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

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

2010-11-02

arxiv.org/abs/1011.0532v1

Mathematics

Probability

Scientific paper

We study a one-dimensional stochastic differential equation driven by a stable L\'evy 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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