Continuous branching processes and spectral positivity

Bingham, Ν. H.

doi:10.1016/0304-4149(76)90011-9

Cited by 86 publications

(78 citation statements)

References 25 publications

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“…From the relation (2.11) we know that 19) and hence the properties of continuity, almost everywhere differentiability and strict monotonicity carry over to the case q > 0.…”

Section: Scale Functions and The Excursion Measurementioning

confidence: 99%

“…(In fact the same bijection characterises all the continuous-state branching processes with monotone non-decreasing paths when X is replaced by a subordinator). A classic result due to Bingham [19] gives (under very mild conditions) the law of the maximum of the continuous-state branching process with non-monotone paths as follows. For all x ≥ y > 0,…”

Section: Scale Functions and Applied Probabilitymentioning

confidence: 99%

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The Theory of Scale Functions for Spectrally Negative Lévy Processes

2012

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The purpose of this review article is to give an up to date account of the theory and application of scale functions for spectrally negative Lévy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Lévy processes, in particular a reasonable understanding of the Lévy-Khintchine formula and its relationship to the Lévy-Itô decomposition. We shall also touch on more general topics such as excursion theory and semi-martingale calculus. However, wherever possible, we shall try to focus on key ideas taking a selective stance on the technical details. For the reader who is less familiar with some of the mathematical theories and techniques which are used at various points in this review, we note that all the necessary technical background can be found in the following texts on Lévy processes;

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“…From the relation (2.11) we know that 19) and hence the properties of continuity, almost everywhere differentiability and strict monotonicity carry over to the case q > 0.…”

Section: Scale Functions and The Excursion Measurementioning

confidence: 99%

Section: Scale Functions and Applied Probabilitymentioning

confidence: 99%

The Theory of Scale Functions for Spectrally Negative Lévy Processes

2012

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“…From the PSSMP Lamperti representation of (Y, P x ) and Proposition 4, we deduce that T 0 = x α−1 I . From [3], it is known that Continuous-state branching processes and self-similarity 1153 Therefore, u t (∞) = (c + (α − 1)t) −1/(α−1) and, hence,…”

Section: Lemma 1 the Distribution Ofmentioning

confidence: 99%

Continuous-State Branching Processes and Self-Similarity

Kyprianou

Pardo

2008

J. Appl. Probab.

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In this paper we study the α-stable continuous-state branching processes (for α ∈ (1, 2]) and the α-stable continuous-state branching processes conditioned never to become extinct in the light of positive self-similarity. Understanding the interaction of the Lamperti transformation for continuous-state branching processes and the Lamperti transformation for positive, self-similar Markov processes gives access to a number of explicit results concerning the paths of α-stable continuous-state branching processes and α-stable continuous-state branching processes conditioned never to become extinct.

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“…Such classes of processes have been introduced by Jirina [14] and studied by many authors included Bingham [3], Grey [11], Grimvall [12], Lamperti [19; 20], to name but a few. A continuous state branching process Y = (Y t , t ≥ 0) is a Markov process taking values in [0, ∞], where 0 and ∞ are two absorbing states.…”

Section: Cb-and Cbi-processesmentioning

confidence: 99%

Continuous-State Branching Processes and Self-Similarity

Kyprianou

Pardo

2008

Journal of Applied Probability

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In this paper we study the α-stable continuous-state branching processes (for α ∈ (1, 2]) and the latter process conditioned never to become extinct in the light of positive self-similarity. Understanding the interaction of the Lamperti transformation for continuous state branching processes and the Lamperti transformation for positive self-similar Markov processes gives access to a number of explicit results concerning the paths of α-stable continuousstate branching processes and α-stable continuous-state branching processes conditioned never to become extinct.

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Continuous branching processes and spectral positivity

Cited by 86 publications

References 25 publications

The Theory of Scale Functions for Spectrally Negative Lévy Processes

The Theory of Scale Functions for Spectrally Negative Lévy Processes

Continuous-State Branching Processes and Self-Similarity

Continuous-State Branching Processes and Self-Similarity

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