Pathwise uniqueness of one-dimensional SDEs driven by one-sided stable processes

Abstract: For $\alpha\in (0,1)$, we consider stochastic differential equations driven by one-sided stable processes of order $\alpha$: [dX_t= \phi(X_{t-})\ dZ_t.] We prove that pathwise uniqueness holds for this equation under the assumptions that $\phi$ is continuous, non-decreasing and positive on $\R$. A counterexample is given to show that the positivity of $\phi$ is crucial for pathwise uniqueness to hold.