A dual Yamada–Watanabe theorem for Lévy driven stochastic differential equations

David Criens

doi:10.1214/21-ecp384

Abstract

A dual Yamada–Watanabe theorem for Lévy driven stochastic differential equations

Highlights

We prove a dual Yamada–Watanabe theorem for one-dimensional stochastic differential equations driven by quasi-left continuous semimartingales with independent increments
The classical Yamada–Watanabe theorem [22] for Brownian stochastic differential equations (SDEs) tells us that strong uniqueness and weak existence implies weak uniqueness and strong existence
Jacod [11] lifted this result to SDEs driven by semimartingales and extended it by showing that strong uniqueness and weak existence is equivalent to weak joint uniqueness and strong existence

Summary

Introduction

The classical Yamada–Watanabe theorem [22] for Brownian stochastic differential equations (SDEs) tells us that strong uniqueness (i.e. pathwise uniqueness) and weak existence implies weak uniqueness (i.e. uniqueness in law) and strong existence. The purpose of this short paper is to close the gap in the literature and to prove a dual theorem for SDEs driven by quasi-left continuous semimartingales with independent increments (SIIs), which is a large class of drivers including in particular all Lévy processes. Similar to Cherny’s proof for Brownian SDEs, the main idea is to recover the driver L from the solution process X and an independent SII V .

Results

Conclusion