Stability criteria for rough systems

TL;DR

This paper introduces a quantitative method for establishing local exponential stability of stationary solutions in rough differential equations and their discretizations, based on linearization and growth conditions.

Contribution

It provides new data-driven stability criteria using Doss-Sussmann technique and stopping times, applicable to both continuous and discretized systems.

Findings

Stability criteria depend on boundedness and small growth of diffusion derivatives

Criteria are applicable to both continuous systems and discretizations with small step size

Method uses linearization and is data-driven

Abstract

We propose a quantitative direct method to prove the local stability of a stationary solution for a rough differential equation and its regular discretization scheme. Using Doss-Sussmann technique and stopping time analysis, we provide stability criteria for a stationary solution of the continuous system to be exponentially stable, provided the diffusion term is bounded and its derivatives exhibit small growth. The same conclusions hold for the regular discretization scheme with a sufficiently small step size, but one needs to apply the sewing lemma and stopping times for the discrete time set. Our stability criteria are based on the linearization of the drift and require only information about the bound and growth rates of the diffusion, making them data-driven criteria.