A Contraction Theory Approach to Stochastic Incremental Stability

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.

💡 Research Summary

The paper extends nonlinear contraction theory to Itô stochastic differential equations, establishing a rigorous framework for incremental stability in the presence of random disturbances. Starting from the deterministic contraction concept—where the distance between any two trajectories decays exponentially—the authors introduce the notion of stochastic contraction. They consider a system of the form (dx = f(x,t)dt + \sigma(x,t)dW_t) and define a state‑dependent metric (M(x,t)) that induces a Riemannian‑like distance. By applying Itô’s lemma to the squared norm of the transformed error (\eta = M^{1/2}(x-y)), they derive a differential inequality for the expected value of the error energy:

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