Unsafe Probabilities and Risk Contours for Stochastic Processes using Convex Optimization

TL;DR

This paper introduces a convex optimization-based algorithm to estimate the maximum probability of unsafety in stochastic process trajectories, providing interpretable risk contours and leveraging semidefinite programming.

Contribution

It formulates unsafe probability estimation as an infinite-dimensional linear program linked to barrier certificates, with a practical finite-dimensional approximation using Moment-Sum-of-Squares hierarchy.

Findings

Algorithm effectively estimates unsafe probabilities for stochastic processes.

Risk contours visually represent stochastic safety for initial conditions.

Method is demonstrated on example stochastic processes.

Abstract

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.