# 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.

## Key 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.