Subordinators: Examples and Applications

Bertoin, Jean

doi:10.1007/978-3-540-48115-7_1

Cited by 240 publications

(300 citation statements)

References 120 publications

(144 reference statements)

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“…The condition (2) implies that for each η ∈ (0, 1), there exists a sufficiently large integer n such that…”

Section: Lemma 34 Recall the Definition I (δ)mentioning

confidence: 99%

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Fractal-Dimensional Properties of Subordinators

Barker

2018

J Theor Probab

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This work looks at the box-counting dimension of sets related to subordinators (non-decreasing Lévy processes). It was recently shown in Savov (Electron Commun Probab 19:1-10, 2014) that almost surely lim δ→0 U (δ)N (t, δ) = t, where N (t, δ) is the minimal number of boxes of size at most δ needed to cover a subordinator's range up to time t, and U (δ) is the subordinator's renewal function. Our main result is a central limit theorem (CLT) for N (t, δ), complementing and refining work in Savov (2014). Box-counting dimension is defined in terms of N (t, δ), but for subordinators we prove that it can also be defined using a new process obtained by shortening the original subordinator's jumps of size greater than δ. This new process can be manipulated with remarkable ease in comparison with N (t, δ), and allows better understanding of the box-counting dimension of a subordinator's range in terms of its Lévy measure, improving upon Savov (2014, Corollary 1). Further, we shall prove corresponding CLT and almost sure convergence results for the new process.

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“…The condition (2) implies that for each η ∈ (0, 1), there exists a sufficiently large integer n such that…”

Section: Lemma 34 Recall the Definition I (δ)mentioning

confidence: 99%

“…The limsup and liminf, respectively, define the upper and lower box-counting dimensions. For further background reading, we refer to [1,2] for subordinators, [7,21,23] for Lévy processes, and [9,26] for fractals.…”

Section: Introductionmentioning

confidence: 99%

Fractal-Dimensional Properties of Subordinators

Barker

2018

J Theor Probab

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“…Subordination, i.e. where the chronometer is a Lévy process independent of the base process and the base process is a Lévy process or, more generally, a time-homogeneous Markov process, is a classical area, initiated by Bochner (1949Bochner ( ,1955; some recent references are Bertoin (1996Bertoin ( ,1997, Sato (1999), and Barndorff-Nielsen, Pedersen and Sato (2001). There is a wide range of Lévy processes, obtained by subordination of Brownian motion, which are of interest as models in mathematical finance.…”

Section: Introductionmentioning

confidence: 99%

Infinite Divisibility for Stochastic Processes and Time Change

Barndorff–Nielsen

Maejima

Sato³

2006

J Theor Probab

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General results concerning infinite divisibility, selfdecomposability, and the class L m property as properties of stochastic processes are presented. A new concept called temporal selfdecomposability of stochastic processes is introduced. Lévy processes, additive processes, selfsimilar processes, and stationary processes of Ornstein-Uhlenbeck type are studied in relation to these concepts. In the latter half of the paper time change of stochastic processes is studied, where chronometers (stochastic processes that serve to change time) and base processes (processes to be time-changed) are independent but do not, in general, have independent increments. Conditions for inheritance of infinite divisibility and selfdecomposability under time change are given.

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“…On an entirely different line of research, probabilists developed the notion of stochastic time changes (also called stochastic subordination) as a way of understanding jump processes, see [11], [12], [14]. This work gave rise to a representation of Levy processes, a family of translation invariant jump processes, as subordinated Brownian motions whereby the time change is uncorrelated to the underlying process.…”

Section: Introductionmentioning

confidence: 99%

“…This equation constrains the form of the functions φ(λ) as not all functions admit a one-parameter family of positive measures µ t (ds) such as the equation above is satisfied. The measures µ t (ds), if well-defined, are called the renewal measures [11]. Notice that Bochner subordination also applies to general positivity preserving contraction semigroups R t , as the subordinated semigroupR t can be consistently defined as follows:…”

Section: Introductionmentioning

confidence: 99%