On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 1, p. 138-159

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.

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.

DOI : https://doi.org/10.1214/11-AIHP420
Classification:  60H10,  60H30,  60J75
Keywords: stable processes, stochastic differential equations with jumps
@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},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {1},
     year = {2013},
     pages = {138-159},
     doi = {10.1214/11-AIHP420},
     zbl = {1273.60069},
     mrnumber = {3060151},
     language = {en},
     url = {http://www.numdam.org/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, Volume 49 (2013) no. 1, pp. 138-159. doi : 10.1214/11-AIHP420. http://www.numdam.org/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