Stochastic hybrid inclusions with diffusive flows

Recent work has considered a class of stochastic hybrid inclusions that allows for the interaction of random and worst case effects through the combination of constrained non-stochastic differential inclusions and constrained stochastic difference inclusions. In this work, we extend those models by allowing the flows to come from constrained stochastic differential inclusions (SDIs). As part of this program, we present hybrid filtrations, hybrid stopping times, a hybrid Itô's rule, and a hybrid Dynkin's formula. Subsequently, these results are used to establish Lyapunov-based sufficient conditions for uniform global asymptotic stability in probability of compact sets and uniform global recurrence in probability of open, bounded sets. The conclusion emphasizes the need for sequential compactness results for constrained SDIs, in order to extend recent sequential compactness results for a class of stochastic hybrid inclusions to the case with diffusive flows.