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Journal articles on the topic 'Processus de Lévy'

Author: Grafiati

Published: 4 June 2021

Last updated: 8 June 2024

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1

Fourati, S. "Points de croissance des processus de Lévy et théorie générale des processus." Probability Theory and Related Fields 110, no. 1 (January 16, 1998): 13–49. http://dx.doi.org/10.1007/s004400050143.

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2

Simon, Thomas. "Petites déviations et support d'un processus de Lévy." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 329, no. 4 (August 1999): 331–34. http://dx.doi.org/10.1016/s0764-4442(00)88576-6.

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3

Fourati, Sonia. "Fluctuation des processus de Lévy et dispersion ( « scattering »)." Comptes Rendus Mathematique 342, no. 2 (January 2006): 135–39. http://dx.doi.org/10.1016/j.crma.2005.11.012.

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4

Pinhas, Max. "Critères macroéconomiques actuariels et processus de Paul Lévy." Rivista di Matematica per le Scienze Economiche e Sociali 9, no. 2 (September 1986): 143–47. http://dx.doi.org/10.1007/bf02086872.

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5

Chaumont, L. "Sur certains processus de lévy conditionnés à rester positifs." Stochastics and Stochastic Reports 47, no. 1-2 (March 1994): 1–20. http://dx.doi.org/10.1080/17442509408833880.

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6

Carmona, P., F. Petit, and M. Yor. "Sur les fonctionnelles exponentielles de certains processus de lévy." Stochastics and Stochastic Reports 47, no. 1-2 (March 1994): 71–101. http://dx.doi.org/10.1080/17442509408833883.

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7

Abbassi, Mohamed. "Processus de Lévy autosimilaires sur les groupes de Lie." Comptes Rendus Mathematique 348, no. 21-22 (November 2010): 1207–10. http://dx.doi.org/10.1016/j.crma.2010.10.014.

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8

Alili, Larbi, and Loïc Chaumont. "Quelques nouvelles identités de fluctuation pour les processus de Lévy." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 328, no. 7 (April 1999): 613–16. http://dx.doi.org/10.1016/s0764-4442(99)80256-0.

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9

Louhichi, Sana, and Emmanuel Rio. "Convergence du processus de sommes partielles vers un processus de Lévy pour les suites associées." Comptes Rendus Mathematique 349, no. 1-2 (January 2011): 89–91. http://dx.doi.org/10.1016/j.crma.2010.12.001.

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10

Fourati, S. "Inversion de l'espace et du temps des processus de Lévy stables." Probability Theory and Related Fields 135, no. 2 (November 10, 2005): 201–15. http://dx.doi.org/10.1007/s00440-005-0455-2.

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11

Simon, Thomas. "Fonctions de Mittag–Leffler et processus de Lévy stables sans sauts négatifs." Expositiones Mathematicae 28, no. 3 (2010): 290–98. http://dx.doi.org/10.1016/j.exmath.2009.12.002.

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12

Hlebec, Boris. "Factors and steps in translating." Babel. Revue internationale de la traduction / International Journal of Translation 35, no. 3 (January 1, 1989): 129–41. http://dx.doi.org/10.1075/babel.35.3.02hle.

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L'article comprend la clé de voûte d'une théorie unifiée de la traduction, qui est sémiotique et communicationnelle, dont l'essentiel réside dans un catalogage des facteurs distinctifs et des pas conduisant à une traduction excellente. C'est ainsi qu'a été developpée la demande de Lévy concernant la traduction comme un processus de décision. Une nouvelle définition de la traduction a été proposée, en gardant séparées les recréations et les modifications. La notion "intention" a été suggérée comme terme général pour toutes sortes de sens dans le texte, tandis que les fonctions communicationnelles (y inclus le style) sont considérées comme des types étendus d'intentions. Finalement, le processus de traduire a été illustré par l'exemple d'un passage du roman d'Ivo Andric Le Pont sur la Drina et les démarches éventuelles, dépendant de facteurs changeants, sont discutées.

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13

El Karoui, Nicole, and Sylvie Roelly. "Propriétés de martingales, explosion et représentation de Lévy—Khintchine d'une classe de processus de branchement à valeurs mesures." Stochastic Processes and their Applications 38, no. 2 (August 1991): 239–66. http://dx.doi.org/10.1016/0304-4149(91)90093-r.

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14

Ramseier, Aude, and Sabine Oppliger. "Apprendre et pratiquer sa créativité : des dispositifs en actes inspirés de la démarche des arbres de connaissances." Voix Plurielles 13, no. 1 (May 14, 2016): 113–24. http://dx.doi.org/10.26522/vp.v13i1.1374.

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Cet article décrit et analyse l’émergence de la créativité dans un contexte scolaire dans une classe d’élèves de neuf à onze ans dans le canton de Vaud en Suisse. Il s’appuie sur les bases d’une recherche-action menée par une des auteures durant l’année scolaire 2011-2012. Les dispositifs mis en place, inspirés de la démarche des arbres de connaissances (Authier et Lévy 1996) et des principes des communautés d’apprenants (Brown et Campione 1995), proposent des conditions-cadre propices à une culture de créativité partagée qui institutionnalise cette dimension en l’intégrant dans un processus continu qui la légitime comme objet d’apprentissage. Cette recherche permet de questionner les relations entre les conditions-cadre proposées par les dispositifs présentés et des éléments liés à la créativité, à savoir ce qui peut la promouvoir dans un contexte d’apprentissage scolaire. La question de recherche a été formulée de la manière suivante : en quoi les dispositifs mis en acte proposent des conditions-cadre favorables à l’émergence de la créativité des élèves ? Les résultats mettent en évidence une posture d’auteur-acteur endossée par les élèves. La posture de l’enseignant est aussi abordée ainsi que son rôle en tant qu’accompagnateur facilitateur qui intègre de manière intentionnelle et transversale la créativité dans son enseignement. Il est le garant de la conduite de cette démarche qui mobilise les potentialités individuelles au service du collectif et de la co-création d’une culture de la reconnaissance des compétences des élèves en classe par la création d’espaces qui la rendent possible.

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15

Wu, Lan, and Xiao Zhang. "Occupation times of Lévy processes." International Journal of Financial Engineering 08, no. 03 (May 6, 2021): 2142003. http://dx.doi.org/10.1142/s2424786321420032.

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In this paper, we give a complete and succinct proof that an explicit formula for the occupation time holds for all Lévy processes, which is important to the pricing problems of various occupation-time-related derivatives such as step options and corridor options. We construct a sequence of Lévy processes converging to a given Lévy process to obtain our conclusion. Besides financial applications, the mathematical results about occupation times of a Lévy process are of interest in applied probability.

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16

Dieker, A. B. "Applications of factorization embeddings for Lévy processes." Advances in Applied Probability 38, no. 3 (September 2006): 768–91. http://dx.doi.org/10.1239/aap/1158685001.

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We give three applications of the Pecherskii-Rogozin-Spitzer identity for Lévy processes. First, we find the joint distribution of the supremum and the epoch at which it is ‘attained’ if a Lévy process has phase-type upward jumps. We also find the characteristics of the ladder process. Second, we establish general properties of perturbed risk models, and obtain explicit fluctuation identities in the case that the Lévy process is spectrally positive. Third, we study the tail asymptotics for the supremum of a Lévy process under different assumptions on the tail of the Lévy measure.

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17

Dieker, A. B. "Applications of factorization embeddings for Lévy processes." Advances in Applied Probability 38, no. 03 (September 2006): 768–91. http://dx.doi.org/10.1017/s0001867800001269.

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We give three applications of the Pecherskii-Rogozin-Spitzer identity for Lévy processes. First, we find the joint distribution of the supremum and the epoch at which it is ‘attained’ if a Lévy process has phase-type upward jumps. We also find the characteristics of the ladder process. Second, we establish general properties of perturbed risk models, and obtain explicit fluctuation identities in the case that the Lévy process is spectrally positive. Third, we study the tail asymptotics for the supremum of a Lévy process under different assumptions on the tail of the Lévy measure.

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18

Hellmund, Gunnar, Michaela Prokešová, and Eva B. Vedel Jensen. "Lévy-based Cox point processes." Advances in Applied Probability 40, no. 3 (September 2008): 603–29. http://dx.doi.org/10.1239/aap/1222868178.

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In this paper we introduce Lévy-driven Cox point processes (LCPs) as Cox point processes with driving intensity function Λ defined by a kernel smoothing of a Lévy basis (an independently scattered, infinitely divisible random measure). We also consider log Lévy-driven Cox point processes (LLCPs) with Λ equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes, and log shot noise Cox processes. We study the theoretical properties of Lévy-based Cox processes, including moment properties described by nth-order product densities, mixing properties, specification of inhomogeneity, and spatio-temporal extensions.

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19

Hellmund, Gunnar, Michaela Prokešová, and Eva B. Vedel Jensen. "Lévy-based Cox point processes." Advances in Applied Probability 40, no. 03 (September 2008): 603–29. http://dx.doi.org/10.1017/s0001867800002718.

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In this paper we introduce Lévy-driven Cox point processes (LCPs) as Cox point processes with driving intensity function Λ defined by a kernel smoothing of a Lévy basis (an independently scattered, infinitely divisible random measure). We also consider log Lévy-driven Cox point processes (LLCPs) with Λ equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes, and log shot noise Cox processes. We study the theoretical properties of Lévy-based Cox processes, including moment properties described by nth-order product densities, mixing properties, specification of inhomogeneity, and spatio-temporal extensions.

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20

Gajda, Janusz, and Luisa Beghin. "Prabhakar Lévy processes." Statistics & Probability Letters 178 (November 2021): 109162. http://dx.doi.org/10.1016/j.spl.2021.109162.

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21

Crane, Harry. "Combinatorial Lévy processes." Annals of Applied Probability 28, no. 1 (February 2018): 285–339. http://dx.doi.org/10.1214/17-aap1306.

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22

Çinlar, E. "Conditional lévy processes." Computers & Mathematics with Applications 46, no. 7 (October 2003): 993–97. http://dx.doi.org/10.1016/s0898-1221(03)90113-1.

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23

Vigon, Vincent. "Abrupt Lévy processes." Stochastic Processes and their Applications 103, no. 1 (January 2003): 155–68. http://dx.doi.org/10.1016/s0304-4149(02)00186-2.

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24

Ratanov, Nikita. "Kac–Lévy Processes." Journal of Theoretical Probability 33, no. 1 (November 29, 2018): 239–67. http://dx.doi.org/10.1007/s10959-018-0873-6.

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25

Kyprianou, A. E., and R. L. Loeffen. "Refracted Lévy processes." Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 46, no. 1 (February 2010): 24–44. http://dx.doi.org/10.1214/08-aihp307.

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26

Lindstrøm, Tom. "Hyperfinite Lévy Processes." Stochastics and Stochastic Reports 76, no. 6 (December 2004): 517–48. http://dx.doi.org/10.1080/10451120412331315797.

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27

HUANG, ZHI YUAN, and PEI YAN LI. "GENERALIZED FRACTIONAL LÉVY PROCESSES: A WHITE NOISE APPROACH." Stochastics and Dynamics 06, no. 04 (December 2006): 473–85. http://dx.doi.org/10.1142/s0219493706001839.

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In this paper, with a simple condition on Lévy measure, we construct the (tempered) generalized fractional Lévy processes (GFLP) as Lévy white noise functionals and investigate their distribution and sample properties through this white noise approach. We also give some sufficient conditions under which the usual fractional Lévy processes (FLP) are well defined.

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28

Chang, Tongkeun. "Absolute continuity of distributions of one-dimensional Lévy processes." Journal of Applied Probability 54, no. 3 (September 2017): 852–72. http://dx.doi.org/10.1017/jpr.2017.38.

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Abstract In this paper we study the existence of Lebesgue densities of one-dimensional Lévy processes. Equivalently, we show the absolute continuity of the distributions of one-dimensional Lévy processes. Compared with the previous literature, we consider Lévy processes with Lévy symbols of a logarithmic behavior at ∞.

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29

Dia, E. H. A. "Error Bounds for Small Jumps of Lévy Processes." Advances in Applied Probability 45, no. 1 (March 2013): 86–105. http://dx.doi.org/10.1239/aap/1363354104.

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The pricing of options in exponential Lévy models amounts to the computation of expectations of functionals of Lévy processes. In many situations, Monte Carlo methods are used. However, the simulation of a Lévy process with infinite Lévy measure generally requires either truncating or replacing the small jumps by a Brownian motion with the same variance. We will derive bounds for the errors generated by these two types of approximation.

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30

Dia, E. H. A. "Error Bounds for Small Jumps of Lévy Processes." Advances in Applied Probability 45, no. 01 (March 2013): 86–105. http://dx.doi.org/10.1017/s0001867800006200.

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The pricing of options in exponential Lévy models amounts to the computation of expectations of functionals of Lévy processes. In many situations, Monte Carlo methods are used. However, the simulation of a Lévy process with infinite Lévy measure generally requires either truncating or replacing the small jumps by a Brownian motion with the same variance. We will derive bounds for the errors generated by these two types of approximation.

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31

Wagner, Vanja. "Censored symmetric Lévy-type processes." Forum Mathematicum 31, no. 6 (November 1, 2019): 1351–68. http://dx.doi.org/10.1515/forum-2018-0076.

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AbstractWe examine three equivalent constructions of a censored symmetric purely discontinuous Lévy process on an open set D; via the corresponding Dirichlet form, through the Feynman–Kac transform of the Lévy process killed outside of D and from the same killed process by the Ikeda–Nagasawa–Watanabe piecing together procedure. By applying the trace theorem on n-sets for Besov-type spaces of generalized smoothness associated with complete Bernstein functions satisfying certain scaling conditions, we analyze the boundary behavior of the corresponding censored Lévy process and determine conditions under which the process approaches the boundary {\partial D} in finite time. Furthermore, we prove a stronger version of the 3G inequality and its generalized version for Green functions of purely discontinuous Lévy processes on κ-fat open sets. Using this result, we obtain the scale invariant Harnack inequality for the corresponding censored process.

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32

HUANG, ZHIYUAN, XUEBIN LÜ, and JIANPING WAN. "FRACTIONAL LÉVY PROCESSES AND NOISES ON GEL′FAND TRIPLE." Stochastics and Dynamics 10, no. 01 (March 2010): 37–51. http://dx.doi.org/10.1142/s0219493710002838.

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In this paper, we construct a class of infinitely divisible distributions on Gel′fand triple. Based on this construction, we define Lévy processes on Gel′fand triple and give their Lévy–Itô decompositions. Then, we construct the general Lévy white noises on Gel′fand triple. By using the Riemann–Liouville fractional integral method, we define the general fractional Lévy noises on Gel′fand triple and investigate their distribution properties.

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33

FARKAS, WALTER, NILS REICH, and CHRISTOPH SCHWAB. "ANISOTROPIC STABLE LEVY COPULA PROCESSES — ANALYTICAL AND NUMERICAL ASPECTS." Mathematical Models and Methods in Applied Sciences 17, no. 09 (September 2007): 1405–43. http://dx.doi.org/10.1142/s0218202507002327.

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We consider the valuation of derivative contracts on baskets of risky assets whose prices are Lévy-like Feller processes of tempered stable type. The dependence among the marginals' jump structure is parametrized by a Lévy copula. For marginals of regular, exponential Lévy type in the sense of Ref. 7 we show that the infinitesimal generator [Formula: see text] of the resulting Lévy copula process is a pseudo-differential operator whose principal symbol is a distribution of anisotropic homogeneity. We analyze the jump measure of the corresponding Lévy copula processes. We prove that the domains of their infinitesimal generators [Formula: see text] are certain anisotropic Sobolev spaces. In these spaces and for a large class of Lévy copula processes, we prove a Gårding inequality for [Formula: see text]. We design a wavelet-based dimension-independent tensor product discretization for the efficient numerical solution of the parabolic Kolmogorov equation [Formula: see text] arising in valuation of derivative contracts under possibly stopped Lévy copula processes. In the wavelet basis, diagonal preconditioning yields a bounded condition number of the resulting matrices.

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34

Brockwell, Peter J. "Representations of continuous-time ARMA processes." Journal of Applied Probability 41, A (2004): 375–82. http://dx.doi.org/10.1239/jap/1082552212.

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Using the kernel representation of a continuous-time Lévy-driven ARMA (autoregressive moving average) process, we extend the class of nonnegative Lévy-driven Ornstein–Uhlenbeck processes employed by Barndorff-Nielsen and Shephard (2001) to allow for nonmonotone autocovariance functions. We also consider a class of fractionally integrated Lévy-driven continuous-time ARMA processes obtained by a simple modification of the kernel of the continuous-time ARMA process. Asymptotic properties of the kernel and of the autocovariance function are derived.

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35

Brockwell, Peter J. "Representations of continuous-time ARMA processes." Journal of Applied Probability 41, A (2004): 375–82. http://dx.doi.org/10.1017/s0021900200112422.

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Using the kernel representation of a continuous-time Lévy-driven ARMA (autoregressive moving average) process, we extend the class of nonnegative Lévy-driven Ornstein–Uhlenbeck processes employed by Barndorff-Nielsen and Shephard (2001) to allow for nonmonotone autocovariance functions. We also consider a class of fractionally integrated Lévy-driven continuous-time ARMA processes obtained by a simple modification of the kernel of the continuous-time ARMA process. Asymptotic properties of the kernel and of the autocovariance function are derived.

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36

Gad, Kamille Sofie Tågholt, and Jesper Lund Pedersen. "Variance Optimal Stopping for Geometric Lévy Processes." Advances in Applied Probability 47, no. 1 (March 2015): 128–45. http://dx.doi.org/10.1239/aap/1427814584.

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The main result of this paper is the solution to the optimal stopping problem of maximizing the variance of a geometric Lévy process. We call this problem the variance problem. We show that, for some geometric Lévy processes, we achieve higher variances by allowing randomized stopping. Furthermore, for some geometric Lévy processes, the problem has a solution only if randomized stopping is allowed. When randomized stopping is allowed, we give a solution to the variance problem. We identify the Lévy processes for which the allowance of randomized stopping times increases the maximum variance. When it does, we also solve the variance problem without randomized stopping.

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37

Gad, Kamille Sofie Tågholt, and Jesper Lund Pedersen. "Variance Optimal Stopping for Geometric Lévy Processes." Advances in Applied Probability 47, no. 01 (March 2015): 128–45. http://dx.doi.org/10.1017/s0001867800007734.

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The main result of this paper is the solution to the optimal stopping problem of maximizing the variance of a geometric Lévy process. We call this problem the variance problem. We show that, for some geometric Lévy processes, we achieve higher variances by allowing randomized stopping. Furthermore, for some geometric Lévy processes, the problem has a solution only if randomized stopping is allowed. When randomized stopping is allowed, we give a solution to the variance problem. We identify the Lévy processes for which the allowance of randomized stopping times increases the maximum variance. When it does, we also solve the variance problem without randomized stopping.

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38

Bansaye, Vincent, Juan Carlos Pardo, and Charline Smadi. "Extinction rate of continuous state branching processes in critical Lévy environments." ESAIM: Probability and Statistics 25 (2021): 346–75. http://dx.doi.org/10.1051/ps/2021014.

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We study the speed of extinction of continuous state branching processes in a Lévy environment, where the associated Lévy process oscillates. Assuming that the Lévy process satisfies Spitzer’s condition, we extend recent results where the associated branching mechanism is stable. The study relies on the path analysis of the branching process together with its Lévy environment, when the latter is conditioned to have a non-negative running infimum. For that purpose, we combine the approach developed in Afanasyev et al. [2], for the discrete setting and i.i.d. environments, with fluctuation theory of Lévy processes and a result on exponential functionals of Lévy processes due to Patie and Savov [28].

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39

Horton, Emma L., and Andreas E. Kyprianou. "More on hypergeometric Lévy processes." Advances in Applied Probability 48, A (July 2016): 153–58. http://dx.doi.org/10.1017/apr.2016.47.

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AbstractKuznetsov and co-authors in 2011‒14 introduced the family of hypergeometric Lévy processes. They appear naturally in the study of fluctuations of stable processes when one analyses stable processes through the theory of positive self-similar Markov processes. Hypergeometric Lévy processes are defined through their characteristic exponent, which, as a complex-valued function, has four independent parameters. In 2014 it was shown that the definition of a hypergeometric Lévy process could be taken to include a greater range of the aforesaid parameters than originally specified. In this short article, we push the parameter range even further.

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40

BOYARCHENKO, SVETLANA I., and SERGEI Z. LEVENDORSKIǏ. "OPTION PRICING FOR TRUNCATED LÉVY PROCESSES." International Journal of Theoretical and Applied Finance 03, no. 03 (July 2000): 549–52. http://dx.doi.org/10.1142/s0219024900000541.

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A general class of truncated Lévy processes is introduced, and possible ways of fitting parameters of the constructed family of truncated Lévy processes to data are discussed. For a market of a riskless bond and a stock whose log-price follows a truncated Lévy process, TLP-analogs of the Black–Scholes equation, the Black–Scholes formula, the Dynkin derivative and the Leland's model are obtained, a locally risk-minimizing portfolio is constructed, and an optimal exercise price for a perpetual American put is computed.

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41

He, Kai. "Stochastic calculus for fractional Lévy processes." Infinite Dimensional Analysis, Quantum Probability and Related Topics 17, no. 01 (March 2014): 1450006. http://dx.doi.org/10.1142/s0219025714500064.

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In this paper, we construct fractional Lévy processes for any parameter H ∈ (0, 1), as the generalization of the fractional Brownian motion. By using Malliavin calculus, we also define the stochastic integral for fractional Lévy processes.

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42

Ng, Siu Ah. "A nonstandard Lévy-Khintchine formula and Lévy processes." Acta Mathematica Sinica, English Series 24, no. 2 (February 2008): 241–52. http://dx.doi.org/10.1007/s10114-007-1024-7.

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43

Caballero, M. E., and L. Chaumont. "Conditioned stable Lévy processes and the Lamperti representation." Journal of Applied Probability 43, no. 4 (December 2006): 967–83. http://dx.doi.org/10.1239/jap/1165505201.

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By variously killing a stable Lévy process when it leaves the positive half-line, conditioning it to stay positive, and conditioning it to hit 0 continuously, we obtain three different, positive, self-similar Markov processes which illustrate the three classes described by Lamperti (1972). For each of these processes, we explicitly compute the infinitesimal generator and from this deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0, which we describe here. As an application, for a certain class of Lévy processes we give the law of the minimum before an independent exponential time. This provides the explicit form of the spatial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.

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44

Caballero, M. E., and L. Chaumont. "Conditioned stable Lévy processes and the Lamperti representation." Journal of Applied Probability 43, no. 04 (December 2006): 967–83. http://dx.doi.org/10.1017/s0021900200002369.

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By variously killing a stable Lévy process when it leaves the positive half-line, conditioning it to stay positive, and conditioning it to hit 0 continuously, we obtain three different, positive, self-similar Markov processes which illustrate the three classes described by Lamperti (1972). For each of these processes, we explicitly compute the infinitesimal generator and from this deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0, which we describe here. As an application, for a certain class of Lévy processes we give the law of the minimum before an independent exponential time. This provides the explicit form of the spatial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.

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45

FRANZ, UWE. "CLASSICAL MARKOV PROCESSES FROM QUANTUM LÉVY PROCESSES." Infinite Dimensional Analysis, Quantum Probability and Related Topics 02, no. 01 (March 1999): 105–29. http://dx.doi.org/10.1142/s0219025799000060.

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We show how classical Markov processes can be obtained from quantum Lévy processes. It is shown that quantum Lévy processes are quantum Markov processes, and sufficient conditions for restrictions to subalgebras to remain quantum Markov processes are given. A classical Markov process (which has the same time-ordered moments as the quantum process in the vacuum state) exists whenever we can restrict to a commutative subalgebra without losing the quantum Markov property.8 Several examples, including the Azéma martingale, with explicit calculations are presented. In particular, the action of the generator of the classical Markov processes on polynomials or their moments are calculated using Hopf algebra duality.

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46

FAJARDO, JOSÉ, and ERNESTO MORDECKI. "PRICING DERIVATIVES ON TWO-DIMENSIONAL LÉVY PROCESSES." International Journal of Theoretical and Applied Finance 09, no. 02 (March 2006): 185–97. http://dx.doi.org/10.1142/s0219024906003536.

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The aim of this work is to use a duality approach to study the pricing of derivatives depending on two stocks driven by a bidimensional Lévy process. The main idea is to apply Girsanov's Theorem for Lévy processes, in order to reduce the posed problem to a problem with one Lévy driven stock in an auxiliary market, baptized as "dual market". In this way, we extend the results obtained by Gerber and Shiu [5] for two-dimensional Brownian motion.

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47

Bekker, R., O. J. Boxma, and J. A. C. Resing. "Lévy processes with adaptable exponent." Advances in Applied Probability 41, no. 1 (March 2009): 177–205. http://dx.doi.org/10.1239/aap/1240319581.

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In this paper we consider Lévy processes without negative jumps, reflected at the origin. Feedback information about the level of the Lévy process (‘workload level’) may lead to adaptation of the Lévy exponent. Examples of such models are queueing models in which the service speed or customer arrival rate changes depending on the workload level, and dam models in which the release rate depends on the buffer content. We first consider a class of models where information about the workload level is continuously available. In particular, we consider dam processes with a two-step release rule and M/G/1 queues in which the arrival rate, service speed, and/or jump size distribution may be adapted depending on whether the workload is above or below some level K. Secondly, we consider a class of models in which the workload can only be observed at Poisson instants. At these Poisson instants, the Lévy exponent may be adapted based on the amount of work present. For both classes of models, we determine the steady-state workload distribution.

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48

Bekker, R., O. J. Boxma, and J. A. C. Resing. "Lévy processes with adaptable exponent." Advances in Applied Probability 41, no. 01 (March 2009): 177–205. http://dx.doi.org/10.1017/s0001867800003189.

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Abstract:

In this paper we consider Lévy processes without negative jumps, reflected at the origin. Feedback information about the level of the Lévy process (‘workload level’) may lead to adaptation of the Lévy exponent. Examples of such models are queueing models in which the service speed or customer arrival rate changes depending on the workload level, and dam models in which the release rate depends on the buffer content. We first consider a class of models where information about the workload level is continuously available. In particular, we consider dam processes with a two-step release rule and M/G/1 queues in which the arrival rate, service speed, and/or jump size distribution may be adapted depending on whether the workload is above or below some levelK. Secondly, we consider a class of models in which the workload can only be observed at Poisson instants. At these Poisson instants, the Lévy exponent may be adapted based on the amount of work present. For both classes of models, we determine the steady-state workload distribution.

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PÉREZ-ABREU, VÍCTOR, and ALFONSO ROCHA-ARTEAGA. "COVARIANCE-PARAMETER LÉVY PROCESSES IN THE SPACE OF TRACE-CLASS OPERATORS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 08, no. 01 (March 2005): 33–54. http://dx.doi.org/10.1142/s0219025705001846.

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The paper deals with Lévy processes with values in L1(H), the Banach space of trace-class operators in a Hilbert space H. Lévy processes with values and parameter in a cone K of L1(H) are introduced and several properties are established. A family of L1(H)-valued Lévy processes is obtained via the subordination of K-parameter, L1(H)-valued Lévy processes, identifying explicitly their generating triplets.

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50

Konstantopoulos, Takis, and Gregory S. Richardson. "Conditional limit theorems for spectrally positive Lévy processes." Advances in Applied Probability 34, no. 1 (March 2002): 158–78. http://dx.doi.org/10.1239/aap/1019160955.

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We consider spectrally positive Lévy processes with regularly varying Lévy measure and study conditional limit theorems that describe the way that various rare events occur. Specifically, we are interested in the asymptotic behaviour of the distribution of the path of the Lévy process (appropriately scaled) up to some fixed time, conditionally on the event that the process exceeds a (large) positive value at that time. Another rare event we study is the occurrence of a large maximum value up to a fixed time, and the corresponding asymptotic behaviour of the (scaled) Lévy process path. We study these distributional limit theorems both for a centred Lévy process and for one with negative drift. In the latter case, we also look at the reflected process, which is of importance in applications. Our techniques are based on the explicit representation of the Lévy process in terms of a two-dimensional Poisson random measure and merely use the Poissonian properties and regular variation estimates. We also provide a proof for the asymptotic behaviour of the tail of the stationary distribution for the reflected process. The work is motivated by earlier results for discrete-time random walks (e.g. Durrett (1980) and Asmussen (1996)) and also by their applications in risk and queueing theory.

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