## Electronic Journal of Probability

### Jump type SDEs for self-similar processes

#### Abstract

We present a new approach to positive self-similar Markov processes (pssMps) by reformulating Lamperti's transformation via jump type SDEs. As applications, we give direct constructions of pssMps (re)started continuously at zero if the Lamperti transformed Lévy process is spectrally negative. Our paper can be seen as a continuation of similar studies for continuous state branching processes.

#### Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 94, 39 pp.

Dates
Accepted: 30 October 2012
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062416

Digital Object Identifier
doi:10.1214/EJP.v17-2402

Mathematical Reviews number (MathSciNet)
MR2994842

Zentralblatt MATH identifier
1286.60036

Rights

#### Citation

Döring, Leif; Barczy, Matyas. Jump type SDEs for self-similar processes. Electron. J. Probab. 17 (2012), paper no. 94, 39 pp. doi:10.1214/EJP.v17-2402. https://projecteuclid.org/euclid.ejp/1465062416

#### References

• Aldous, David. Stopping times and tightness. II. Ann. Probab. 17 (1989), no. 2, 586–595.
• J. Berestycki, L. Döring, L. Mytnik and L. Zambotti: Hitting properties and non-uniqueness for SDE driven by stable processes. ARXIV1111.4388, (2011).
• Bertoin, Jean; Caballero, Maria-Emilia. Entrance from $0+$ for increasing semi-stable Markov processes. Bernoulli 8 (2002), no. 2, 195–205.
• Bertoin, Jean; Savov, Mladen. Some applications of duality for Lévy processes in a half-line. Bull. Lond. Math. Soc. 43 (2011), no. 1, 97–110.
• Bertoin, Jean; Yor, Marc. The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal. 17 (2002), no. 4, 389–400.
• Bertoin, Jean; Yor, Marc. Exponential functionals of Lévy processes. Probab. Surv. 2 (2005), 191–212.
• Caballero, M. E.; Chaumont, L. Conditioned stable Lévy processes and the Lamperti representation. J. Appl. Probab. 43 (2006), no. 4, 967–983.
• Caballero, M. E.; Chaumont, L. Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes. Ann. Probab. 34 (2006), no. 3, 1012–1034.
• Caballero, Ma. Emilia; Lambert, Amaury; Uribe Bravo, Gerónimo. Proof(s) of the Lamperti representation of continuous-state branching processes. Probab. Surv. 6 (2009), 62–89.
• Chaumont, Loíc; 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.
• Dawson, D. A.; Li, Zenghu. Skew convolution semigroups and affine Markov processes. Ann. Probab. 34 (2006), no. 3, 1103–1142.
• Dawson, Donald A.; Li, Zenghu. Stochastic equations, flows and measure-valued processes. Ann. Probab. 40 (2012), no. 2, 813–857.
• Di Nunno, Giulia; Øksendal, Bernt; Proske, Frank. Malliavin calculus for Lévy processes with applications to finance. Universitext. Springer-Verlag, Berlin, 2009. xiv+413 pp. ISBN: 978-3-540-78571-2
• Doney, Ronald A. Fluctuation theory for Lévy processes. Lectures from the 35th Summer School on Probability Theory held in Saint-Flour, July 6-23, 2005. Edited and with a foreword by Jean Picard. Lecture Notes in Mathematics, 1897. Springer, Berlin, 2007. x+147 pp. ISBN: 978-3-540-48510-0; 3-540-48510-4
• Doney, R. A.; Maller, R. A. Stability of the overshoot for Lévy processes. Ann. Probab. 30 (2002), no. 1, 188–212.
• Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8
• Fitzsimmons, P. J. On the existence of recurrent extensions of self-similar Markov processes. Electron. Comm. Probab. 11 (2006), 230–241.
• Fu, Zongfei; Li, Zenghu. Stochastic equations of non-negative processes with jumps. Stochastic Process. Appl. 120 (2010), no. 3, 306–330.
• Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. xiv+464 pp. ISBN: 0-444-86172-6
• Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 2003. xx+661 pp. ISBN: 3-540-43932-3
• Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8
• Kyprianou, Andreas E. Introductory lectures on fluctuations of Lévy processes with applications. Universitext. Springer-Verlag, Berlin, 2006. xiv+373 pp. ISBN: 978-3-540-31342-7; 3-540-31342-7
• Lamperti, John. Continuous state branching processes. Bull. Amer. Math. Soc. 73 1967 382–386.
• Lamperti, John. Semi-stable Markov processes. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 205–225.
• Li, Zenghu; Mytnik, Leonid. Strong solutions for stochastic differential equations with jumps. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), no. 4, 1055–1067.
• Z.H. Li and F. Pu: Strong solutions of jump-type stochastic equations Electron. Commun. Probab. 17, Article 33, 1–13, (2012).
• Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7
• Rivero, Víctor. Recurrent extensions of self-similar Markov processes and Cramér's condition. Bernoulli 11 (2005), no. 3, 471–509.
• Rivero, Víctor. Recurrent extensions of self-similar Markov processes and Cramér's condition. II. Bernoulli 13 (2007), no. 4, 1053–1070.
• Sato, Ken-iti. Lévy processes and infinitely divisible distributions. Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999. xii+486 pp. ISBN: 0-521-55302-4
• Situ, Rong. Theory of stochastic differential equations with jumps and applications. Mathematical and analytical techniques with applications to engineering. Mathematical and Analytical Techniques with Applications to Engineering. Springer, New York, 2005. xx+434 pp. ISBN: 978-0387-25083-0; 0-387-25083-2
• Stroock, Daniel W.; Varadhan, S. R. Srinivasa. Multidimensional diffusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 233. Springer-Verlag, Berlin-New York, 1979. xii+338 pp. ISBN: 3-540-90353-4
• Vuolle-Apiala, J. Itô excursion theory for self-similar Markov processes. Ann. Probab. 22 (1994), no. 2, 546–565.
• Williams, David. Diffusions, Markov processes, and martingales. Vol. 1. Foundations. Probability and Mathematical Statistics. John Wiley & Sons, Ltd., Chichester, 1979. xiii+237 pp. ISBN: 0-471-99705-6
• Yamada, Toshio; Watanabe, Shinzo. On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 1971 155–167.
• Watanabe, Shinzo; Yamada, Toshio. On the uniqueness of solutions of stochastic differential equations. II. J. Math. Kyoto Univ. 11 1971 553–563.