Strong completeness of SDEs and non-explosion for RDEs with coefficients having unbounded derivatives

TL;DR

This paper proves non-explosion results for rough differential equations with unbounded coefficients and their derivatives, and establishes the existence of global solutions for stochastic differential equations, with sharpness demonstrated through counterexamples.

Contribution

It introduces non-explosion criteria for RDEs with unbounded coefficients and proves global solution existence for SDEs, extending previous results to more general settings.

Findings

Non-explosion results for RDEs with unbounded coefficients

Existence of global bi-continuous solutions for SDEs

Counterexamples showing sharpness of non-explosion conditions

Abstract

We establish a non-explosion result for rough differential equations (RDEs) in which the noise and drift coefficients, together with their derivatives, may grow unboundedly at infinity. In addition, we prove the existence of a global bi-continuous solution flow for stochastic differential equations (SDEs). Finally, the non-explosion results for RDEs are shown to be sharp by constructing counterexamples.