Unsafe Probabilities and Risk Contours for Stochastic Processes using Convex Optimization

This paper proposes an algorithm to calculate the maximal probability of unsafety with respect to trajectories of a stochastic process and a hazard set. The unsafe probability estimation problem is cast as a primal-dual pair of infinite-dimensional linear programs in occupation measures and continuous functions. This convex relaxation is nonconservative (to the true probability of unsafety) under compactness and regularity conditions in dynamics. The continuous-function linear program is linked to existing probability-certifying barrier certificates of safety. Risk contours for initial conditions of the stochastic process may be generated by suitably modifying the objective of the continuous-function program, forming an interpretable and visual representation of stochastic safety for test initial conditions. All infinite-dimensional linear programs are truncated to finite dimension by the Moment-Sum-of-Squares hierarchy of semidefinite programs. Unsafe-probability estimation and risk contours are generated for example stochastic processes.