Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes

Nicolas Fournier

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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 $\sigma$.


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.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 1 (2013), 138-159.

First available in Project Euclid: 29 January 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.) 60J75: Jump processes

Stable processes Stochastic differential equations with jumps


Fournier, Nicolas. On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 1, 138--159. doi:10.1214/11-AIHP420.

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