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 $\alpha$: \[dX_t= ϕ(X_{t-})\ dZ_t.\] We prove that pathwise uniqueness holds for this equation under the assumptions that $\varphi$ is continuous, non-decreasing and positive on $\mathbb{R}$. A counterexample is given to show that the positivity of $\varphi$ is crucial for pathwise uniqueness to hold.

10 pages

References in corpus (2)

• On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes
• On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations with jumps