Author
Victor Rivero
Other affiliations: Paris West University Nanterre La Défense, Pierre-and-Marie-Curie University, University of Bath
Bio: Victor Rivero is an academic researcher from Centro de Investigación en Matemáticas. The author has contributed to research in topics: Markov process & Subordinator. The author has an hindex of 17, co-authored 63 publications receiving 1394 citations. Previous affiliations of Victor Rivero include Paris West University Nanterre La Défense & Pierre-and-Marie-Curie University.
Papers published on a yearly basis
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TL;DR: In this article, the authors give an up-to-date account of the theory and applications of scale functions for spectrally negative Levy processes, including the first extensive overview of how to work numerically with scale functions.
Abstract: The purpose of this review article is to give an up to date account of the theory and applications of scale functions for spectrally negative Levy 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 Levy processes, in particular a reasonable understanding of the Levy–Khintchine formula and its relationship to the Levy–Ito 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 Levy processes; (Bertoin, Levy Processes (1996); Sato, Levy Processes and Infinitely Divisible Distributions (1999); Kyprianou, Introductory Lectures on Fluctuations of Levy Processes and Their Applications (2006); Doney, Fluctuation Theory for Levy Processes (2007)), Applebaum Levy Processes and Stochastic Calculus (2009).
288 citations
01 Jan 2012
TL;DR: In this paper, the authors give an up-to-date account of the theory and applications of scale functions for spectrally negative L´ evy processes, including the first extensive overview of how to work numerically with scale functions.
Abstract: The purpose of this review article is to give an up to date account of the theory and applications of scale functions for spectrally negative L´ evy 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 L´ evy processes, in particular a reasonable understanding of the L´ evy-Khintchine formula and its relationship to the L´ˆ o 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 L´ evy
186 citations
TL;DR: In this paper, a self-similar Markov process X is constructed via its associated entrance law, which can be viewed as X conditioned never to hit 0, and then the process is constructed similarly to the way in which the Brownian excursion measure is constructed through the law of a Bessel(3) process.
Abstract: Let ξ be a real-valued Levy process that satisfies Cramer's condition, and X a self-similar Markov process associated with ξ via Lamperti's transformation. In this case, X has 0 as a trap and satisfies the assumptions set out by Vuolle-Apiala. We deduce from the latter that there exists a unique excursion measure \exc, compatible with the semigroup of X and such that \exc(X0+>0)=0. Here, we give a precise description of \exc via its associated entrance law. To this end, we construct a self-similar process X
atural, which can be viewed as X conditioned never to hit 0, and then we construct \exc similarly to the way in which the Brownian excursion measure is constructed via the law of a Bessel(3) process. An alternative description of \exc is given by specifying the law of the excursion process conditioned to have a given length. We establish some duality relations from which we determine the image under time reversal of \exc.
129 citations
TL;DR: In this article, the authors provide new families of scale functions for spectrally negative Levy processes which are completely explicit and allow feeding the theory of Bernstein functions directly into the Wiener-Hopf factorization.
Abstract: Following from recent developments in Hubalek and Kyprianou [28], the objective of this paper is to provide further methods for constructing new families of scale functions for spectrally negative Levy processes which are completely explicit. This is the result of an observation in the aforementioned paper which permits feeding the theory of Bernstein functions directly into the Wiener-Hopf factorization for spectrally negative Levy processes. Many new, concrete examples of scale functions are offered although the methodology in principle delivers still more explicit examples than those listed.
80 citations
TL;DR: In this paper, the authors continue the recent work of Avram et al. by showing that whenever the Levy measure of a spectrally negative Levy process has a density which is log-convex then the solution of the associated actuarial control problem of de Finetti is solved by a barrier strategy.
Abstract: We continue the recent work of Avram et al. (Ann. Appl. Probab. 17:156–180, 2007) and Loeffen (Ann. Appl. Probab., 2007) by showing that whenever the Levy measure of a spectrally negative Levy process has a density which is log-convex then the solution of the associated actuarial control problem of de Finetti is solved by a barrier strategy. Moreover, the level of the barrier can be identified in terms of the scale function of the underlying Levy process. Our method appeals directly to very recent developments in the theory of potential analysis of subordinators and their application to convexity and smoothness properties of the relevant scale functions.
59 citations
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TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.
5,689 citations
Book•
01 Jan 2013
TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.
Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.
1,957 citations
1,620 citations
1,188 citations
TL;DR: In this paper, applied probability and queuing in the field of applied probabilistic analysis is discussed. But the authors focus on the application of queueing in the context of road traffic.
Abstract: (1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.
1,121 citations