Stochastic reachability as an exit problem

For stochastic hybrid systems, safety verification methods are very little supported mainly because of complexity and difficulty of the associated mathematical problems. The key of the methods that succeeded in solving various instances of this problem is to prove the equivalence of these instances with known problems. In this paper, we apply the same pattern to the most general model of stochastic hybrid systems. Stochastic reachability problem can be treated as an exit problem for a suitable class of Markov processes. The solutions of this problem can be characterised using Hamilton Jacobi theory.