p-Safe Analysis of Stochastic Hybrid Processes

We develop a method for determining whether a stochastic system is safe, i.e., whether its trajectories reach unsafe states. Specifically, we define and solve a probabilistic safety problem for Markov processes. Based on the knowledge of the extended generator, we are able to develop an evolution equation, as a system of integral equations, describing the connection between unsafe and initial states. Subsequently, using the moment method, we approximate the infinite-dimensional optimisation problem searching for the largest set of safe states by a finite dimensional polynomial optimisation problem. In particular, we address the above safety problem to a special class of stochastic hybrid processes, namely piecewise-deterministic Markov processes. These are characterized by deterministic dynamics and stochastic jumps, where both the time and the destination of the jumps are stochastic. In addition, the jumps can be both spontaneous (in the style of a Poisson process) or forced (governed by guards). In this case, the extended generator of this process and its corresponding martingale problem turn out to be defined on a rather restricted domain.