A jump-type SDE approach to real-valued self-similar Markov processes

Döring, Leif

doi:10.1090/S0002-9947-2015-06270-9

A jump-type SDE approach to real-valued self-similar Markov processes
HTML articles powered by AMS MathViewer

by Leif Döring PDF

Trans. Amer. Math. Soc. 367 (2015), 7797-7836 Request permission

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