Reachability Computation for Switching Diffusions: Finite Abstractions with Certifiable and Tuneable Precision

We consider continuous time stochastic hybrid systems with no resets and continuous dynamics described by linear stochastic differential equations -- models also known as switching diffusions. We show that for this class of models reachability (and dually, safety) properties can be studied on an abstraction defined in terms of a discrete time and finite space Markov chain (DTMC), with provable error bounds. The technical contribution of the paper is a characterization of the uniform convergence of the time discretization of such stochastic processes with respect to safety properties. This allows us to newly provide a complete and sound numerical procedure for reachability and safety computation over switching diffusions.