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

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.

DOI : 10.1214/11-AIHP420
Classification : 60H10, 60H30, 60J75
Mots clés : stable processes, stochastic differential equations with jumps
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     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},
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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) no. 1, pp. 138-159. doi : 10.1214/11-AIHP420. http://www.numdam.org/articles/10.1214/11-AIHP420/

[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 | Zbl

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

[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 | Zbl

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

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

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

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

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