Safety Verification of Stochastic Systems: A Set-Erosion Approach

We study the safety verification problem for discrete-time stochastic systems. We propose an approach for safety verification termed set-erosion strategy that verifies the safety of a stochastic system on a safe set through the safety of its associated deterministic system on an eroded subset. The amount of erosion is captured by the probabilistic bound on the distance between stochastic trajectories and their associated deterministic counterpart. Building on recent development of stochastic analysis, we establish a sharp probabilistic bound on this distance. Combining this bound with the set-erosion strategy, we establish a general framework for the safety verification of stochastic systems. Our method is versatile and can work effectively with any deterministic safety verification techniques. We exemplify our method by incorporating barrier functions designed for deterministic safety verification, obtaining barrier certificates much tighter than existing results. Numerical experiments are conducted to demonstrate the efficacy and superiority of our method.