A Dual Yamada-Watanabe Theorem for Levy driven stochastic differential equations

TL;DR

This paper establishes a dual Yamada-Watanabe theorem for one-dimensional SDEs driven by Levy processes, showing that weak uniqueness implies joint uniqueness in law for solutions and their drivers.

Contribution

It extends the Yamada-Watanabe theorem to Levy-driven SDEs, covering time-inhomogeneous Levy processes and establishing joint uniqueness from weak uniqueness.

Findings

Weak uniqueness implies joint uniqueness for Levy-driven SDEs.

The theorem applies to quasi-left continuous semimartingales with independent increments.

It broadens the theoretical understanding of solution uniqueness in stochastic differential equations.

Abstract

We prove a dual Yamada-Watanabe theorem for one-dimensional stochastic differential equations driven by quasi-left continuous semimartingales with independent increments. In particular, our result covers stochastic differential equations driven by (time-inhomogeneous) Levy processes. More precisely, we prove that weak uniqueness, i.e. uniqueness in law, implies weak joint uniqueness, i.e. joint uniqueness in law for the solution process and its driver.