Estimating infinitesimal generators of stochastic systems with formal error bounds

Abstract

In this work, we propose a data-driven technique for a formal estimation of infinitesimal generators of continuous-time stochastic systems with unknown dynamics. In the proposed framework, we first approximate the infinitesimal generator of the solution process via a set of data collected from solution processes of unknown systems. We then put some proper assumptions on dynamics of systems and quantify the closeness between the infinitesimal generator and its approximation while providing a priori guaranteed confidence bound. We show that both the time discretization and the number of data play significant roles in providing a reasonable closeness precision. Motivations and Related Works. Infinitesimal generator of a stochastic process (i.e., a partial differential operator that encodes a great deal of information about the process) plays a principal role in many analysis frameworks including: (i) stability analysis of continuous-time stochastic systems via Lyapunov functions [1], (ii) establishing similarity relations between two continuous-time stochastic systems via stochastic simulation functions [2], (iii) constructing finite abstractions of continuous-time stochastic control systems [3], and (iv) safety analysis of continuous-time stochastic systems via barrier certificates [4], to name a few. In the relevant literature, only [5] studies the estimation of infinitesimal generators, whose approach is based on the assumption of knowing the precise model and discretizing both time and state. To the best of our knowledge, our work is the first to propose a method for estimating the infinitesimal generator of stochastic systems with unknown dynamics while providing formal guarantees. The complete version of this work can be found in [6].