Concentration of Stochastic System Trajectories with Time-varying Contraction Conditions
This work provides theoretical guarantees for stochastic system analysis, particularly in applications like safe control, but appears incremental as it builds on existing methods like incremental stability and martingale techniques.
The paper tackles the problem of bounding deviations between stochastic system trajectories and their deterministic counterparts under time-varying contraction conditions, achieving tight bounds of order O(√log(1/δ)) for the deviation.
We establish two concentration inequalities for nonlinear stochastic system under time-varying contraction conditions. The key to our approach is an energy function termed Averaged Moment Generating Function (AMGF). By combining it with incremental stability analysis, we develop a concentration inequality that bounds the deviation between the stochastic system state and its deterministic counterpart. As this inequality is restricted to single time instance, we further combine AMGF with martingale-based methods to derive a concentration inequality that bounds the fluctuation of the entire stochastic trajectory. Additionally, by synthesizing the two results, we significantly improve the trajectory-level concentration inequality for strongly contractive systems. Given the probability level $1-δ$, the derived inequalities ensure an $\mO(\sqrt{\log(1/δ))}$ bound on the deviation of stochastic trajectories, which is tight under our assumptions. Our results are exemplified through a case study on stochastic safe control.