of the Feller diffusion) is driven by a subordinator then the logistic branching processes in a Brownian environment converges to a specified distribution; otherwise, it becomes extinct a.s. In the latter scenario, and following a similar approach to Lambert [14], we provide the expectation and the Laplace transform of the absorption time, as a functional of the solution to a Ricatti differential equation. In particular, the latter characterises the law of the process coming down from infinity.

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“…Indeed, the authors showed that the killing time is finite almost surely and the left-limit at the killing time is 0. Applications of the conditioned processes have been found for instance in the study of entrance and exit at infinity of stochastic differential equations driven by stable processes, see Döring and Kyprianou [7].…”

Section: Introductionmentioning

confidence: 99%