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

TL;DR

This paper proves pathwise uniqueness for one-dimensional SDEs driven by one-sided stable processes of order alpha, assuming the coefficient is continuous, non-decreasing, and positive, highlighting the importance of positivity.

Contribution

It establishes pathwise uniqueness for a class of SDEs driven by one-sided stable processes with specific coefficient conditions, and demonstrates the necessity of positivity.

Findings

Pathwise uniqueness holds under the given conditions.

Positivity of the coefficient is crucial for uniqueness.

Counterexample shows failure of uniqueness if positivity is not assumed.

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.