A fundamental relation is derived governing the Lagrangian kinematics of fluid particles on the surface of nonlinear ocean waves which may be known only stochastically. The horizontal trajectories of fluid particles on the free surface are shown to obey a pair of coupled nonlinear Ricatti-type ordinary differential equations driven by the temporal and spatial gradients of the free-surface elevation defined relative to an Eulerian frame. This equation is explicit in that it does not require the solution of a fully nonlinear potential flow free-surface problem and may be viewed as a deterministic or stochastic equation depending on the interpretation of the definition of the free-surface elevation. It is free of empirical corrections often used to estimate the particle kinematics above the calm water surface, is valid in potential flow and for waves of large steepness in two and three dimensions and in waters of all depths and may be used for the evaluation of the extreme unsteady loads exerted on surface piercing vertical circular cylinders by steep random waves.