Neural Drift Estimation for Ergodic Diffusions: Non-parametric Analysis and Numerical Exploration

Simone Di Gregorio and Francesco Iafrate Abstract We take into consideration generalization bounds in [3] for the problem of the estimation of the drift component for ergodic stochasti c differential equations, when the estimator is a ReLU neural network and the estimatio n is non-parametric with respect to the statistical model. We show a practical wa y to enforce the theoretical estimation procedure, enabling inference on noisy and roug h functional data. Results are shown for a simulated It ˆ o-Taylor approximation of the sample paths. The problem of statistical modeling for multi-dimensional ergodic diffusions from discrete observations has a long-standing literature that largely focused on parametric approaches to the estimation of the coefficient set [4, 5]. In this work, we pick a different approach stemming from the analysis in [3], focus ing on neural network estimators of the drift coefficient and without parametric as sumptions on the statistical model.