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 $\sigma$.

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