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Relevant bibliographies by topics / Lévy subordinators

Academic literature on the topic 'Lévy subordinators'

Author: Grafiati

Published: 11 March 2023

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Journal articles on the topic "Lévy subordinators"

1

Barndorff-Nielsen, Ole E., Jan Pedersen, and Ken-Iti Sato. "Multivariate subordination, self-decomposability and stability." Advances in Applied Probability 33, no. 1 (March 2001): 160–87. http://dx.doi.org/10.1017/s0001867800010685.

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Multivariate subordinators are multivariate Lévy processes that are increasing in each component. Various examples of multivariate subordinators, of interest for applications, are given. Subordination of Lévy processes with independent components by multivariate subordinators is defined. Multiparameter Lévy processes and their subordination are introduced so that the subordinated processes are multivariate Lévy processes. The relations between the characteristic triplets involved are established. It is shown that operator self-decomposability and the operator version of the class Lm property are inherited from the multivariate subordinator to the subordinated process under the condition of operator stability of the subordinand.

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Sun, Yunpeng, Rafael Mendoza-Arriaga, and Vadim Linetsky. "Marshall–Olkin distributions, subordinators, efficient simulation, and applications to credit risk." Advances in Applied Probability 49, no. 2 (June 2017): 481–514. http://dx.doi.org/10.1017/apr.2017.10.

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Abstract In the paper we present a novel construction of Marshall–Olkin (MO) multivariate exponential distributions of failure times as distributions of the first-passage times of the coordinates of multidimensional Lévy subordinator processes above independent unit-mean exponential random variables. A time-inhomogeneous version is also given that replaces Lévy subordinators with additive subordinators. An attractive feature of MO distributions for applications, such as to portfolio credit risk, is its singular component that yields positive probabilities of simultaneous defaults of multiple obligors, capturing the default clustering phenomenon. The drawback of the original MO fatal shock construction of MO distributions is that it requires one to simulate 2n-1 independent exponential random variables. In practice, the dimensionality is typically on the order of hundreds or thousands of obligors in a large credit portfolio, rendering the MO fatal shock construction infeasible to simulate. The subordinator construction reduces the problem of simulating a rich subclass of MO distributions to simulating an n-dimensional subordinator. When one works with the class of subordinators constructed from independent one-dimensional subordinators with known transition distributions, such as gamma and inverse Gaussian, or their Sato versions in the additive case, the simulation effort is linear in n. To illustrate, we present a simulation of 100,000 samples of a credit portfolio with 1,000 obligors that takes less than 18 seconds on a PC.

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3

Covo, Shai. "One-dimensional distributions of subordinators with upper truncated Lévy measure, and applications." Advances in Applied Probability 41, no. 2 (June 2009): 367–92. http://dx.doi.org/10.1239/aap/1246886616.

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Given a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with continuous Lévy measure ν, we derive a formula for the distribution function Fs (x; t) at time t of the associated subordinator whose Lévy measure is the restriction of ν to (0,s]. It will be expressed in terms of ν and the marginal distribution function F (⋅; t) of the original process. A generalization concerning an arbitrary truncation of ν will follow. Under certain conditions, an analogous formula will be obtained for the nth derivative, ∂nFs (x; t) ∂ xn. The requirement that ν is continuous is shown to have no intrinsic meaning. A number of interesting results involving the size ordered jumps of subordinators will be derived. An appropriate approximation for the small jumps of a gamma process will be considered, leading to a revisiting of the generalized Dickman distribution.

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Covo, Shai. "One-dimensional distributions of subordinators with upper truncated Lévy measure, and applications." Advances in Applied Probability 41, no. 02 (June 2009): 367–92. http://dx.doi.org/10.1017/s0001867800003347.

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Given a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with continuous Lévy measure ν, we derive a formula for the distribution functionFs(x;t) at timetof the associated subordinator whose Lévy measure is the restriction of ν to (0,s]. It will be expressed in terms of ν and the marginal distribution functionF(⋅;t) of the original process. A generalization concerning an arbitrary truncation of ν will follow. Under certain conditions, an analogous formula will be obtained for thenth derivative, ∂nFs(x;t) ∂xn. The requirement that ν is continuous is shown to have no intrinsic meaning. A number of interesting results involving the size ordered jumps of subordinators will be derived. An appropriate approximation for the small jumps of a gamma process will be considered, leading to a revisiting of the generalized Dickman distribution.

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5

Levajković, Tijana, Hermann Mena, and Martin Zarfl. "Lévy processes, subordinators and crime modelling." Novi Sad Journal of Mathematics 46, no. 2 (August 26, 2016): 65–86. http://dx.doi.org/10.30755/nsjom.03903.

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6

Al Masry, Zeina, Landy Rabehasaina, and Ghislain Verdier. "Change-level detection for Lévy subordinators." Stochastic Processes and their Applications 147 (May 2022): 423–55. http://dx.doi.org/10.1016/j.spa.2022.01.022.

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7

Beghin, Luisa, and Costantino Ricciuti. "Lévy Processes Linked to the Lower-Incomplete Gamma Function." Fractal and Fractional 5, no. 3 (July 17, 2021): 72. http://dx.doi.org/10.3390/fractalfract5030072.

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We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes that are time-changed by these subordinators with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion produces a model of anomalous diffusion, which exhibits a sub-diffusive behavior.

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8

Hering, Christian, Marius Hofert, Jan-Frederik Mai, and Matthias Scherer. "Constructing hierarchical Archimedean copulas with Lévy subordinators." Journal of Multivariate Analysis 101, no. 6 (July 2010): 1428–33. http://dx.doi.org/10.1016/j.jmva.2009.10.005.

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9

Schneider, Jan, and Roman Urban. "Lévy Subordinators in Cones of Fuzzy Sets." Journal of Theoretical Probability 32, no. 4 (August 9, 2018): 1909–24. http://dx.doi.org/10.1007/s10959-018-0853-x.

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10

Covo, Shai. "On Approximations of Small Jumps of Subordinators with Particular Emphasis on a Dickman-Type Limit." Journal of Applied Probability 46, no. 3 (September 2009): 732–55. http://dx.doi.org/10.1239/jap/1253279849.

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Let X be a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with infinite Lévy measure, let Xε be the sum of jumps not exceeding ε, and let µ(ε)=E[Xε(1)]. We study the question of weak convergence of Xε/µ(ε) as ε ↓0, in terms of the limit behavior of µ(ε)/ε. The most interesting case reduces to the weak convergence of Xε/ε to a subordinator whose marginals are generalized Dickman distributions; we give some necessary and sufficient conditions for this to hold. For a certain significant class of subordinators for which the latter convergence holds, and whose most prominent representative is the gamma process, we give some detailed analysis regarding the convergence quality (in particular, in the context of approximating X itself). This paper completes, in some respects, the study made by Asmussen and Rosiński (2001).

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Dissertations / Theses on the topic "Lévy subordinators"

1

Cecchetti, Sara. "An analysis of credit risk financial indicators." Doctoral thesis, Luiss Guido Carli, 2011. http://hdl.handle.net/11385/200885.

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Marsalle, Laurence. "Applications des subordinateurs à l'étude de trois familles de temps exceptionnels." Paris 6, 1997. http://www.theses.fr/1997PA066763.

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Rivero, Mercado Victor Manuel. "Récouvrements aléatoires et processus de Markov auto-similaires." Paris 6, 2004. https://tel.archives-ouvertes.fr/tel-00007346.

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Winkel, Matthias. "Quelques contributions à la théorie des processus de Lévy et des applications en turbulence et en économétrie." Paris 6, 2001. http://www.theses.fr/2001PA066492.

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RIVERO, MERCADO Victor. "Recouvrements Aléatoires et Processus de Markov Auto-Similaires." Phd thesis, Université Pierre et Marie Curie - Paris VI, 2004. http://tel.archives-ouvertes.fr/tel-00007346.

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Cette thèse comprend deux parties. La première traite de la construction d'un ensemble aléeatoire qui a la propriété de régénération. Plus précisement, on construit des intervalles aléatoires issus des maxima locaux d'un processus de Poisson ponctuel. Ceux-ci sont utilisés pour recouvrir partiellement la semi--droite des réels positifs et on s'intéresse alors à l'ensemble résiduel $\Rs,$ des points qui n'ont pas été recouverts. On donne des critères intégrales pour déterminer si l'ensemble $\Rs$ a une mesure de Lebesgue non nulle, si il est discret ou encore si il est borné. On montre que l'ensemble $\Rs$ est régenératif et on caractérise le subordinateur associé via sa mesure potentiel. On donne des formules pour calculer quelques dimensions fractales pour $\Rs.$ La deuxième partie est constituée de quelques contributions à la théorie des processus de Markov auto--similaires positifs. Pour obtenir les résultats de cette partie on utilise amplement la transformation de Lamperti qui permet de rélier les processus de Markov auto--similaires positif aux processus de Lévy à valeurs dans $\re.$ On s' interesse d'abord, au comportement à l'infini d'un processus de Markov auto--similaire croissant. On détermine, sous certaines hypothèses, une fonction déterministe $f$ telle que la limite inférieure, lorsque $t$ tend vers l'infini, du quotient $X_t/f(t)$ est finie et non nulle avec probabilité $1.$ Un résultat analogue est obtenu pour déterminer le comportement près de 0 du processus $X$ issu de 0. Ensuite, on étudie les différentes manières de construire un processus de Markov auto--similaire $\widetilde(X)$ pour lequel 0 est un point régulier et récurrent. En premier lieu, on donne des conditions qui nous permettent d'assurer qu'un tel processus existe et d'expliciter sa résolvante. En second lieu, on fait une étude systématique de la mesure d'excursions d'Itô $\exc$ pour le processus $\widetilde(X)$. On donne en particulier une description à la Imhof de $\exc,$ on determine la loi sous $\exc$ de l'excursion normalisée et l'image sous retournement de temps de $\exc$. De plus, on construit et on décrit un processus qui est en dualité faible avec le processus $\widetilde(X).$ On obtient diverses estimations de la queue de probabilité de la loi d'une variable aléatoire fonctionnelle exponentielle d'un processus de Lévy.

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6

Mai, Jan-Frederik [Verfasser]. "Extendibility of Marshall-Olkin distributions via Lévy subordinators and an application to portfolio credit risk / Jan-Frederik Mai." 2010. http://d-nb.info/1004856806/34.

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7

Ibrahim, Rabï. "Étude empirique de distributions associées à la Fonction de Pénalité Escomptée." Thèse, 2010. http://hdl.handle.net/1866/3798.

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On présente une nouvelle approche de simulation pour la fonction de densité conjointe du surplus avant la ruine et du déficit au moment de la ruine, pour des modèles de risque déterminés par des subordinateurs de Lévy. Cette approche s'inspire de la décomposition "Ladder height" pour la probabilité de ruine dans le Modèle Classique. Ce modèle, déterminé par un processus de Poisson composé, est un cas particulier du modèle plus général déterminé par un subordinateur, pour lequel la décomposition "Ladder height" de la probabilité de ruine s'applique aussi. La Fonction de Pénalité Escomptée, encore appelée Fonction Gerber-Shiu (Fonction GS), a apporté une approche unificatrice dans l'étude des quantités liées à l'événement de la ruine été introduite. La probabilité de ruine et la fonction de densité conjointe du surplus avant la ruine et du déficit au moment de la ruine sont des cas particuliers de la Fonction GS. On retrouve, dans la littérature, des expressions pour exprimer ces deux quantités, mais elles sont difficilement exploitables de par leurs formes de séries infinies de convolutions sans formes analytiques fermées. Cependant, puisqu'elles sont dérivées de la Fonction GS, les expressions pour les deux quantités partagent une certaine ressemblance qui nous permet de nous inspirer de la décomposition "Ladder height" de la probabilité de ruine pour dériver une approche de simulation pour cette fonction de densité conjointe. On présente une introduction détaillée des modèles de risque que nous étudions dans ce mémoire et pour lesquels il est possible de réaliser la simulation. Afin de motiver ce travail, on introduit brièvement le vaste domaine des mesures de risque, afin d'en calculer quelques unes pour ces modèles de risque. Ce travail contribue à une meilleure compréhension du comportement des modèles de risques déterminés par des subordinateurs face à l'éventualité de la ruine, puisqu'il apporte un point de vue numérique absent de la littérature.
We discuss a simulation approach for the joint density function of the surplus prior to ruin and deficit at ruin for risk models driven by Lévy subordinators. This approach is inspired by the Ladder Height decomposition for the probability of ruin of such models. The Classical Risk Model driven by a Compound Poisson process is a particular case of this more generalized one. The Expected Discounted Penalty Function, also referred to as the Gerber-Shiu Function (GS Function), was introduced as a unifying approach to deal with different quantities related to the event of ruin. The probability of ruin and the joint density function of surplus prior to ruin and deficit at ruin are particular cases of this function. Expressions for those two quantities have been derived from the GS Function, but those are not easily evaluated nor handled as they are infinite series of convolutions with no analytical closed form. However they share a similar structure, thus allowing to use the Ladder Height decomposition of the Probability of Ruin as a guiding method to generate simulated values for this joint density function. We present an introduction to risk models driven by subordinators, and describe those models for which it is possible to process the simulation. To motivate this work, we also present an application for this distribution, in order to calculate different risk measures for those risk models. An brief introduction to the vast field of Risk Measures is conducted where we present selected measures calculated in this empirical study. This work contributes to better understanding the behavior of subordinators driven risk models, as it offers a numerical point of view, which is absent in the literature.

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Book chapters on the topic "Lévy subordinators"

1

Kyprianou, Andreas E. "Subordinators at First Passage and Renewal Measures." In Fluctuations of Lévy Processes with Applications, 115–52. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-642-37632-0_5.

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2

Marchal, Philippe. "First Passage Times of Subordinators and Urns." In A Lifetime of Excursions Through Random Walks and Lévy Processes, 343–55. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-83309-1_18.

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