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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Abstract

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 $\sigma$.

Résumé

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

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

Dates
First available in Project Euclid: 29 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1359470129

Digital Object Identifier
doi:10.1214/11-AIHP420

Mathematical Reviews number (MathSciNet)
MR3060151

Zentralblatt MATH identifier
1273.60069

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

Keywords
Stable processes Stochastic differential equations with jumps

Citation

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. https://projecteuclid.org/euclid.aihp/1359470129


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