Adaptive step size numerical integration for stochastic differential equations with discontinuous drift and diffusion

Abstract

Stochastic hybrid systems (SHSs) are a modelling framework for a cyber-physical system (CPS), used to simulate, validate, and verify safety critical controllers under uncertainty. Popular simulation tools can miss detecting discontinuities when simulating SHS, thereby producing incorrect outputs during simulation. We propose a novel adaptive step size simulation/integration technique for a subset of SHS—stochastic differential equations (SDEs) with discontinuous drift and diffusion coefficients. Each integration step, of the Euler-Maruyama numerical solution of the SDEs, is made dependent upon the values of the continuous variables inducing the discontinuity. This in turn guarantees convergence of the system trajectory towards the discontinuity without missing it. A thorough analysis and extensive benchmarking of the proposed integration technique shows the efficacy of the approach when simulating complex SHSs.