A jump-type SDE approach to real-valued self-similar Markov processes
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Abstract:
In his 1972 paper, John Lamperti characterized all positive self-similar Markov processes as time-changes of exponentials of Lévy processes. In the past decade the problem of representing all non-negative self-similar Markov processes that do not necessarily have zero as a trap has been solved gradually via connections to ladder height processes and excursion theory.
Motivated by a recent article of Chaumont, Panti, and Rivero, we represent via jump-type SDEs the symmetric real-valued self-similar Markov processes that only decrease the absolute value by jumps and leave zero continuously.
Our construction of these self-similar processes involves a pseudo excursion construction and singular stochastic calculus arguments ensuring that solutions to the SDEs spend zero time at zero to avoid problems caused by a “bang-bang” drift.
References
- David Aldous, Stopping times and tightness, Ann. Probability 6 (1978), no. 2, 335–340. MR 474446, DOI 10.1214/aop/1176995579
- David Applebaum, Lévy processes and stochastic calculus, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 116, Cambridge University Press, Cambridge, 2009. MR 2512800, DOI 10.1017/CBO9780511809781
- Mátyás Barczy and Leif Döring, On entire moments of self-similar Markov processes, Stoch. Anal. Appl. 31 (2013), no. 2, 191–198. MR 3021485, DOI 10.1080/07362994.2013.741397
- Richard F. Bass, Krzysztof Burdzy, and Zhen-Qing Chen, Pathwise uniqueness for a degenerate stochastic differential equation, Ann. Probab. 35 (2007), no. 6, 2385–2418. MR 2353392, DOI 10.1214/009117907000000033
- J. Berestycki, L. Döring, L. Mytnik and L. Zambotti, On existence and uniqueness for self-similar SDEs driven by stable processes, arXiv:1111.4388 (2012)
- Jean Bertoin and Maria-Emilia Caballero, Entrance from $0+$ for increasing semi-stable Markov processes, Bernoulli 8 (2002), no. 2, 195–205. MR 1895890
- Jean Bertoin and Mladen Savov, Some applications of duality for Lévy processes in a half-line, Bull. Lond. Math. Soc. 43 (2011), no. 1, 97–110. MR 2765554, DOI 10.1112/blms/bdq084
- Jean Bertoin and Marc Yor, The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes, Potential Anal. 17 (2002), no. 4, 389–400. MR 1918243, DOI 10.1023/A:1016377720516
- Jean Bertoin and Marc Yor, On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes, Ann. Fac. Sci. Toulouse Math. (6) 11 (2002), no. 1, 33–45 (English, with English and French summaries). MR 1986381, DOI 10.5802/afst.1016
- Jean Bertoin and Marc Yor, Exponential functionals of Lévy processes, Probab. Surv. 2 (2005), 191–212. MR 2178044, DOI 10.1214/154957805100000122
- R. M. Blumenthal, On construction of Markov processes, Z. Wahrsch. Verw. Gebiete 63 (1983), no. 4, 433–444. MR 705615, DOI 10.1007/BF00533718
- M. E. Caballero and L. Chaumont, Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes, Ann. Probab. 34 (2006), no. 3, 1012–1034. MR 2243877, DOI 10.1214/009117905000000611
- Loïc Chaumont, Henry Pantí, and Víctor Rivero, The Lamperti representation of real-valued self-similar Markov processes, Bernoulli 19 (2013), no. 5B, 2494–2523. MR 3160562, DOI 10.3150/12-BEJ460
- Loïc Chaumont, Andreas Kyprianou, Juan Carlos Pardo, and Víctor Rivero, Fluctuation theory and exit systems for positive self-similar Markov processes, Ann. Probab. 40 (2012), no. 1, 245–279. MR 2917773, DOI 10.1214/10-AOP612
- Oleksandr Chybiryakov, The Lamperti correspondence extended to Lévy processes and semi-stable Markov processes in locally compact groups, Stochastic Process. Appl. 116 (2006), no. 5, 857–872. MR 2218339, DOI 10.1016/j.spa.2005.11.009
- Claude Dellacherie and Paul-André Meyer, Probabilités et potentiel. Chapitres V à VIII, Revised edition, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1385, Hermann, Paris, 1980 (French). Théorie des martingales. [Martingale theory]. MR 566768
- Leif Döring and Mátyás Barczy, A jump type SDE approach to positive self-similar Markov processes, Electron. J. Probab. 17 (2012), no. 94, 39. MR 2994842, DOI 10.1214/EJP.v17-2402
- Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085, DOI 10.1002/9780470316658
- P. J. Fitzsimmons, On the existence of recurrent extensions of self-similar Markov processes, Electron. Comm. Probab. 11 (2006), 230–241. MR 2266714, DOI 10.1214/ECP.v11-1222
- Zongfei Fu and Zenghu Li, Stochastic equations of non-negative processes with jumps, Stochastic Process. Appl. 120 (2010), no. 3, 306–330. MR 2584896, DOI 10.1016/j.spa.2009.11.005
- Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. MR 637061
- Jean Jacod and Albert N. Shiryaev, Limit theorems for stochastic processes, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 2003. MR 1943877, DOI 10.1007/978-3-662-05265-5
- Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR 1121940, DOI 10.1007/978-1-4612-0949-2
- Sun Wah Kiu, Two dimensional semi-stable Markov processes, Ann. Probability 3 (1975), no. 3, 440–448. MR 388563, DOI 10.1214/aop/1176996351
- Sun Wah Kiu, Semistable Markov processes in $\textbf {R}^{n}$, Stochastic Process. Appl. 10 (1980), no. 2, 183–191. MR 587423, DOI 10.1016/0304-4149(80)90020-4
- A. Kuznetsov and J. C. Pardo, Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes, Acta Appl. Math. 123 (2013), 113–139. MR 3010227, DOI 10.1007/s10440-012-9718-y
- John Lamperti, Semi-stable Markov processes. I, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 205–225. MR 307358, DOI 10.1007/BF00536091
- Zenghu Li and Fei Pu, Strong solutions of jump-type stochastic equations, Electron. Commun. Probab. 17 (2012), no. 33, 13. MR 2965746, DOI 10.1214/ECP.v17-1915
- Pierre Patie, Exponential functional of a new family of Lévy processes and self-similar continuous state branching processes with immigration, Bull. Sci. Math. 133 (2009), no. 4, 355–382 (English, with English and French summaries). MR 2532690, DOI 10.1016/j.bulsci.2008.10.001
- Philip E. Protter, Stochastic integration and differential equations, 2nd ed., Applications of Mathematics (New York), vol. 21, Springer-Verlag, Berlin, 2004. Stochastic Modelling and Applied Probability. MR 2020294
- Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
- Víctor Rivero, Recurrent extensions of self-similar Markov processes and Cramér’s condition, Bernoulli 11 (2005), no. 3, 471–509. MR 2146891, DOI 10.3150/bj/1120591185
- Víctor Rivero, Recurrent extensions of self-similar Markov processes and Cramér’s condition. II, Bernoulli 13 (2007), no. 4, 1053–1070. MR 2364226, DOI 10.3150/07-BEJ6082
- Ken-iti Sato, Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original; Revised by the author. MR 1739520
- Toshio Yamada and Shinzo Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971), 155–167. MR 278420, DOI 10.1215/kjm/1250523691
Additional Information
- Leif Döring
- Affiliation: Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, 4, Place Jussieu, 75005 Paris, France
- Email: leif.doering@googlemail.com
- Received by editor(s): October 27, 2012
- Received by editor(s) in revised form: August 9, 2013
- Published electronically: February 18, 2015
- Additional Notes: The author was supported by the Fondation Science Matématiques de Paris
- © Copyright 2015
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 7797-7836
- MSC (2010): Primary 60G18; Secondary 60G55
- DOI: https://doi.org/10.1090/S0002-9947-2015-06270-9
- MathSciNet review: 3391900