Unbounded-Time Safety Verification of Stochastic Differential Dynamics

Shenghua Feng,Bai Xue,Sriram Sankaranarayanan,Naijun Zhan,Mingshuai Chen

doi:10.1007/978-3-030-53291-8_18

Abstract

In this paper, we propose a method for bounding the probability that a stochastic differential equation (SDE) system violates a safety specification over the infinite time horizon. SDEs are mathematical models of stochastic processes that capture how states evolve continuously in time. They are widely used in numerous applications such as engineered systems (e.g., modeling how pedestrians move in an intersection), computational finance (e.g., modeling stock option prices), and ecological processes (e.g., population change over time). Previously the safety verification problem has been tackled over finite and infinite time horizons using a diverse set of approaches. The approach in this paper attempts to connect the two views by first identifying a finite time bound, beyond which the probability of a safety violation can be bounded by a negligibly small number. This is achieved by discovering an exponential barrier certificate that proves exponentially converging bounds on the probability of safety violations over time. Once the finite time interval is found, a finite-time verification approach is used to bound the probability of violation over this interval. We demonstrate our approach over a collection of interesting examples from the literature, wherein our approach can be used to find tight bounds on the violation probability of safety properties over the infinite time horizon.

Highlights

We investigate the problem of verifying probabilistic safety properties for continuous stochastic dynamics modeled by stochastic differential equations (SDEs)
We proposed a constructive method, based on the synthesis of stochastic barrier certificates, for computing an exponentially decreasing upper bound, if existent, on the tail probability that an SDE system violates a given safety specification
We showed that such an upper bound facilitates a reduction of the verification problem over an unbounded temporal horizon to that over a bounded one

Summary

Introduction

We investigate the problem of verifying probabilistic safety properties for continuous stochastic dynamics modeled by stochastic differential equations (SDEs). We propose a novel reduction-based method to verify unbounded-time safety properties of stochastic systems modeled as nonlinear polynomial SDEs. We employ a similar idea as in [11] (for verifying delay differential equations) that reduces the safety verification problem over the infinite time horizon to the one over a finite time interval. We employ a similar idea as in [11] (for verifying delay differential equations) that reduces the safety verification problem over the infinite time horizon to the one over a finite time interval This is achieved by computing an exponential stochastic barrier certificate which witnesses an exponentially decreasing upper bound on the probability that a target system violates a given safety specification. We present an alternative method to address the reduced finite-time horizon verification problem based on the discovery of a time-dependent stochastic barrier certificate.

Problem Formulation

Exponentially Decreasing Bound on the Tail Failure Probability

Synthesizing Stochastic Barrier Certificates Using SDP

Implementation and Experimental Results

Conclusion