A Contraction Theory Approach to Stochastic Incremental Stability

TL;DR

This paper develops a stochastic contraction theory for Itô systems, providing bounds on trajectory differences influenced by noise, with applications in observer design and synchronization.

Contribution

It introduces a stochastic contraction framework that quantifies incremental stability in stochastic systems, extending nonlinear contraction theory to stochastic dynamics.

Findings

Derived a stochastic version of contraction theory for Itô systems

Provided bounds on mean square trajectory differences based on noise and contraction rate

Applied results to stochastic observer design and synchronization

Abstract

We investigate the incremental stability properties of It\^o stochastic dynamical systems. Specifically, we derive a stochastic version of nonlinear contraction theory that provides a bound on the mean square distance between any two trajectories of a stochastically contracting system. This bound can be expressed as a function of the noise intensity and the contraction rate of the noise-free system. We illustrate these results in the contexts of stochastic nonlinear observers design and stochastic synchronization.