On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes
Fournier, Nicolas
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013), p. 138-159 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Nous étudions une équation différentielle stochastique de dimension 1 dirigée par un processus de Lévy stable. Lorsque α∈(1,2), nous examinons l’unicité trajectorielle pour cette équation. Quand α∈(0,1), nous étudions une autre équation, équivalente en loi, mais pour laquelle l’unicité trajectorielle s’avère vraie sous des hypothèses bien plus faibles. Nous obtenons des résultats variés, selon que α∈(0,1) ou α∈(1,2) et selon que le processus stable dirigeant l’équation est symétrique ou non. Nos hypothèses concernent la régularité et la monotonie des coefficients de dérive et de diffusion.

We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order α with drift and diffusion coefficients b, σ. When α∈(1,2), we investigate pathwise uniqueness for this equation. When α∈(0,1), we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether α∈(0,1) or α∈(1,2) and on whether the driving stable process is symmetric or not. Our assumptions involve the regularity and monotonicity of b and σ.

Publié le : 2013-01-01
DOI : https://doi.org/10.1214/11-AIHP420
Classification:  60H10,  60H30,  60J75 
@article{AIHPB_2013__49_1_138_0,
     author = {Fournier, Nicolas},
     title = {On pathwise uniqueness for stochastic differential equations driven by stable L\'evy processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {49},
     year = {2013},
     pages = {138-159},
     doi = {10.1214/11-AIHP420},
     mrnumber = {3060151},
     zbl = {1273.60069},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2013__49_1_138_0}
}

Fournier, Nicolas. On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) pp. 138-159. doi : 10.1214/11-AIHP420. http://gdmltest.u-ga.fr/item/AIHPB_2013__49_1_138_0/

[1] R. F. Bass. Stochastic differential equations driven by symmetric stable processes. In Séminaire de Probabilités XXXVI 302-313. Lecture Notes in Math. 1801. Springer, Berlin, 2003. | Numdam | MR 1971592 | Zbl 1039.60056

[2] R. F. Bass. Stochastic differential equations with jumps. Probab. Surv. 1 (2004) 1-19. | MR 2095564 | Zbl 1189.60114

[3] R. F. Bass, K. Burdzy and Z. Q. Chen. Stochastic differential equations driven by stable processes for which pathwise uniqueness fails. Stochastic Process. Appl. 111 (2004) 1-15. | MR 2049566 | Zbl 1111.60038

[4] J. Bertoin. Lévy Processes. Cambridge Tracts in Math. 121. Cambridge Univ. Press, Cambridge, 1996. | MR 1406564 | Zbl 0861.60003

[5] N. Fournier. Jumping SDEs: Absolute continuity using monotonicity. Stochastic Process. Appl. 98 (2002) 317-330. | MR 1887538 | Zbl 1058.60043

[6] N. Fournier and J. Printems. Stability of the stochastic heat equation in L1([0,1]). Preprint, 2010. | MR 2819657 | Zbl 1225.60104

[7] Z. Fu and Z. Li. Stochastic equations of non-negative processes with jumps. Stochastic Process. Appl. 120 (2010) 306-330. | MR 2584896 | Zbl 1184.60022

[8] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes, 2nd edition. North-Holland, Amsterdam, 1989. | MR 1011252 | Zbl 0495.60005

[9] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Springer, Berlin, 2003. | MR 1943877 | Zbl 0635.60021

[10] T. Komatsu. On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations of jump type. Proc. Japan Acad. Ser. A 58 (1982) 383-386. | MR 683262 | Zbl 0511.60057

[11] J. F. Le Gall. Applications du temps local aux équations différentielles stochastiques unidimensionnelles. In Séminaire de Probabilités XVII 15-31. Lecture Notes in Math. 986. Springer, Berlin, 1983. | Numdam | MR 770393 | Zbl 0527.60062

[12] Z. Li and L. Mytnik. Strong solutions for stochastic differential equations with jumps. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 1055-1067. | Numdam | MR 2884224 | Zbl 1273.60070

[13] P. E. Protter. Stochastic Integration and Differential Equations, 2nd edition. Stoch. Model. Appl. Probab. 21. Springer, Berlin, 2005. | MR 2273672 | Zbl 0694.60047

[14] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer, Berlin, 1999. | MR 1725357 | Zbl 1087.60040

[15] K. I. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Stud. Adv. Math. 68. Cambridge Univ. Press, Cambridge, 1999. | MR 1739520 | Zbl 0973.60001

[16] R. Situ. Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering. Springer, New York, 2005. | MR 2160585 | Zbl 1070.60002