Extended Jacobson density theorem for rings with derivations and automorphisms

A deterministic version of the Itô calculus is presented. We consider a modelY _t=H(N _t ,t) with a deterministic Brownian N _t and an unknown functionH. We predictY _c from the observation {Y _t;t ∈ [a, b]}, wherea. We prove that there exists an estimatorY _t based on the observation such thatE[(? _t?Y _c)²]=O((c?b)²) asc ↓b.