Research

In my way of understanding life there are random cycles that rule our behavior, they help us to learn, take decisions and solve problems. For instance, every morning, month, season, year... we restart some of our activities and start taking decisions about what we want and what we have to do. Every other summer day we like to go out with friends to have beer and enjoy of the company and the nice weather, we do not know what is gone happen next time, but it is rewarding to do it. Based in previous experiences we build routines and forge our character. Each of our decisions is based in previous experiences that help us to get the best of what we do. In this way, we build some probability rules that help us decide the next movement, get the best of the moment and reduce our risk of failure. Not all random patterns appear as time moves on, of course, look at a Brassica oleracea and you will know of what I mean.  Random cycles and patterns are present in many phenomena in life. 

I think that is one of the reasons why I became a reasercher specialized in stochastic processes and you will often find renewal theory, the theory of excursions from a set, and Markov processes theory as some of my favourite tools. Among other, these are powerful tools that can be  applied in almost any area of stochastic modelling, as for instance to build fluctuation theory of real valued Lévy processes and of Markov additive processes. 

Self-similarity is at the basis of theory of fractals and chaos, and it rules the behavior of various fundamental stochastic processes that appear as scaling limits of other stochastic processes. Examples of these are stable processes and self-similar Markov processes. Self-similar Markov processes are related to Lévy and Markov additive processes through the so-called Lamperti's transform.  

Local times are random clocks that are useful to describe the amount of time spent in particular spots, in particular they are helpful to understand the lengths of the above mentioned random cycles. We can study the dependence of the time spent in two or more neighboring spots, this leads to the study of local times as stochastic processes in the space variable.  They are closely related to branching processes and to various of the topics mentionned above.

The topics marked in bold letters and the questions that appear related to them are the core of my research, their rich mathematical structure connects them to many area of mathematics, and are useful in modelling various phenomena from nature.  

Go to the research page to know more about the output of my research

1. Caballero, María Emilia; Chaumont, Loïc; Rivero, Víctor. Lévy processes resurrected in the positive half-line. Preprint 2024 https://arxiv.org/abs/2404.19399

2. Kyprianou, Andreas; Motala, Mehar; Rivero, Víctor. Williams' path decomposition for self-similar Markov processes in Rd. Preprint 2023 http://arxiv.org/submit/5215611/pdf

3. Contreras, Jesús; Rivero Víctor. Ray-Knight Theorems for Spectrally Negative Lévy Processes. Preprint 2023 https://arxiv.org/abs/2306.12407

4. Contreras, Jesús ;  Rivero Víctor. Generalized scale functions for spectrally negative Lévy processes. ALEA, Lat. Am. J. Probab. Math. Stat. 20, 645–663 (2023). https://arxiv.org/abs/2209.15576

5. Arista, Jonas ; Rivero, Víctor. Implicit renewal theory for exponential functionals of Lévy processes. Stochastic Processes and their Applications, to appear (2023). https://arxiv.org/abs/1510.01809

6. Kyprianou, Andreas E. ;  Rivero, Victor ;  Satitkanitkul, Weerapat . Stable Lévy processes in a cone. Ann. Inst. Henri Poincaré Probab. Stat.  57  (2021),  no. 4, 2066--2099. https://arxiv.org/abs/1804.08393

7. Behme, Anita ;  Lindner, Alexander ;  Reker, Jana ;  Rivero, Victor . Continuity properties and the support of killed exponential functionals. Stochastic Process. Appl. 140 (2021), 115--146. https://arxiv.org/abs/1912.03052 

8. Kyprianou, Andreas E. ;  Rivero, Victor ;  Satitkanitkul, Weerapat . Deep factorisation of the stable process III: the view from radial excursion theory and the point of closest reach. Potential Anal.  53  (2020),  no. 4, 1347--1375. https://arxiv.org/abs/1706.09924

9. Kyprianou, Andreas E. ;  Rivero, Victor ;  Şengül, Batı ;  Yang, Ting . Entrance laws at the origin of self-similar Markov processes in high dimensions. Trans. Amer. Math. Soc.  373  (2020),  no. 9, 6227--6299. https://arxiv.org/abs/1812.01926

10. Pantí, H. ;  Pardo, J. C. ;  Rivero, V. M.  Recurrent extensions of real-valued self-similar Markov processes. Potential Anal.  53  (2020),  no. 3, 899--920. https://arxiv.org/abs/1808.00129

11. Kyprianou, Andreas E. ;  Rivero, Víctor M. ;  Satitkanitkul, Weerapat . Conditioned real self-similar Markov processes. Stochastic Process. Appl.  129  (2019),  no. 3, 954--977. https://arxiv.org/abs/1510.01781 

12. Kyprianou, Andreas E. ;  Rivero, Victor ;  Şengül, Batı . Deep factorisation of the stable process II: Potentials and applications. Ann. Inst. Henri Poincaré Probab. Stat.  54  (2018),  no. 1, 343--362. https://arxiv.org/abs/1511.06356 

13. Pardo, J. C. ;  Pérez, J. L. ;  Rivero, V. M.  The excursion measure away from zero for spectrally negative Lévy processes. Ann. Inst. Henri Poincaré Probab. Stat.  54  (2018),  no. 1, 75--99. https://arxiv.org/abs/1507.05225 

14. Blancas, Airam ;  Rivero, Víctor . On branching process with rare neutral mutation. Bernoulli  24  (2018),  no. 2, 1576--1612. https://arxiv.org/abs/1508.01901 

15. Kyprianou, Andreas E. ;  Rivero, Victor ;  Şengül, Batı . Conditioning subordinators embedded in Markov processes. Stochastic Process. Appl.  127  (2017),  no. 4, 1234--1254. https://arxiv.org/abs/1506.07870 

16. Doney, R. A. ;  Rivero, V.  Erratum to: Asymptotic behaviour of first passage time distributions for Lévy processes. Probab. Theory Related Fields  164  (2016),  no. 3-4, 1079--1083. 

17. Rivero, Víctor . Entrance laws for positive self-similar Markov processes. Mathematical Congress of the Americas, 119--140, Contemp. Math., 656, Amer. Math. Soc., Providence, RI,  2016.  https://arxiv.org/abs/1507.05229 

18. Doney, Ronald A. ;  Rivero, Víctor M.  Asymptotic behaviour of first passage time distributions for subordinators. Electron. J. Probab. 20 (2015), no. 91, 28 pp. https://arxiv.org/abs/1306.1503 

19. Alili, Larbi ;  Jedidi, Wissem ;  Rivero, Víctor . On exponential functionals, harmonic potential measures and undershoots of subordinators. ALEA Lat. Am. J. Probab. Math. Stat.  11  (2014),  no. 1, 711--735. https://arxiv.org/abs/1310.4955 

20. Pardo, Juan Carlos ;  Rivero, Víctor . Self-similar Markov processes. Bol. Soc. Mat. Mexicana (3)  19  (2013),  no. 2, 201--235. 

21. Chaumont, Loïc ;  Pantí, Henry ;  Rivero, Víctor . The Lamperti representation of real-valued self-similar Markov processes. Bernoulli  19  (2013),  no. 5B, 2494--2523. https://arxiv.org/abs/1111.1272 

22. Pardo, J. C. ;  Rivero, V. ;  van Schaik, K.  On the density of exponential functionals of Lévy processes. Bernoulli  19  (2013),  no. 5A, 1938--1964. https://arxiv.org/abs/1107.3760 

23. Doney, R. A. ;  Rivero, V.  Asymptotic behaviour of first passage time distributions for Lévy processes. Probab. Theory Related Fields  157  (2013),  no. 1-2, 1--45. https://arxiv.org/abs/1107.4415 

24. Rivero, Víctor . Tail asymptotics for exponential functionals of Lévy processes: the convolution equivalent case. Ann. Inst. Henri Poincaré Probab. Stat.  48  (2012),  no. 4, 1081--1102. https://arxiv.org/abs/0905.2401 

25. Haas, Bénédicte ;  Rivero, Víctor . Quasi-stationary distributions and Yaglom limits of self-similar Markov processes. Stochastic Process. Appl.  122  (2012),  no. 12, 4054--4095. https://arxiv.org/abs/1110.4795 

26. Chaumont, Loïc ;  Kyprianou, Andreas ;  Pardo, Juan Carlos ;  Rivero, Víctor . Fluctuation theory and exit systems for positive self-similar Markov processes. Ann. Probab.  40  (2012),  no. 1, 245--279. https://arxiv.org/abs/0812.2506 

27. Kyprianou, A. E. ;  Pardo, J. C. ;  Rivero, V.  Exact and asymptotic n-tuple laws at first and last passage. Ann. Appl. Probab.  20  (2010),  no. 2, 522--564. https://arxiv.org/abs/0811.3075 

28. Kyprianou, Andreas E. ;  Rivero, Víctor ;  Song, Renming . Convexity and smoothness of scale functions and de Finetti's control problem. J. Theoret. Probab.  23  (2010),  no. 2, 547--564. https://arxiv.org/abs/0801.1951 

29. Caballero, Maria Emilia ;  Rivero, Víctor . On the asymptotic behaviour of increasing self-similar Markov processes. Electron. J. Probab. 14 (2009), 865--894. https://arxiv.org/abs/0711.1834 

30. Kyprianou, A. E. ;  Rivero, V.  Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electron. J. Probab. 13 (2008), no. 57, 1672--1701. https://arxiv.org/abs/0712.3588 

31. Chaumont, Loïc ;  Rivero, Víctor . On some transformations between positive self-similar Markov processes. Stochastic Process. Appl.  117  (2007),  no. 12, 1889--1909. https://arxiv.org/abs/math/0601243 

32. Rivero, Víctor . Recurrent extensions of self-similar Markov processes and Cramér's condition. II. Bernoulli  13  (2007),  no. 4, 1053--1070. https://arxiv.org/abs/0711.4442 

33. Rivero, Víctor . Sinaĭ's condition for real valued Lévy processes. Ann. Inst. H. Poincaré Probab. Statist.  43  (2007),  no. 3, 299--319. https://arxiv.org/abs/math/0505495 

34. Rivero, Víctor . Recurrent extensions of self-similar Markov processes and Cramér's condition. Bernoulli  11  (2005),  no. 3, 471--509. 

35. Rivero, Víctor . A law of iterated logarithm for increasing self-similar Markov processes. Stoch. Stoch. Rep.  75  (2003),  no. 6, 443--472.

36. Rivero, Víctor Manuel . On random sets connected to the partial records of Poisson point process. J. Theoret. Probab.  16  (2003),  no. 1, 277--307.

• The theory of scale functions for spectrally negative Levy processes, with A. Kuznetsov and A.E. Kyprianou. Levy Matters II, Recent Progress in Theory and Applications: Fractional Lévy Fields, and Scale Functions. Series: Lecture Notes in Mathematics, Vol. 2061 (2012) Download from Springerlink

• Caballero, M. E. ;  Rivero, V. M. ;  Uribe Bravo, G. ;  Velarde, C.  Cadenas de Markov [Markov chains]. Un enfoque elemental [An elementary approach]. Aportaciones Matemáticas: Textos [Mathematical Contributions: Texts], 29. Sociedad Matemática Mexicana, México,  2004. iv+117 pp. ISBN: 970-32-2189-0 Reimpresión 2017

• XIII Symposium on Probability and Stochastic Processes. UNAM, Mexico, December 4-8, 2017. Series: Progress in Probability, Birkhausser Basel, (2020) Co-Editors: Sergio I. López, Alfonso Rocha-Arteaga, Arno Siri-Jégousse.

• XII SYMPOSIUM ON PROBABILITY AND STOCHASTIC PROCESSES, STOCHASTIC MODELS. Universidad Autonoma de Yucatan, Mérida, , 2015. Series: Progress in Probability, Birkhausser Basel (2018), Co-editors: Daniel Hernández-Hernández, Juan Carlos Pardo, Víctor Rivero.

• XI SYMPOSIUM ON PROBABILITY AND STOCHASTIC PROCESSES, STOCHASTIC MODELS. CIMAT, Mexico, November 18-22, 2013. Series: Progress in Probability, Vol. 69, Birkhausser Basel (2015). Co-editors: Ramses Mena, Juan Carlos Pardo, Víctor Rivero, Gerónimo Uribe.

• X SYMPOSIUM ON PROBABILITY AND STOCHASTIC PROCESSES AND FIRST JOINT METTING FRANCE MEXICO OF PROBABILITY. ESAIM Proceedings, Vol. 31, 2011. Editors: María Emilia Caballero, Loïc Chaumont, Daniel Hernández-Hernández and Víctor Rivero. Available on line: http:// www.esaim-proc.org/

• IX SYMPOSIUM ON PROBABILITY AND STOCHASTIC PROCESSES, STOCHASTIC MODELS. Centro de Investigación en Matemáticas A.C. Guanajuato, Gto. México 20-24/11/2006. Editores Ma. Emilia Caballero Acosta (Instituto de Matemáticas, UNAM México), Víctor Rivero (CIMAT Gto.), Juan Ruiz de Chavez (UAM-I Depto. de Matemáticas, México). Número especial de la revista Stochastic Models, Vol. 24, Suplemento 1, (2008).