Stochastic analysis based on deterministic Brownian motion

A deterministic version of the Ito calculus is presented. We consider a model Y-t = H (N-t, t) with a deterministic Brownian N-t and an unknown function H. We predict Y-c from the observation {Y-t;t is an element of [a,b]}, where a < b < c. We prove that there exists an estimator Yt based on the observation such that E[((Y) over cap (t) - Y-c)(2)] = O((c - b)(2)) as c down arrow b.