Data-Driven Estimation of Infinitesimal Generators of Stochastic Systems

This paper is concerned with a data-driven approach for the estimation of infinitesimal generators of continuous-time stochastic systems with unknown dynamics. We first approximate the infinitesimal generator of the solution process via a set of data collected from trajectories of the unknown system. The approximation utilizes both time discretization and sampling from the solution process. Assuming proper continuity assumptions on dynamics of the system, we then quantify the closeness between the infinitesimal generator and its approximation while providing a priori guaranteed confidence bound. We demonstrate that both the time discretization and the number of data play significant roles in providing a reasonable closeness precision. Moreover, for a fixed size of data, variance of the estimation converges to infinity when the time discretization parameter goes to zero. The formulated error bound shows how to pick proper data size and time discretization jointly to prevent this counter-intuitive behavior. The proposed results are demonstrated on a case study. Copyright (C) 2021 The Authors.