# Multilevel Monte Carlo Method for Statistical Model Checking of Hybrid Systems

## TL;DR

This paper introduces a multilevel Monte Carlo approach for statistical model checking of stochastic hybrid systems, effectively estimating properties like reachability despite simulation challenges.

## Contribution

It develops a novel MLMC-based method with smoothing and adaptive error balancing for verifying continuous-time stochastic hybrid systems.

## Key findings

- Effective estimation of reachability probabilities in hybrid systems
- Quantified error bounds for the MLMC approach
- Successful application to thermostatically controlled loads model

## Abstract

We study statistical model checking of continuous-time stochastic hybrid systems. The challenge in applying statistical model checking to these systems is that one cannot simulate such systems exactly. We employ the multilevel Monte Carlo method (MLMC) and work on a sequence of discrete-time stochastic processes whose executions approximate and converge weakly to that of the original continuous-time stochastic hybrid system with respect to satisfaction of the property of interest. With focus on bounded-horizon reachability, we recast the model checking problem as the computation of the distribution of the exit time, which is in turn formulated as the expectation of an indicator function. This latter computation involves estimating discontinuous functionals, which reduces the bound on the convergence rate of the Monte Carlo algorithm. We propose a smoothing step with tunable precision and formally quantify the error of the MLMC approach in the mean-square sense, which is composed of smoothing error, bias, and variance. We formulate a general adaptive algorithm which balances these error terms. Finally, we describe an application of our technique to verify a model of thermostatically controlled loads.