Strong completeness of SDEs and non-explosion for RDEs with coefficients having unbounded derivatives
Xue-Mei Li, Kexing Ying
TL;DR
This work develops a non-explosion criterion for rough differential equations with coefficients whose derivatives may be unbounded, via a growth framework controlled by a function $f$ and a small parameter $\kappa$. It also establishes strong completeness and a continuous global solution flow for SDEs, including the additive-noise case, by leveraging a derivative-flow (linearization) approach and rough-path techniques. A key ingredient is an interactive regularity bound that translates rough integrals into an ODE-like framework, enabling uniform Hölder-norm control and non-explosion across unbounded regimes. The results are complemented by a sharp counterexample in the additive-noise setting, and by a systematic treatment of Young and rough differential equations, with detailed estimates and a localization-based construction of maximal flows. These findings advance the understanding of global behavior for SDEs and RDEs under weakened growth and regularity assumptions, with implications for stochastic flows and numerical approximations.
Abstract
We establish a non-explosion result for rough differential equations (RDEs) in which both the noise and drift coefficients together with their derivatives are allowed to grow at infinity. Additionally, we prove the existence of a bi-continuous solution flow for stochastic differential equations (SDEs). In the case of RDEs with additive noise, we show that our result is optimal by providing a counterexample.