Figure1

Figure 1. Outline of the theory. Many dynamical processes in nature can be modeled as an SDE, which has a drift and diffusion component. Unlike deterministic systems, the consecutive points on a trajectory are not related deterministically, but have a distribution. The flowchart shows how we can collect $$ (k+1) $$-length snapshots of this process, and apply purely data-driven algorithms to estimate the drift. No presumption is needed on a prior distribution on the coordinates, or on a parametric form for the SDE terms. The article presents the details of these methods, their theoretical foundations in Probability theory, and a rigorous demonstration that they provide a convergent, unbiased estimate.