Estimation of Infinitesimal Generators for Unknown Stochastic Hybrid Systems via Sampling: A Formal Approach

In this letter, we develop a data-driven framework with formal confidence bounds for the estimation of infinitesimal generators for continuous-time stochastic hybrid systems with unknown dynamics. The proposed approximation scheme employs both time discretization and sampling from the solution process, and estimates the infinitesimal generator of the solution process via a set of data collected from trajectories of systems. We assume some mild continuity assumptions on the dynamics of the system and quantify the closeness between the infinitesimal generator and its approximation while ensuring an a-priori guaranteed confidence bound. To provide a reasonable closeness precision, we discuss significant roles of both time discretization and number of data in our approximation scheme. In particular, for a fixed number of data, variance of the estimation converges to infinity when the time discretization goes to zero. The proposed approximation framework guides us how to jointly select a suitable data size and a time discretization parameter to cope with this counter-intuitive behavior. We demonstrate the effectiveness of our proposed results by applying them to a nonlinear jet engine compressor with unknown dynamics.