Implicit renewal theory for exponential functionals of Lévy processes

Arista, Jonas; Rivero, Víctor

doi:10.48550/arxiv.1510.01809

Cited by 5 publications

(9 citation statements)

References 54 publications

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“…The form of the density follows from Theorem 1 of Arista and Rivero [3]. We finish our proof by observing that The asymptotic behaviour of the positive moments of Z t has been studied in Palau et al [29] using a fine study of the negative moments of exponential functional of Lévy processes.…”

Section: Population Model With Competition In a Lévy Random Environmentmentioning

confidence: 65%

“…It is not difficult to see that Z is a weak solution to (3). In order to prove our result, we consider two solutions to (3), Z ′ and Z ′′ , and consider τ ′ p = inf{t ≥ 0 :…”

Section: Stochastic Differential Equationsmentioning

confidence: 92%

“…Since after this time both processes are equal to ∞, we obtain that Z ′ and Z ′′ are indistinguishable. In other words, there is a unique strong solution to (3). (see [31, p.104]).…”

Section: Stochastic Differential Equationsmentioning

confidence: 99%

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Branching processes in a Lévy random environment

Palau¹,

Pardo²

2015

Preprint

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In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by Brownian motions and Poisson random measures which are mutually independent. The existence and uniqueness of strong solutions are established under some general conditions that allows us to consider the case when the strong solution explodes at a finite time. We use the latter result to construct continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a stochastic differential equation. We also study some properties of such processes that extends recent results obtained by Bansaye et al. in (Electron.

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Section: Population Model With Competition In a Lévy Random Environmentmentioning

confidence: 65%

“…It is not difficult to see that Z is a weak solution to (3). In order to prove our result, we consider two solutions to (3), Z ′ and Z ′′ , and consider τ ′ p = inf{t ≥ 0 :…”

Section: Stochastic Differential Equationsmentioning

confidence: 92%

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Branching processes in a Lévy random environment

Palau¹,

Pardo²

2015

Preprint

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“…Let Π − α be the Lévy measure of |α|η − . The behaviour of P − 1 (I ≤ t) as t → 0 is given in Theorem 7 of [1]. More precisely, provided that Π…”

Section: Behaviour Of the Expectationmentioning

confidence: 99%

“…(recall that η + (0) = 0 under P + 1 ). Proposition 2 in [10] gives the finiteness and an expression for the negative moments of I η + , under the assumption that e η + (1) positive moments of all order. However, the proof straightforwardly adapts to E + 1 (I…”

Section: The Upper Boundmentioning

confidence: 99%

Intrinsic area near the origin for self-similar growth-fragmentations and related random surfaces

Ged¹

2019

Preprint

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We study the behaviour of a natural measure defined on the leaves of the genealogical tree of some branching processes, namely self-similar growth-fragmentation processes. Each particle, or cell, is attributed a positive mass that evolves in continuous time according to a positive self-similar Markov process and gives birth to children at negative jumps events. We are interested in the asymptotics of the mass of the ball centered at the root, as its radius decreases to 0. We obtain the almost sure behaviour of this mass when the Eve cell starts with a strictly positive size. This differs from the situation where the Eve cell grows indefinitely from size 0. In this case, we show that, when properly rescaled, the mass of the ball converges in distribution towards a non-degenerate random variable. We then derive bounds describing the almost sure behaviour of the rescaled mass. Those results are applied to certain random surfaces, exploiting the connection between growth-fragmentations and random planar maps obtained in [6]. This allows us to extend a result of Le Gall [24] on the volume of a free Brownian disk close to its boundary, to a larger family of stable disks. The upper bound of the mass of a typical ball in the Brownian map is refined, and we obtain a lower bound as well.

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Martingales in self-similar growth-fragmentations and their connections with random planar maps

Bertoin

Budd

Curien

et al. 2017

Probab. Theory Relat. Fields

195

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The purpose of the present work is twofold. First, we develop the theory of general self-similar growth-fragmentation processes by focusing on martingales which appear naturally in this setting and by recasting classical results for branching random walks in this framework. In particular, we establish many-to-one formulas for growth-fragmentations and define the notion of intrinsic area of a growth-fragmentation. Second, we identify a distinguished family of growth-fragmentations closely related to stable Lévy processes, which are then shown to arise as the scaling limit of the perimeter process in Markovian explorations of certain random planar maps with large degrees (which are, roughly speaking, the dual maps of the stable maps of Le Gall & Miermont [36]). As a consequence of this result, we are able to identify the law of the intrinsic area of these distinguished growthfragmentations. This generalizes a geometric connection between large Boltzmann triangulations and a certain growth-fragmentation process, which was established in [8].

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Implicit renewal theory for exponential functionals of Lévy processes

Cited by 5 publications

References 54 publications

Branching processes in a Lévy random environment

Branching processes in a Lévy random environment

Intrinsic area near the origin for self-similar growth-fragmentations and related random surfaces

Martingales in self-similar growth-fragmentations and their connections with random planar maps

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