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},
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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://archive.numdam.org/articles/10.1214/11-AIHP420/

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