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) no. 1, pp. 138-159.

Nous étudions une équation différentielle stochastique de dimension $1$ dirigée par un processus de Lévy stable. Lorsque $\alpha \in (1,2)$, nous examinons l’unicité trajectorielle pour cette équation. Quand $\alpha \in (0,1)$, 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 $\alpha \in (0,1)$ ou $\alpha \in (1,2)$ 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 $\alpha$ with drift and diffusion coefficients $b$, $\sigma$. When $\alpha \in (1,2)$, we investigate pathwise uniqueness for this equation. When $\alpha \in (0,1)$, 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 $\alpha \in (0,1)$ or $\alpha \in (1,2)$ and on whether the driving stable process is symmetric or not. Our assumptions involve the regularity and monotonicity of $b$ and $\sigma$.

Classification : 60H10, 60H30, 60J75
Mots clés : stable processes, stochastic differential equations with jumps

[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 $L1\left(\right[0,1\left]\right)$. 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