In this paper, a novel framework of system identification is introduced to capture the hybrid features of systems subject to both deterministic unmodeled dynamics and stochastic observation disturbances. Using the concepts of persistent identification, control-oriented system modeling, and stochastic analysis, we investigate the central issues of irreducible identification errors and time complexity in such identification problems, Upper and lower bounds on errors and speed of persistent identification are obtained. The error bounds are expressed as functions of observation lengths, sizes of unmodeled dynamics, and probability distributions of disturbances. Asymptotic normality and complexity lower bounds are investigated when periodic inputs and LS estimation are applied. Generic features of asymptotic normality are further explored to extend the asymptotic lower bounds to a wider range of signals and identification mappings.